In space-time, observers calculate that observed object times, lengths, distances, and masses depend on observer velocity relative to object {relativity}.
space-time
Space has three dimensions, time has one dimension, and space and time dimensions combine into unified four-dimensional space-time.
Objects and observers move through events on a path (world-line) through space-time. Because light is self-propagating and has no medium, zero-rest-mass particles move through space at light speed. By experiment, all objects move through space-time at light speed, so space dimensions and time dimension relate by light speed. Therefore, rest masses move through time at light speed.
In space-time, time dimension and space dimensions have same units. If space dimensions have time units, their time is space-dimension length divided by light speed. If time dimension has space units, its distance/length is time-dimension time times light speed.
absolute space-time
Universe has absolute space-time (and absolute reference frame). Absolute space-time has local curvature depending on mass and energy space-time positions. Absolute space-time has global curvature depending on mass and energy space-time positions, and so has an overall shape.
However, observers move at light speed through space-time and can calculate only observed-object relative-motion properties. Because they have a reference frame relative to space-time, they have no direct knowledge of space-time. Special relativity describes how observers moving with uniform velocity in relation to objects calculate object motions and properties. General relativity describes how observers moving with non-uniform velocity, due to gravitational fields and/or accelerations, in relation to objects calculate object motions and properties.
space and time
In classical physics, space vs. time graphs show object movements through space locations over time intervals. Objects can have any position and time. Space and time are separate and independent variables. Distance vectors sum.
In relativity, space-time unifies space and time coordinates. Space-time graphs show object space-time events. Object events cannot be at all space-time points, because object maximum speed is light speed. Because some events cannot have space-time separations from other events, separation vectors do not sum.
measurement
Stationary observers measure object length using rulers, which are stationary objects with standard length {unit length} {length unit}, such as one meter. Measurements count the number of unit lengths between object ends. In space-time, because all observers calculate that light always travels at light speed through space, observers count light's time of flight between the two object-end space-time events, and then multiply by light speed to find length.
Stationary observers measure object time using clocks, which are stationary objects with standard time intervals {unit time} {time unit} between clock ticks, such as one second. Clocks have frequencies, such as one cycle per second. If time-interval unit increases, frequency decreases. Measurements count the number of unit times between two events. In space-time, because all observers calculate that light always travels at light speed through space, observers count time of flight between space-time events.
accuracy
Experiments show that stationary clocks and their time units, and rulers and their length units, maintain accuracy for stationary observers after movement over time and space. Clocks at different space locations can synchronize. Rulers at different times and space locations can coincide. Therefore, observers can standardize on the same time at different space locations, and standardize on the same length at different times and space positions.
reference frame
Observers and objects have space-time coordinate systems (reference frame) centered on themselves. Observers and objects traveling with same uniform velocity share the same reference frame and are stationary with respect to each other. Reference frames can be stationary, have uniform velocity, or accelerate in relation to other reference frames.
Observers and observed objects travel through absolute space-time. If they travel at same speed and same direction (same velocity), observed object appears stationary, and observers and observed objects have the same space-time and reference frame. If they travel at different velocities, observed object appears to move at velocity difference, and observers and observed objects have different reference frames. Different reference frames have different events in their space-times and so have different lengths and times.
Because space and time unify in space-time, and different reference frames differ by uniform velocity, reference-frame coordinates can linearly transform into each other.
object speed through space-time
By experiment, all observers always observe that electromagnetic waves travel at light speed through empty space, even if light sources have different uniform velocities. Moving light sources do not add to or subtract from light's observed speed.
By experiment, all observers always see that massless particles move through empty space at light speed, even if accelerations start the particles. Accelerations do not add to or subtract from massless-particle observed speed.
By experiment and calculation, all observers always see that stationary, uniformly moving, and accelerating objects move through space-time at light speed.
Universe objects are always moving through space-time. If they have no rest mass, they travel through space at light speed and so do not travel through time, and space-time interval has zero proper time (light-like). If they have positive mass, they travel through less space and more time, so space-time interval has positive proper time (time-like). If they are relatively stationary, they travel through no space and only time. If they have negative mass, they travel through space and backward through time, so space-time interval has negative proper time (space-like).
stationary observers and objects
Assume that stationary-observer reference frame has a positive-upward vertical time coordinate and a positive-right horizontal uniform-motion-direction space coordinate. See Figure 1.
Stationary objects do not move through space, so stationary observers see stationary objects move at light speed through time only. If stationary observers start at space-time coordinate origin, their space position stays at 0, and their events move along positive time coordinate. Their world-line is perpendicular to time coordinate. After one second, they have moved one light-second vertically. See Figure 2.
uniformly moving objects
Moving objects move through space and time at light speed. To stationary observers, if a uniformly moving object starts at space-time coordinate origin, object events move along a straight line up and to the right, at angle less than 45 degrees to time axis. For example, for velocity 0.5 * c, line has constant slope of 2. See Figure 3. After one second, object has moved one-light-second in that direction.
To stationary observers, because moving object has moved through space more, and object speed through space-time is constant light speed, object has moved through time less, object time has shortened, and object time-interval unit has increased (time dilation).
massless particles
Massless particles move through space at light speed. To stationary observers, if particle starts at space-time coordinate origin, particle events move along a straight line equally up and to the right, and so at a 45-degree angle to both vertical time coordinate and horizontal uniform-motion-direction space coordinate. See Figure 4. After one second, they have moved one-light-second in that direction. To stationary observers, because moving object has moved through space maximally, and object speed through space-time is constant light speed, object has moved through time minimally, object time is zero, and object time-interval unit is infinite.
accelerating objects
Accelerating-object velocities change. In space-time, object world-lines curve. See Figure 5.
stationary observers and moving objects, and time dilation
Stationary observers see that moving objects move through space over time. Because objects move through space-time at light speed, and relatively moving objects move through some space, stationary observers calculate that moving objects move less through time than stationary objects do. Stationary observers calculate that moving objects have shorter times. All observers see that light moves through space at light speed, and frequency times wavelength equals speed, so shorter times mean longer time-interval units (time dilation).
Because moving objects move along uniform-motion-direction space coordinate compared no movement for the stationary case, stationary observers calculate that moving objects have moved less along time dimension than stationary objects did. The number of time intervals is less, so time intervals have become longer (and clocks have slowed down).
To stationary observers, if uniformly moving objects start at space-time coordinate origin, their events move along a straight line through space-time in the positive space direction and in the positive time direction. For example, for relative velocity half light speed, straight line has constant slope 2. After one second, objects have moved one-light-second along straight line.
Because moving objects move along positive uniform-motion-direction space coordinate, stationary observers calculate that times are the same (clocks synchronize) along a different space coordinate than for stationary objects. Because light-travel distances are less, observers calculate that times are less. If clock signals come to observer from earlier in time, then they have taken longer. If clock signals come to observer from later in time, then they have taken shorter. Therefore, stationary observers calculate that moving objects receive clock signals that came from later in time. The motion-direction space coordinate has time zero at all events. To stationary observers, the motion-direction space coordinate points in the positive time direction and positive motion-direction space direction. For example, for relative velocity half light speed, uniform-motion-direction space coordinate has constant slope 0.5.
To stationary observers, moving-object reference-frame time coordinate and uniform-motion-direction space coordinate rotate toward each other the same angle. To stationary observers, massless-particle reference-frame time coordinate and uniform-motion-direction space coordinate rotate toward each other until they coincide at angle 45 degrees to both stationary reference-frame time coordinate and stationary reference-frame space coordinate.
Scientifically, time is time interval required for a photon to return to the same position in a cycle. For example, a photon leaves a source, travels to a mirror, reflects back to the source, triggers a click or photon that goes to a detector (for counting), and causes a second photon to repeat the cycle. Time measures number of unit times between time points. Time measures clock oscillations between events. Unit time can use clock frequency or period. Unit time can be oscillation number as light travels unit length.
If observers and objects are relatively stationary, the photon leaves from, and returns to, the same space position, over a time interval. The observed time and scientific time are the same.
If observers move relatively to objects, the photon leaves from, and returns to, different space positions. Because distance is longer, time interval is longer, so time is slower, and frequency is lower. Because people do not see the same path as the calculated path, scientific time is shorter.
synchronized clocks
Stationary-observer reference frame has time coordinate perpendicular to uniform-velocity-direction space coordinate. Stationary observers at an event on the time coordinate can set distant clocks to the same value as at the event (synchronization).
Observers moving at positive uniform-velocity move through positive time and space. Therefore, the time coordinate is at an angle (between 0 and 45 degrees) away from the stationary time coordinate toward the space coordinate.
Moving observers move toward stationary-observer positive-space-direction synchronized clocks. Therefore, they receive signals from those clocks earlier than stationary observer does, so they calculate that those clocks have past/earlier times. To synchronize moving-observer distant clocks with the clock at moving-observer current event, those clocks must have later times than stationary-observer clocks at the same time as moving-observer current event. Therefore, uniform-velocity-direction space coordinate is at an angle (between 0 and 45 degrees) away from stationary-observer space coordinate toward time coordinate.
Observers moving at uniform velocity have an acute angle (not a right angle) between positive time coordinate and positive uniform-velocity-direction space coordinate.
If moving observer has light speed, time and space coordinates merge at the 45-degree line. Massless particles travel only through space. All clocks have same time. Time interval becomes infinite, and time slows to zero. Lengths shorten to zero. Massless particles travel toward synchronized clocks as fast as light from clocks travels toward massless particles, but light can only travel through space at light speed, not higher.
simultaneity
Using synchronized clocks and knowing light speed, observers can calculate that events occur at the same time (simultaneity). Both events are on one reference-frame line that is parallel to time coordinate.
space-time separation
Space-time space and time coordinates use same units, either length units or time units. For time units, space distance changes to time. Because distance traveled equals light speed multiplied by time in motion, space-coordinate time is distance divided by light speed. For example, if distance is 300,000,000 meters, space coordinate has time 1 second.
For space units, time changes to space distance. Because distance traveled equals light speed multiplied by time in motion, time-coordinate distance is time multiplied by light speed. For example, if time is 1/300000000 second, time coordinate has distance 1 meter.
Two space-time points (events) have space-time separation, measured using time units or space units. Space-time separation is not spatial distance and is not time interval, but depends on both time and space coordinates. If first space-time point is at coordinate origin, and second point is (t, x, y, z), space-time separation s = (t^2 - (x/c)^2 - (y/c)^2 - (z/c)^2)^0.5, using time units.
Because unified space-time has light speed as maximum speed, space-time coordinates are not like Cartesian coordinates with time on horizontal axis, distance on vertical axis, and separation equal to (s^2 + t^2)^0.5. Cartesian coordinates are about independent space and time, which allows infinite speed through space.
Simultaneous events have same time on time axis (no temporal separation), and so have only spatial separation. See Figure 6. s2, s3, and s5 have same time. Setting c = 1 for convenience, space-time separation between s2 and s3 is 1.
Stationary particles do not change position, so stationary-particle events have only temporal separation. See Figure 6. Setting c = 1 for convenience, space-time separation between s1 and s2 is (2^2 - 0^2)^0.5 = 2. Because motions shorten times and lengths, stationary particles have maximum space-time separation.
Moving objects have more spatial separation and less temporal separation than stationary particles. See Figure 6. Setting c = 1 for convenience, space-time separation between s1 and s4 is (3^2 - 1^2)^0.5 = 8^0.5, representing slower particles, which have greater separation. Separation between s1 and s3 is (2^2 - 1^2)^0.5 = 3^0.5, representing faster particles, which have lesser separation. Separation between s1 and s5 is (2^2 - 2^2)^0.5 = 0, representing massless particles, which have maximum speed and have no space-time separation.
Negative space-time separation means objects have moved backward in time. Because objects cannot travel faster than light, objects cannot move backward in time and so cannot have negative space-time separation.
time and separation
Space-time separation is time observed by moving object as it travels. Zero-rest-mass objects travel at light speed and feel no time change. Stationary objects feel maximum time. Massive objects travel at less than light speed and have slower time than stationary objects. Faster objects have slower time than slower objects.
time and space relation
Because space coordinates subtract from time coordinate, shortest spatial distance is longest space-time separation. Longest spatial distance is shortest space-time separation.
simultaneity is relative
Stationary observers can synchronize different-space-position clocks to the same time. For stationary observers, if time axis is vertical, simultaneous events are on horizontal three-spatial-dimension hyperplanes. In two dimensions, if time axis is vertical, simultaneous events are on a horizontal line. See Figure 7. Events along horizontal axis are simultaneous. Events at s2, s3, and s5 are simultaneous at later time.
Moving observers can synchronize different-space-position clocks to the same time. If moving observers compare their synchronized clocks to stationary-observer synchronized clocks at the same spatial positions, moving observer clocks have later time in the uniform-motion direction (and earlier time in the opposite direction). Because they are moving closer to the clocks all the time, so the signals arrive quicker, for the same information to arrive at the current space position for both moving and stationary observer, the time at the distant position must be later for moving-observer synchronized clocks. For moving observers, simultaneous events are on three-spatial-dimension hyperplanes at an angle to time axis. In two dimensions, simultaneous events are on angled lines. For stationary observer, moving observer clocks are simultaneous along a positively sloped line, at a less-than-45-degree angle. See Figure 8. Stationary observer sees events s2, s3, and s5.
Simultaneous events are not simultaneous for observers with different velocities.
The three-spatial-dimension hyperplane of simultaneous events, line s2-s3-s5 in the diagram, is at same angle as world-line angle.
To depict moving observer at actual world-line, rather than as stationary, three-spatial-dimension hyperplanes of simultaneous events must transform their coordinate axes. Hyperplanes of simultaneous events must change from right angles to angle between world-line and space axis (limiting angle is 45 degrees). The angled-line series in the diagram represents the hyperplanes. See Figure 9.
stationary observers and moving objects, and length contraction
Because time is shorter, observers must calculate that motion-direction object distances and lengths are shorter (length contraction) so that all observers still see that light moves through space at the same light speed. Stationary observers calculate that both times and motion-direction lengths (and distances) shorten in the same proportion to keep light velocity constant. Lengths shorten in inverse proportion that time dilates.
Scientifically, length is space-time spatial distance between endpoints at space-time same time. Length measures number of unit lengths between space locations. Unit length can be a ruler. Unit length can be how far light travels in unit time.
If observers move relatively to objects, those light rays must leave the endpoints at different times, and scientific length is shorter. Moving-observer simultaneous times are later than stationary-observer ones. Coordinate transformations can calculate scientific lengths.
distances and relativity
Time shortening and length contracting are ratios. Events near each other in space and time have small total shortening. Events far from each other in space and time have large total shortening. For example, people walking and people sitting perceive several-days time difference when observing events in Andromeda galaxy. At large distances, slow relative motions can have measurable relativistic effects. At short distances, only fast relative motions show measurable relativistic effects.
observations
People and cameras observing lengths see light rays that arrive at the same time at iris or aperture. People and clocks observing time see light-ray-path-endpoint light rays that arrive at the same time at iris or detector.
If observers and objects are relatively stationary, those light rays leave the endpoints simultaneously, and observed length and scientific length are the same. Observers see and measure events whose information simultaneously reaches their space-time events. For example, stationary observers receive information from both ends of a stationary ruler perpendicular to sightline between eye and ruler-center at same time. Stationary observers simultaneously see the whole stationary ruler, but light from some positions along the ruler left before other positions. Stationary observers simultaneously see the whole night sky, but light from nearby stars left those stars a few years ago and light from farther stars left those stars longer ago.
Relativity is not about actual observer observations but about calculations based on knowledge of light speed, space-time, and space-time curvature. Length contraction and time dilation involve simultaneous points in space-time at object, not at observer. In stationary-observer reference frame, moving lengths calculate to be shorter, but human observers and instruments do not actually see or measure shorter lengths.
light
All observers see that light moves at light speed through space, no matter if light source moves relatively toward or away from observer. Light has no medium and self-propagates at light speed through empty space. Charge acceleration starts electromagnetic induction. Electric and magnetic fields change. Changing electric and magnetic fields interact to send transversely-changing electric and magnetic fields perpendicular to charge acceleration and velocity direction. Therefore, light-source motion does not supply extra motion to light speed. Light speed is the same for forward, backward, or no source motion.
Observers moving relatively to light sources along light-ray direction see frequency changes, because relative motion causes observer to encounter light-wave peaks and troughs at a different rate than if light sources are relatively stationary. If observer velocity is toward light source, expected light speed is higher than observed light speed. To reduce speed to observed light speed, length must decrease (contract). By Doppler effect, wavelength is shorter, and light frequency becomes higher. If observer velocity is away from light source, expected light speed is lower than observed light speed. To increase speed to observed light speed, length increases. By Doppler effect, wavelength is longer, and light frequency becomes lower.
Observers moving transverse to light-ray direction see light traveling at light speed, because no motion component is along light ray. If observer velocity is transverse to light-ray light source, observer sees relativistic length contraction and time dilation, in the same proportion, so relative velocity remains constant.
Relativity is about relative motion transverse to observer sightline toward object center. If massless particles, such as light photons, travel transversely, they travel only through space, do not travel through time, have infinite time interval (and clocks have stopped), and have no length in transverse direction.
To stationary observers, if massless particles start at space-time coordinate origin, their events move along a straight line at a 45-degree angle to both stationary time coordinate and stationary space coordinate. After one second, they have moved one-light-second in that space-time direction.
From a space-time event, signals can go only to space-time events in positive time direction and positive or negative space direction {light-cone}. All observed events happen in the present, unaffected by time. Only influences from previous events that have simultaneously reached space-time event can affect event. At any instant, observers see a space-time spatial cross-section.
relative distances and times
Observers with different relative velocities see the same observed objects at different events in observer space-time, so objects have different measured/calculated lengths and times. Observations are relative to velocity differences.
relativity principle and invariance
In space-time, all observable physical laws are the same at any constant velocity {principle of relativity} {relativity principle}|. At different relative velocities, observers see different length contractions and time dilations, which cancel to maintain physical laws. Only relative times, masses, and lengths have meaning for physical laws.
Because physical events occur in unified space-time, and all objects move through space-time at light speed, for stationary and moving observers, and for uniform-velocity and stationary (zero uniform velocity) systems and reference frames, physical laws are the same (invariance). Kinetics and dynamics equations, Maxwell's electromagnetism equations, and Newton's and Einstein's gravitation laws are invariant for all reference frames with uniform velocity.
By relativity, stationary observers calculate shortened lengths and times for moving objects. Because both are shorter in the same ratio, velocity is constant, and system kinetics are the same.
Such systems have no forces or accelerations. All system parts and reference-frame points have the same motion.
Calculated length contraction, time dilation, and mass increase change in the same ratio, so physical laws are the same in all reference frames differing only in uniform velocity. Space-time always has energy-momentum conservation.
no absolute velocities
Because systems with different uniform velocities have the same physical laws, uniform velocity has no physical effects, and observers cannot determine their or other object's absolute uniform velocity through space-time. All velocities are relative to observers and reference frames.
no absolute lengths and times
Because systems with different uniform velocities have the same physical laws, observers cannot determine absolute lengths and times. All lengths and times are relative to observers and reference frames.
events and physical laws
Because physical laws are invariant for all uniform-velocity reference frames and under linear coordinate transformations, physical space-time location can have no influence on physical laws, so physical laws are the same at all universe space-time points (events). There are no preferred space-time events.
event order
Before and after are relative concepts, because different observers can see different event sequences. Different observers have different pasts and futures.
Moving and stationary observers see some events in different orders. Compared to stationary observers, for events {spatially separated events} far enough apart that light cannot travel from one event to the other, moving observers see stationary events later than faster moving events.
Moving and stationary observers do not see the same spatially separated simultaneous events. Compared to stationary observers, moving observers see stationary events later than faster moving events.
relativity and space
At large distances and times, by proportionality, small relative velocities have large time-dilation and length-contraction effects. The space of relativistic velocities is hyperbolic space.
hyperbolic space
Distance equals light speed times time: x = c*t. Product of position and inverse-time always equals light speed: x * t^-1 = c. On space-time coordinates, equation graph is a hyperbola. If position is far, time is far. If position is near, time is near. If objects are rulers or clocks, distance separation and time separation are directly proportional.
mass, momentum, energy
In space-time, mass, momentum, and energy unite in a vector (energy-momentum tensor) {4-momentum}. Momentum is movement through space. Energy is movement through time. Energy equals mass times light speed squared. Action relates momentum and space and relates energy and time.
mass increase
Stationary observers calculate that relatively moving objects increase mass.
gravity and acceleration
Relative acceleration and gravity change relative velocity, so observed lengths and times depend on relative acceleration and gravity [Greene, 1999] [Mach, 1885] [Mach, 1906] [Rees, 1997] [Rees, 1999] [Rees, 2001] [Smolin, 2001] [Weyl, 1952].
electric charge
Stationary observers calculate that relatively moving charged objects increase charge density.
electric field and magnetic field
In space-time, electric field and magnetic field unite. To stationary observers, motion makes relativistic charge that has magnetic field. To moving observers, the same motion stands still, and charges have electric field. In space-time, electric field is in space, and magnetic field is in time (electromagnetic-field tensor).
geodesic
Objects travel through space-time along shortest space-time separation, which is the straightest path (geodesic) through space-time. Objects travel through positively curved space-time along shortest spatial distance and longest time. Objects travel through negatively curved space-time along shortest time and longest spatial distance.
curvature
At space-time points, mass, energy, stress, and pressure curve space-time. Around masses, gravity-field gradient (gravitational potential) is space-time curvature. Central masses curve local space-time, and that curvature pulls adjacent space-time points to curve space-time at faraway points.
Alternatively, central masses cause field energy density at far space-time points, and that energy curves space-time. Farther away space-time points have less curvature, because same energy spreads over more surface area, so energy density is less.
Energy and curvature spread to infinity at light speed, because space-time has tension and propagation characteristics the same as light-propagation speed.
free fall
Freely falling observers see no space-time curvature, because they see no acceleration, because they are at rest in the free-fall reference frame. Stationary observers at earth's surface see no space-time curvature, because they see no acceleration, because they are at rest in their reference frame.
equivalence
To stationary observers, gravity and applied force accelerate mass. Acceleration indicates space-time curvature. For both gravity and applied force, stationary observers calculate that objects move along geodesics through space-time curvature. Gravity and applied force acceleration are equivalent.
Space-time curvature is relative to observer, and so is not absolute. Observers detect only accelerations and cannot detect space-time curvature per se. However, acceleration is real and can slow clocks.
curvature and acceleration
In curved space-time, objects increasingly travel through more space, and so decreasingly less time, which means that objects accelerate. Stationary observers see increasing time dilation and time unit, decreasing frequency and time, and decreasing length.
gravitons
Gravitons are quanta of space-time gravitational waves. Because gravitons have zero rest mass, gravity acts out to infinity.
Whereas photons propagate through empty space as traveling waves in field lines, gravitons propagate through empty space as traveling waves in field surfaces. Gravitational field lines and electromagnetic field surfaces have the same tension, which is the maximum tension that they can have. Because of that maximum tension, and because they have no mass, photons and gravitons travel at light speed.
Alternatively, all zero-rest-mass bosons propagate the same because they are really the same at high energy.
tidal effects
Gravity varies inversely with distance, so objects in gravitational fields that have non-negligible diameter feel different forces on near and far sides.
However, gravity does not have to have space-time tidal effects, because local fields can be uniform, two fields can cancel, and pressure can cancel gravity, so that local space-time curvature is zero or constant.
forward time
Spatial directions can go forward or backward, but the time dimension has one direction, forward. Entropy increase, system evolution, or memory can cause time unidirectionality.
universe and relativity
Perhaps, universe is infinitely old and large, with no expansion or evolution. Perhaps, curved time coordinate allows far clocks to run slower or faster. Perhaps, far clocks stop at infinite distance, where red shift is infinite, so red shift does not need universe expansion as explanation.
electromagnetic and gravitational forces
Electromagnetism and gravity have effects to infinity and so transmit force using zero-rest-mass particles, which travel at light speed for all observers. Electromagnetic-force changes and gravitational-force changes propagate through space at light speed. Those forces' strengths determine propagation speeds, which are both light speed. Perhaps, those strengths correspond. Force strength depends on charge and mass quanta, so space-time relates to quanta. Perhaps, space-time space and time dimension relations depend on electromagnetic-force and gravitational-force strengths.
Particle energy varies directly with mass and light-speed squared {E equals m c squared}: E = m * c^2.
gravity and electromagnetism
Gravitation energy relates to electromagnetism energy.
Both electromagnetic force and gravitational force exchange zero-rest-mass particles, so both forces have effects out to infinite distances.
Electromagnetism and gravitation are spatial fields. Electromagnetism makes radial electromagnetic-field lines. Gravity makes radial gravitational-field surfaces. Surfaces have more space than lines, so electromagnetism is stronger than gravity.
Electromagnetic and gravitational waves do not travel through a medium. They propagate by induction along field lines and surfaces. Wave-propagation speed depends on field strength and field type. Electromagnetism is stronger than gravity, but in same proportion gravity uses more space, making both field lines and surfaces have maximum tension, so both electromagnetic and gravitational waves travel at light speed.
energy
Object total energy equals rest energy plus kinetic energy plus potential energy: E = RE + KE + PE. Kinetic energy varies directly with mass and velocity squared: KE = 0.5 * m * v^2. Potential-energy change PE varies directly with local force-field force and position change d in field: PE = F * d. Rest energy is constant.
energy and momentum
In classical physics, particle energy and momentum are separate physical properties, with separate conservation laws. Energy conservation depends on time symmetry. Momentum conservation depends on space symmetry.
In relativity, space and time unite in four-dimensional space-time. By experiment and calculation, all particles and objects travel at light speed through space-time. Particle motion through space-time has momentum and energy, but energy is through time and momentum is through space. In space-time, momentum and energy unite into one four-dimensional vector {energy-momentum four-vector} (4-momentum). Energy is time-like component, and momentum is space-like component.
energy and momentum conservation
For constant particle rest energy, energy conservation means that potential-energy change equals negative of kinetic-energy change. In space-time, potential energy changes through space, and kinetic energy changes through time. Kinetic-energy change changes velocity and so changes momentum. Because energy and momentum stay constant, energy-momentum four-vector separation is invariant for any inertial space-time reference frame and under any linear coordinate transformations. For potential energy (including rest energy) and momentum changes, 4-momentum-vector space-time separation has equation, in space units: s^2 = E^2 / c^2 - p^2, where E is change in potential energy and rest energy, and p is kinetic-energy change in space units. (Dividing by c makes time units into space units.)
rest-mass energy
Resting masses {proper mass} (rest mass) have no speed through space dimensions, and so travel through time dimension at light speed c. Along time dimension, rest-mass 4-momentum-vector separation s is m0 * c, where m0 is rest mass. Because rest masses do not change space position, potential energy is zero, and rest mass is constant. Because rest masses have no velocity, kinetic energy is zero, and momentum is zero. Therefore, s^2 = (m0 * c)^2 = E^2/c^2 - (0)^2, so E = m0 * c^2.
Rest mass has available energy. Rest masses are like energy concentrations. Mass densities are like energy fields.
moving masses
Moving masses have increased positive kinetic energy. Increased kinetic energy is similar to concentrated mass, so stationary observers calculate that moving masses have mass increase (relativistic mass).
Moving mass goes through space-time separation m * c, where m is total mass (rest mass and relativistic mass). Moving mass has momentum total-mass times velocity. Rest-mass energy is rest mass times light speed. For example, if potential energy is zero (with no gravity), and velocity is 0.75 * c, - s^2 = - (m * c)^2 = - (m0 * c^2)^2/c^2 - (m * 0.75 * c)^2, then - m^2 * c^2 = - m0^2 *c^2 - (0.75)^2 * m^2 * c^2, and then - 0.25 * m^2 = - m0^2, and total mass m = 2 * m0.
In empty space, energy E depends on rest energy, in time dimension, and kinetic energy KE, in space dimensions. Rest energy = m0 * c^2. KE depends on momentum p: KE = p * c. Total energy sums rest energy and kinetic energy {relativistic energy-momentum equation}: E^2 = (m0 * c^2)^2 + (p * c)^2.
For zero-rest-mass particles, E = p*c.
For resting masses, p = 0, and E = m0 * c^2.
For moving masses, total energy is total mass m times light-speed squared: E = m * c^2. If velocity is near zero, total mass is almost the same as rest mass. If velocity is near light speed, total mass is very large, much greater than rest mass.
relativistic mass increase
Objects traveling through space have momentum and kinetic energy. Higher-velocity objects travel more through space and less through time, causing more time dilation and length contraction. Objects traveling more through space increase 4-momentum momentum and kinetic energy, in the same proportion that time dilates. Therefore, total mass m increases with velocity: m = m0 / (1 - (v^2 / c^2))^0.5, where c is light speed, v is velocity, and m0 is rest mass. For example, if velocity is 0.75 light speed, total observed mass is twice rest mass.
equivalence
Energy in time dimension can go into momentum in space dimensions, and vice versa {mass-energy equivalence}.
relativistic energy in series format
In series format, in empty space, total energy E = m0 * c^2 + m0 * v^2 / 2 + (3 * m0 * v^4) / (8 * c^2) + ...., where m0 is rest mass. See Figure 1. The first term is the rest energy. The sum of the higher-power terms is the kinetic energy. For slow particles, later terms are very small, so kinetic energy is m0 * v^2 / 2, matching the classical value of 0.5 * m * v^2.
mass and energy equivalence
Particle decomposition and composition experiments show that mass and energy are equivalent and depend only on reference frame.
cases
An unstable particle with mass can become two zero-rest-mass particles that travel at light speed in opposite directions from particle position. Zero-rest-mass particles have no potential energy. Moving particles have kinetic energy. Mass changes into kinetic energy, to conserve mass-energy. Momentum in one direction equals momentum in opposite direction, to conserve momentum.
An unstable particle with mass can become two particles, one with mass and one with no mass. Total mass is less than before, to conserve energy. The zero-rest-mass particle travels at light speed in opposite direction from new particle with mass, which travels at less than light speed. Momentum in one direction equals momentum in opposite direction, to conserve momentum.
Two particles with mass can collide to make one particle with mass and one zero-rest-mass particle. The zero-rest-mass particle travels at light speed in opposite direction from new particle with mass, which travels at less than light speed, to conserve energy. Momentum in one direction equals momentum in opposite direction, to conserve momentum.
Two equal particles can collide and stop, so kinetic energy becomes mass. Total mass is then more than sum of rest masses before.
Physical objects and events occur in unified space-time. Uniformly moving observers observe no forces and no accelerations and have space-time coordinates (reference frames) with no curvatures. All system parts and reference-frame points have the same motion.
Because uniformly moving reference frames can linearly transform into each other, and objects move through space-time at light speed, physical laws are the same {invariance, uniform motion} for all uniformly moving observers and objects. Locally, kinetics and dynamics equations, Maxwell's electromagnetism equations, and Newton's and Einstein's gravitation laws are invariant for all reference frames with uniform velocity.
conservation laws
Motion equations relate local momentum and energy exchanges between particles and fields. Energy and momentum conservation laws are examples of invariance. Energy conservation is about time symmetry. Momentum conservation is about space symmetries. Space-time unites space and time, so space-time has one energy-momentum conservation law.
cause
By relativity, stationary observers calculate shortened lengths and times for moving objects, in the same ratio. Therefore, velocity is constant, and system kinetics remain the same.
no absolute velocities
Because systems with different uniform velocities have the same physical laws, uniform velocity has no physical effects, and observers cannot determine their or object absolute uniform velocity through space-time. All velocities are relative to observers and reference frames.
no absolute lengths and times
Because systems with different uniform velocities have the same physical laws, observers cannot determine absolute lengths and times. All lengths and times are relative to observers and reference frames.
linear coordinate transformations
Uniform velocities relate reference-frame coordinates linearly. All uniform-velocity reference frames can transform to all other uniform-velocity reference frames by linear coordinate transformations. Therefore, physical laws are invariant for linear coordinate transformations. For example, linear coordinate transformations can derive Maxwell's equations from Coulomb's law.
events and physical laws
Because physical laws are invariant for all uniform-velocity reference frames and under linear coordinate transformations, physical space-time location can have no influence on physical laws, so physical laws are the same at all universe space-time points (events).
space-time separation
Space-time separation is invariant for linear coordinate transformations.
physical constants
Because physical laws are invariant for all uniform-velocity reference frames and under linear coordinate transformations, fundamental physical values are constant for all uniform-velocity reference frames and under linear coordinate transformations. For example, angular momentum and other quanta remain constant.
accelerating objects
In local space-time regions, and for small accelerations, reference frames can approximate uniform-motion reference frames, so physical laws are invariant over linear coordinate transformations. Large space-time regions and large accelerations break physical-law invariance.
Physical laws do not have action-at-a-distance. Physics laws are about what happens at space-time points {local interaction}. Except for large forces, local interactions are approximately linear, allowing linear coordinate transformations and energy-momentum-field tensors. Spaces with local metrics include Minkowski space, Einstein space, and Lorentzian space.
Global physical effects are typically non-linear.
Observers can measure relative speed and direction {relative velocity}| with respect to physical objects (and measurement reference frames), using clocks for time units, rulers for length units, and/or light signals for time and length units.
observers
Observers can measure space-time separation from current space-time event to future space-time event using light signals. Observer clocks measure time for light to go from observer to reflector and return from reflector to observer. Observer rulers measure length for light to go from observer to reflector and return from reflector to observer.
Observers and objects can be stationary or moving. Observers observe themselves as stationary and observe objects with no relative velocity as stationary, and relatively stationary things have no special-relativity effects.
Observers can move relative to objects, and objects can move relative to observers. If relative velocities are the same, the two situations are physically equivalent.
Relatively moving things have relativistic effects, which are space-time calculations about simultaneous distant space-time events. Relativity is about what must be true at distant events and so depends on calculations.
In relativity, only calculations change. Objects do not change, because observations do not affect their space-time event.
observations
People and instruments receive simultaneous signals at their current space-time event. Signals came from different points on lengths and from different phases of time. Their observations do not have time dilation or length contraction.
uniform velocity
Relative velocity measures proportion of light speed through space-time that is through space compared to that through time. Light has maximum speed through space.
Special relativity is about relative uniform velocity, with no acceleration. Special relativity is about empty space.
acceleration
Changing velocity speed or direction requires gravitational and/or electromagnetic force (including mechanical force) fields.
Zero-rest-mass particles, such as photons and gravitons, have no inertia and have no gravity, so they do not interact with other masses and gravitational energies. Adding gravitational or mechanical energy to zero-rest-mass particles does not increase or decrease their velocities. Zero-rest-mass particles travel at maximum-velocity light speed, cannot travel faster or slower than light speed, and cannot be at rest.
Zero-rest-mass and zero-charge particles, such as photons, have no inertia and have no electromagnetism, so they do not interact with other charges and electromagnetic energies. Adding electromagnetic energy to zero-rest-mass particles does not increase or decrease their velocities. Zero-rest-mass and zero-charge particles travel at maximum-velocity light speed, cannot travel faster or slower than light speed, and cannot be at rest.
direction
Two objects can move toward or away from each other (radial motion) and/or right-left and/or up-down with respect to each other (transverse motion). Special-relativity relative-measurement differences are about relative transverse motion. Motion toward or away does not change measured/calculated length or time.
observers and absolute velocity
Universe has absolute space-time. Space-time unites space and time symmetrically. Objects travel through space-time at light speed. Because light travels at light speed no matter observer or object motion, observers cannot observe absolute space-time or absolute velocity. Observers measure that moving objects have length contraction and time dilation, in the same proportion, so relative velocity stays constant, and light has constant light-sped velocity. Relative velocity does not affect physical laws, so observers cannot use experiments to find absolute velocity.
velocity, length, time
For relative velocity v, length is x * (1 - (v/c)^2)^0.5, and time interval is t / (1 - (v/c)^2)^0.5. For example, if relative velocity is half light speed, length is 0.86 * x, and time interval is 1.16 * t. If relative velocity is 99.9% light speed, length is 0.01 * x, and time interval is 99 * t. If relative velocity is light speed, length is zero, and time interval is infinite.
relative velocity maximum
If objects travel faster than light, relativistic length becomes less than zero, so physical objects cannot travel faster than light speed. If objects travel faster than light, relativistic mass becomes more than infinite.
If objects travel faster than light, relativistic time interval becomes more than infinite, and time goes backward. Traveling backward in time violates causality. Space-time events can only receive signals from finite space-time event regions, whose space-time events are near enough so signals from them can reach the space-time event. See Figure 1.
faster than light
Because light always travels at light speed, faster-than-light signals from stationary objects appear to go backward in time to observer moving away. See Figure 2. If signals continue to second stationary object, toward which observer is moving, second stationary object can reflect signal back to first stationary object. Faster-than-light signals from second stationary object appear to go backward in time to observer moving toward it. The signal returns to first stationary object before it sends original signal. If signal can travel faster than light speed, observer can have event knowledge before event happens.
See Figure 3. Moving observers have tilted hyperplanes of space-like space-time events relative to object going from s0 to s1 to s2. They see signals from s2 as going backward in time. Moving observers have different tilted hyperplanes of space-like space-time events relative to object going from s0 to s3. They see signals from s3 as going backward in time. Object going from s0 to s1 to s2 appears to receive reflected signals from s3 at s1, before signals left s2.
Relatively stationary objects have energy {rest energy}| due only to rest mass. Relative motion does not affect rest energy.
Universe has three continuous space dimensions and one continuous time dimension, and they unite symmetrically in four-dimensional space-time {space-time, relativity}|. Time and space are not separate and independent physical properties.
events
All particles and objects move through space-time at light speed. Observers and objects move through time as well as space.
Space-time has events, not independent times and spatial positions. Observers and objects move through space-time events. Objects travel through space-time along four-dimensional-vector paths {world line}, along geodesics.
units
Space-time coordinates have the same units for time and space. Because distance equals time times velocity, distance unit can be time unit times light speed: c*t. Because time equals distance divided by velocity, time unit can be distance unit divided by light speed: x/c. Time unit can use one oscillation over one distance unit: 1 / cm = cm^-1.
relativity
Though relatively moving observers calculate different lengths and times, because lengths and times shorten in same proportion, space-time separation between two space-time events is constant for all uniformly moving observers.
regions
Different space-time regions behave differently. Regions can be inside gravity or electromagnetic sources. Vacuum regions can be near sources. Regions, such as flat space-time, can have weak fields. Regions can have weak fields but have radiation.
time coordinate as imaginary numbers
Relatively moving observers have different reference frames and relative space-times. Relative space-times differ by relative velocity (boost), which causes three-dimensional rotations into time. The complex plane and space-time coordinate system have the same properties, so time coordinate is like imaginary-number coordinate.
Simultaneity requires an observer and two space-time events. By direct observation, two space-time events are simultaneous {simultaneity, relativity}| if light signals from both events reach observer at same time. See Figure 1. To another observer at a different space-time event, the two events are not simultaneous. (If an observer sees that two events happen at the same spatial location, relatively moving observers do not calculate that they coincide.)
synchronized clocks
Two space-time events can be simultaneous for an observer if they occur at the same time on observer's synchronized clocks. Because this simultaneity occurs at distant space-time events, observer can only measure and calculate this simultaneity.
Observer reference-frame space coordinates show all space locations, with clocks synchronized to same time. Events that happen on a space-time space coordinate are all at the same time. For example, at space-time origin, time is 0, and space coordinate shows all space locations, whose events have time 0.
space coordinates
Space coordinates for relatively moving observers are different. Compared to stationary observers, uniformly moving observers move toward stationary-observer synchronized clocks, and receive light rays sooner than stationary observers. Because they come sooner, relatively moving observers calculate that those light rays came from later-time events. See Figure 1.
examples
At s1, observer has time 0 and position 0. s2 and s3 happen at same time on time axis, and information about them reaches s4 at same time.
Information from s1 reaches s2 and s3 at same time on time axis, so they are simultaneous. s2 and s3 are two different observers.
Because signals travel at light speed, information from s1 can reach s6.
Information from s1 cannot reach s4, s5 or s7.
Information from s3 reaches s6 later.
At s4, observer is at time 2 and position 0. Information from s4 reaches s7.
s4, s5, and s6 happen at same time on time axis.
No information about s1, s2, s3, s4, and s6 reaches s5.
Information about s3 reaches s6.
Information about s5 comes to observer at position 0 sooner than information from s6, because s5 is closer in space.
s7 happens later than s4, s5, and s6 on time axis.
space-time separation
Observers and objects move through space-time events. Space-time events have space-time separations. In space-time, neither time separation nor space separation exists independently.
Simultaneous events for observer have same space-time separation from observer. Observers cannot detect events from space-time points outside their light cone. See Figure 2. s2 and s3 are simultaneous for observer at space-time position s4. s4 and s6 are simultaneous for observer at space-time position s7. s4, s5, and s6 are not simultaneous for any observer.
event order
Relatively moving observers do not agree on event locations or times, and can calculate that the same space-time events happen in different orders.
absolute time
Because relatively moving observers calculate different times, spatial positions, and event orders for the same space-time events, observers cannot detect absolute time (or absolute location).
Local space-time coordinate systems can transform by first-power functions into all other local space-time coordinate systems {Lorentz transformation} {linear transformation}.
Distinct space-time points have different pasts and futures {space-time point set}. Because maximum speed is light speed, space-time points have possible past points {past-set} and possible future points {future-set}. Point past-sets and future-sets are unique {indecomposable}. Indecomposable past-sets can affect the space-time point. The space-time point can affect indecomposable future-sets.
Geometries can have points at infinity (ideal point). Space-time should not have ideal points or singularities.
All space-time points {event} have past-set and future-set, so space-time has possible causes and effects {causal structure}. Space-time events change causal structure over time and space.
Space-time points have a space-time region {global causal structure} that light can reach in the future. A space-time point can only affect those events. A space-time point has a space-time region whose events can affect it.
Space paths cannot reverse time, so no event can happen at two times. Between past point and future point reachable from past point, all space-time points are reachable {hyperbolic space-time, global}, so space-time has no singularities.
All light rays from a space-time point make a space-time cone {light cone}|. All light rays to a space-time point make a light cone.
If light rays from a space-time point later converge {converging light cone}, convergence point is a singularity.
In zero gravity, object translations, rotations, vibrations, scale changes, and inversions in space do not change object geometric shape. Zero-gravity four-dimensional space-time has symmetry {conformal symmetry} {conformal symmetry group} that preserves geometric shape, because metric-scale changes {Weyl transformation} leave proper time and proper length unchanged. Mathematically, the Poincaré group, scale invariance (dilation or dilatation), and inversion-translation-inversion (special conformal transformation) have conformal symmetry and preserve geometric shape.
Observers moving uniformly in unified space-time in relation to objects calculate that object length in uniform-velocity direction is shorter than for relatively stationary objects {distance contraction} {length contraction}|.
relativity
Stationary observers calculate that moving objects have shorter lengths in movement direction than stationary objects. Moving observers calculate that stationary objects are moving and have shorter lengths in movement direction. In both cases, observer and object have relative velocity. See Figure 1.
direction
Length contraction happens only in movement direction. Length contraction depends on relative transverse velocity. The radial velocity component has no effect, and directions perpendicular to movement direction have no length contraction.
distance from observer
Because contraction direction is perpendicular to distance direction, distance away does not affect length-contraction ratio.
calculation
When stationary observers look at moving rulers, ruler points do not have same time. See Figure 1.
space-time reference frame
On space-time reference frames, moving events trace vectors. Stationary objects trace vectors parallel to time coordinate. See Figure 2.
space-time separation
Space-time events are separate in both time and space.
Compared to stationary rulers, moving-ruler leading end is earlier in time and behind in space. Trailing end is later in time and ahead in space. Observer calculates that object length and time are shorter. Length contraction and time dilation have same percentage, so physical laws do not change, and space-time separation is same as before. For all uniformly moving observers, physical laws are the same, and space-time separations are the same.
See Figure 3. In space-time, space gain causes time loss, so space-time separation s depends on space separation x and time separation t (ignoring y and z dimensions). Because distance x is light speed c times time t, s^2 = x^2 - (c*t)^2, using distance units, or s^2 = t^2 - x^2/c^2, using time units.
For constant motion, t = x/v, so s^2 = (x/v)^2 - (x/c)^2 = x^2 * (1/v^2 - 1/c^2) = (x^2 / v^2) *(v^2 / v^2 - v^2 / c^2) = (x^2 / v^2) * (1 - v^2 / c^2) = (x^2 / c^2) * (c^2 - v^2) / v^2. Therefore, s = (x/v) * (1 - v^2 / c^2)^0.5 or s = (x/c) * (c^2 - v^2)^0.5 / v. Stationary observers calculate that moving-object length is shorter than stationary length.
length-contraction percentage
If moving object has velocity 0.5 * c (half light speed), space-time separation s = (x/c) * (((c^2 - 0.5 * c)^2)^0.5 / (0.5 * c)) = (x/c) * ((c^2 - 0.25 * c^2)^0.5 / (0.5 * c)) = (x/c) * ((0.75*c^2)^0.5 / (0.5 * c)) = (x/c) * 0.865/0.5 = 0.43 * (x/c).
If moving object has velocity 0.9 * c (nine-tenths light speed), space-time separation s = (x/c) * ((c^2 - (0.9 * c)^2)^0.5 / (0.9 * c)) = (x/c) * 0.19/0.9 = 0.21 * (x/c).
If moving object has velocity 0.99 * c (99% light speed), space-time separation s = (x/c) * ((c^2 - (0.99 * c)^2)^0.5 / (0.99 * c)) = (x/c) * 0.02/0.99 = 0.02 * (x/c).
As moving object approaches light speed, stationary observer sees that length decreases toward zero. Stationary objects have maximum space-time separation.
maximum speed
Length less than zero is impossible. Therefore, nothing can go faster than light speed, and nothing can go backward in time.
length measurement
To measure stationary rulers, stationary observers at one ruler end can send signals to a reflector at other end. See Figure 4. Signal travels from end to end and back. Time to go is same as time to return. Travel time is directly proportional to length.
To measure moving rulers, stationary observers at one ruler end can send signals to a reflector at other end. Ruler reflector moves closer as signal travels and reflects earlier. Observer measures shorter time and measures that ruler has shorter length.
Stationary observers calculate that stationary rulers spread over space only. Stationary observers calculate that moving rulers spread over space and time. Stationary and moving rulers have same space-time separation. See Figure 5.
For moving rulers, for ends to seem simultaneous, ends lie along line tilted away from vertical, not on vertical. See Figure 5. Leading end is further along in space, and trailing end is behind in space. Middle moves toward where leading end was, and away from where trailing end was. For signals to reach middle simultaneously, leading end must signal later in time, and trailing end must signal earlier in time.
time
When moving object passes stationary observer, one end reaches observer before other end. Other end lags behind in time, because ends are traveling through time at less than light speed, and it takes time for other end to reach observer. When moving object moves through space faster, lengths appear shorter, and moving object moves through time slower, so time slows. See Figure 2. At less than light speed, angle is less than 45 degrees. At light speed, angle is 45 degrees.
analogies
Length contraction is like looking at rulers rotated away from perpendicular to sightline. For space-time, rotation is into time dimension.
Length contraction is like looking at rulers from farther away.
Length contraction is like light rays curving inward from both ruler ends, like a concave lens (opposite from gravitational lensing).
Because space-time separation has a negative sign under the square root, length contraction is like using imaginary numbers. Space-time time coordinate is like imaginary axis, so space-time is like complex plane.
Observers moving uniformly in unified space-time in relation to objects calculate that object time in uniform-velocity direction is shorter than for relatively stationary objects {time dilation}|, and that unit time interval takes longer, so time slows down.
relativity
Stationary observers calculate that moving objects have shorter times than stationary objects. Moving observers calculate that stationary objects are moving and have shorter times. In both cases, observer and object have relative velocity. See Figure 1.
direction
Time dilation depends on relative transverse velocity. The radial velocity component has no effect.
distance from observer
Because transverse relative velocity is perpendicular to distance direction, distance away does not affect time-dilation ratio.
observation
When stationary observers look at moving clocks, times are not at same positions. See Figure 1.
space-time reference frame
On space-time reference frames, moving events trace vectors. Stationary objects trace vectors parallel to time coordinate. See Figure 2.
space-time separation
Space-time events are separate in both time and space.
Compared to stationary clock, moving-clock first tick is behind in space and so earlier in time, and latest tick is ahead in space and so later in time. Observer calculates that object length and time are shorter. Length contraction and time dilation have same percentage, so physical laws do not change, and space-time separation is same as before. For all uniformly moving observers, physical laws are the same, and space-time separations are the same.
See Figure 3. In space-time, space gain causes time loss, so space-time separation s depends on space separation x and time separation t (ignoring y and z dimensions).
Because distance x is light speed c times time t, s^2 = x^2 - (c*t)^2, using distance units, or s^2 = t^2 - x^2/c^2, using time units.
For constant motion, x = v*t = (c*t)^2 - (v*t)^2 = t^2 * (c^2 - v^2) = c^2 * t^2 * (1 - v^2 / c^2). Therefore, s = c * t * (1 - v^2 / c^2)^0.5. s/c = t * (1 - v^2 / c^2)^0.5. Stationary observers calculate that moving-object time is shorter than stationary time.
time-dilation percentage
If moving object has velocity 0.5 * c (half light speed), time s = t * (1 - (0.5 * c)^2 / c^2)^0.5 = t * (1 - 0.25) = 0.75 * t.
If moving object has velocity 0.9 * c (nine-tenths light speed), time s = t * (1 - (0.9 * c)^2 / c^2)^0.5 = t * (1 - 0.81) = 0.19 * t.
If moving object has velocity 0.99 * c (99% light speed), time s = t * (1 - (0.99 * c)^2 / c^2)^0.5 = t * (1 - 0.98) = 0.02 * t.
As moving object approaches light speed, stationary observer calculates that time decreases toward zero. Stationary objects have maximum space-time separation.
maximum speed
Negative time is impossible. Therefore, nothing can go faster than light speed, and nothing can go backward in time.
time measurement
To measure stationary clocks, stationary observers observe light from clock when it reaches same resonating-wave oscillation, spin, or revolution phase. Clocks have wavelengths, frequencies, and periods. See Figure 4. Signal travels from end to end and back. Time to go is same as time to return for stationary observers. Travel time is directly proportional to length.
To measure moving clocks, stationary observers can send first-clock-beat signal to reflector at other end. Clock reflector moves closer as signal travels and reflects earlier. Observers measure that resonance cavity is longer and time interval is longer.
When moving object moves past stationary observer, one part reaches observer before other parts. Other parts lag behind in time, because they are traveling through time at less than light speed. It takes time from other parts to reach observer. Moving object goes through space faster, so lengths appear shorter. Moving object goes through time slower, so time slows. See Figure 2. At less than light speed, angle is less than 45 degrees. At light speed, angle is 45 degrees.
Frequencies are clocks. Time interval unit is time between beats or ticks, such as one second. Time is number of beats or ticks, such as 60 cycles. When time slows, frequency decreases, wavelength increases, time unit increases, and cycles decrease. Time dilation makes time unit become longer, so number of ticks is fewer, so time passes more slowly.
analogies
Time dilation is like looking at a repeating process (clock) rotated away from perpendicular to sightline. For space-time, rotation is into space dimension.
Because space-time separation has a negative sign under the square root, time dilation is like using imaginary numbers. Space-time time coordinate is like imaginary axis, so space-time is like complex plane.
Observers moving uniformly in unified space-time in relation to objects calculate that object mass is greater than for relatively stationary objects {mass increase}| {apparent mass} {relativistic mass}.
relativity
Stationary observers calculate that moving objects have greater masses than stationary objects. Moving observers calculate that stationary objects are moving and have greater masses. In both cases, observer and object have relative velocity. See Figure 1.
direction
Mass increase depends on relative transverse velocity. The radial velocity component has no effect.
distance from observer
Because transverse relative velocity is perpendicular to distance direction, distance away does not affect mass-increase ratio.
measurement
Observers measure mass using standard mass {unit mass} {mass unit}, such as one kilogram. Mass measurements use forces, energies, distances, and times. To count mass, observers measure number of unit masses.
comparison to length and time
Stationary observers calculate that length contracts, time dilates, and mass increases. See Figure 2. See Figure 3.
cause
Stationary mass (rest mass) travels only through time and has no kinetic energy or potential-energy change. Moving mass travels through space (and time) and so has kinetic energy and may have potential-energy change.
Because space and time unite in space-time, momentum and energy unite. Momentum and energy both vary directly with mass. Momentum is along space coordinate, and energy is along time coordinate. As velocity increases, object moves more through space and less through time, so relative momentum increases more than velocity, so mass increases.
zero-rest-mass-particle relativistic mass and frequency
For zero-rest-mass particles, rest mass stays zero, but relativistic mass increases. Zero-rest-mass-particle energy E is directly proportional to frequency v: E = h * v, where h is Planck's constant. Zero-rest-mass-particle energy is E = m * c^2. Therefore, relativistic mass is m = h * v / c^2. Adding energy to zero-rest-mass particles increases frequency. Removing energy from zero-rest-mass particles decreases frequency.
non-zero-rest-mass particle relativistic mass
Particles with mass move through space and time, so length contracts, and time dilates. See Figure 4. Relativistic mass m is rest mass m0 plus space-dilation mass mr due to kinetic energy: mr = m0 / (1 - v^2 / c^2)^0.5. As relative velocity increases, stationary observers calculate mass increase.
Relative speed greater than 80% light speed makes object relativistic-mass kinetic energy exceed object rest-mass energy: E = m * c^2 = 0.5 * (3*m) * (0.82 * c)^2.
maximum speed
As objects approach light speed, mass increases toward infinity. As mass increases, inertia resists further acceleration, so nothing can have infinite mass or energy. No object with mass can move at light speed.
Space-time curvature describes motions of accelerating objects and objects in gravitational fields {general relativity}| (geometrodynamics).
space-time
Three spatial dimensions and one time dimension unify into space-time. Space-time has no preferred time direction, no preferred spatial direction, and no handedness.
local space-time
Physical laws are about what happens at space-time points. With small gravity and/or acceleration, space-time-point reference frames locally approximate uniform-velocity reference frames, which have linear coordinate transformations. Their space and time coordinates are straight lines.
Distant galaxies have negligible gravitational effects on local space-time, so empty space has no gravitational fields and no space-time curvature.
Observers traveling with relative uniform velocity to objects calculate that objects shorten time and contract length, whose amount corresponds to angle between time coordinate and motion-direction space coordinate. Angle varies directly with relative velocity.
space-time curvature
Objects accelerate by mechanical force or by gravitation. Observers accelerating with respect to objects increase relative velocity, so length contraction and time dilation change. When they change, reference-frame space-time coordinates change angle between time coordinate and motion-direction space coordinate. This coordinate angle change is space-time curvature. Space curvature alone and time curvature alone cannot happen, because curvature is the angle change between space and time coordinates.
Therefore, models using circle curvature (1/r), sphere curvature (1/r^2), or 4-sphere curvature (1/r^3) do not show the essence of the story. Neither do models showing a flat surface with curvature in the middle, for example, a trampoline with a weight on it.
In space-time, all objects move at light speed. Objects at rest move through time only. Objects moving at light speed move equally through time and space. (Objects cannot move only through space, because motion requires time by definition. Objects cannot move through space more than time, because experiment shows that light speed is maximum speed.)
Space-time plots for motions through flat space-time have object trajectories that are straight lines. Coordinates show equally spaced units of space and time. Coordinate positions are number of space units (meters) and number of time units (seconds or light-seconds).
If coordinates show equally spaced units of space and time, space-time plots for motions through curved space-time have object trajectories that are curved lines, because the relation between space and time is always changing. Note: Using log-log plots, with ln (y) and ln (x), makes power law functions, y = a * x^b, become straight lines. Using semi-log plots, with ln (y) and x, makes exponential functions, y = a * e^(b*x), become straight lines. However, the relation between space and time coordinates is not a power law or exponential function.
Space-time curvature is not about changes to coordinate units. Time dilation and length contraction are about simultaneity relations between objects and observers in different coordinate systems (reference frames). Space-time curvature is about intrinsic properties of space, and how motion partitions between time and space. In curved space-time, motion cannot be purely through time, because the time and space coordinates are not orthogonal, so motion must have both time and space components. Objects originally at rest in a gravitational field must move through space, since all objects move through space-time at light speed. The more space-time curves, the more the space component increases compared to the time component, so objects move faster through space the closer they get to a (larger) mass. A ball thrown upward slows down as space-time curvature decreases, until it is at rest at the top of its trajectory, where upward and downward motions are equal.
global
Non-locally, time coordinate and motion-direction space coordinate angle changes make global reference frames non-linear.
non-linearity
Objects with mass have gravitational fields and curve space-time. Because the objects pass through this curved space-time, their own gravitational field affects their motions. In general relativity, mass acts on itself through its gravitational field. In general relativity, therefore, total force is not the vector sum of forces. Non-local motions are non-linear. Non-local curved space-time is non-linear.
absolute effects
Objects start with no acceleration and in negligible gravitational fields. After objects mechanically accelerate and/or pass through gravitational fields, they return to no acceleration and negligible gravitational fields. Stationary observers calculate that objects have permanently shorter times, so passing through curved space-time has absolute physical effects for stationary observers.
energy-momentum tensor
Energy conservation is due to space-time time symmetry. Momentum conservation is due to space-time spatial symmetry. Angular-momentum conservation is due to space-time right-left symmetry. Because space-time unifies distance and time, space-time unifies energy, momentum, and angular momentum into an energy-momentum tensor.
relativity tests
Relativity tests have all proved that general relativity is correct, and other metric and non-metric theories are not correct. Measurements agree with general-relativity theory to within 10^-12 percent.
For example, the sun bends light rays that come from stars behind Sun at calculated rate.
Uniform-velocity observers calculate that accelerated and then decelerated clocks have lost time and aged less at calculated rate.
Mercury's perihelion precesses around Sun at calculated rate.
Earth and Moon change separation distance periodically at calculated rate.
Distant-star spectral lines red-shift at calculated rate.
Spectral lines red-shift as they pass through Earth gravity at calculated rate.
Accelerating masses, and objects changing mass, make gravity waves. Gravity-wave emission causes binary pulsars to have smaller orbits and shorter orbital periods at calculated rate.
other physics theories
Besides gravity and accelerations, general relativity applies to thermodynamics, hydrodynamics, electrodynamics, and geometric optics.
Because of the Big Bang, universe space is expanding uniformly and linearly {space expansion} {expansion, inflation}. For example, twice as far away, space expands twice as fast. Far enough away, space can expand faster than light.
relativity
Spatial expansion is about space itself expanding. Spatial expansion is not movement through space, so relativity does not apply.
effects on objects
Space expansion is less strong than electromagnetic and nuclear forces, so objects only stretch slightly. Space expansion is less strong than gravity, except between galaxies.
universe inflation
Universe began with low entropy. Before cosmic inflation, universe had little mass-energy, only 10 kilograms in 10^-28 meter diameter sphere, or 10^-8 kilograms in 10^-35 meter diameter sphere if minimum entropy. Only one trapped inflaton can start inflation. Cosmic inflation expanded space faster than light. During cosmic inflation, inflaton field gained potential energy, because space expansion reduces kinetic energy and increases potential energy. Energy density was constant, because energy grew equally with volume.
Perhaps, there are many inflated universes.
Stationary observers in gravitational-force fields calculate the same local object motions that stationary observers calculate for local accelerating objects. Accelerating observers feel the same effects as if they were stationary in gravitational-force field. Local accelerations and gravitational-field effects curve space the same. Uniform-velocity observers cannot distinguish whether object motions are due to gravitational force or acceleration {equivalence principle}| {principle of equivalence} {background independence, acceleration}.
To observers, accelerations caused by gravitation and accelerations caused by mechanical forces are equivalent. Observers cannot distinguish between gravity-caused accelerations and rocket, elevator, or collision accelerations. For example, people inside an elevator cannot distinguish if elevator has accelerated upward or gravitational field is greater, because locally they feel the same stress on their feet.
non-local
Except for high gravity and/or acceleration, space-time points approximate reference frames with linear coordinate transformations. Observers apply special relativity.
Over space-time regions, gravitational fields vary with distance and accelerations vary, so reference frames have non-linear coordinate transformations. Over space-time regions, because space-time curvature differs over space-time points, observers can distinguish object motions due to accelerations or to gravitational-force fields. Observers apply general relativity.
inertial and gravitational mass
Mass has two properties. Mass (gravitational mass) causes gravity. Mass {inertial mass} resists acceleration. Because space-time unifies space and time, gravitational mass is the same as inertial mass, because they both curve space-time the same amount. For example, in gravitational fields, all objects, no matter what their mass, accelerate (free fall) at same rate. Same-diameter lead balls fall at same rate as cloth balls. Object acceleration depends only on gravitational-field strength, not on object mass. This is because, gravity from object and object resistance to motion are equal. Objects in free fall feel no force. Observers in free fall observing objects in free fall see no relative motion. Space-time curvature is not an outside force but sets the field of motion.
Particles and objects have gravitational interactions with universe (fixed) distant galaxies. Particles and objects resist accelerations because of these gravitational interactions. Accelerations are absolute (not relative) with respect to the fixed distant galaxies {Mach's principle} {Mach principle}. Universe distant galaxies make an absolute reference frame, and gravitational mass and inertial mass are equivalent because of these interactions.
However, general relativity does not use Mach's principle. In general relativity, gravitational mass and inertial mass are locally equivalent to observers, because they both curve space-time the same.
By special relativity, object-observer relative motion causes observers to calculate that object has time dilation and motion-direction length contraction. For uniform-velocity observers and objects, time-dilation (and length-contraction) ratio does not change. Reference-frame time coordinate and motion-direction space coordinate maintain same angle to each other. Because coordinates maintain same relation, observed space-time does not curve.
Observers accelerating at same rate and direction as accelerating objects have no relative motion, so space-time time coordinate and motion-direction space coordinate are the same for both observer and object. Observed space-time does not curve.
acceleration
Observers accelerating in relation to objects change relative velocity. Observers calculate that time-dilation and motion-direction length-contraction ratio changes. If relative velocity increases, observers calculate that positive space-time time coordinate rotates toward positive motion-direction space coordinate, and motion-direction space coordinate rotates toward positive time coordinate. (The two space coordinates perpendicular to the motion-direction space coordinate have no changes.) Because the angle between the two coordinates changes, space-time curves {curvature, space-time}. Space-time curvature means that objects traveling along space-time events change relative travel amounts through time and space. If space-time curvature changes, outside observers see acceleration along geodesic direction.
space-time
Because space-time unifies space and time, space-time curvature is not about space curvature or time curvature separately. Coordinates do not curve. Only angle between coordinates changes.
gravity
In classical physics, masses have gravitational fields around them and attract each other by gravity. Gravity varies inversely with squared distance from mass.
Energy conservation is about time symmetry. Momentum conservation is about space symmetry. Energy and momentum vary directly with mass. In general relativity, because space and time unify into space-time, mass, energy, and momentum unify into momentum-energy. Mass-energy curves space-time over all space and time, making a field. General relativity is a field theory.
Because field varies inversely with distance squared, both a time-dilation gradient and a length-contraction gradient are at every space-time point. Space-time curvature is the unified time-dilation and length-contraction gradient. Gradients, curvatures, and accelerations are larger nearer to mass-energies.
tidal force
Objects moving in gravitational fields feel different forces at different distances from central mass. Object near side has more force than object far side (tidal force). Space-time curvature and object acceleration differ at different distances from central mass-energy.
no torsion
Space-time curvature fields have time coordinate, radial space coordinate, and two space coordinates perpendicular to radial coordinate. Because general relativity has no torsion, mass-energy does not affect the two space coordinates perpendicular to the radial space coordinate.
light
Photons and massless particles move through space at light speed. Because all observers calculate constant light speed, observers calculate no space-time curvature along light-ray direction. However, curved space-time can make light rays move transversely to light-ray direction, so photon trajectories bend toward mass-energies.
congruency
Spaces with constant curvature allow congruent figures.
universe curvature
Riemann geometry models spherical, hyperbolic, and no-curvature (flat) space-times. If universe space-time has no overall curvature, universe average mass-energy density and local space-time curvature are everywhere the same. Average mass-energy increases as distance cubed. Space and time coordinate relations do not change.
Euclid's postulates apply to flat space. 1. Only one straight line goes through any two points. Unified space has no curvature. 2. Straight lines can extend indefinitely. Space is continuous and infinite. 3. Circles can be anywhere and have any radius. Space is continuous and infinite. 4. All right angles are equal. Figures can be congruent, and space is homogeneous and isotropic. 5. Two straight lines that intersect a line, so that interior angles add to less than pi, will intersect. Space has no curvature, and parallelograms can exist. Playfair's axiom is another way of stating the fifth postulate.
If universe space-time is hyperbolic {concave space-time}, universe average mass-energy increases more than distance cubed, and average mass-energy density increases with distance. Universe has a saddle-shaped surface, with constant negative curvature, on which geodesics have infinite numbers of parallels. Initially parallel motions and so geodesics diverge.
If universe space-time is spherical {convex space-time}, universe average mass-energy increases less than distance cubed, and average mass-energy density decreases with distance. Universe has a spherical-shaped surface, with constant positive curvature, on which geodesics converge. In spherical space-time, because universe is like a lens, objects halfway around universe appear focused at normal size, and objects one-quarter around spherical universe appear minimum size.
Elliptic geometry is for ellipsoids, including spheres, which have positive curvature and on which geodesics have no parallels. Initially parallel motions and so geodesics converge.
universe shape
Because space is homogeneous, universe shape must be completely symmetric. Possible symmetric shapes are Euclidean, torus, sphere, or hyperboloids. Because universe has mass and energy, it has space-time curvature. Infinite three-dimensional space can have zero curvature, with all three spatial dimensions equivalent. Three-dimensional torus has zero curvature with no boundary. Sphere has positive curvature. Hyperboloid has negative-curvature "saddle". Hyperbolic "torus" has negative curvature "saddle" with no boundary. Universe average mass-energy density determines overall universe shape.
infinite or finite universe
If space is infinite, as it expands, it stays infinite. If space is infinite, as it contracts, it becomes finite and changes shape.
If space is finite, as it expands, it stays finite. Expanding space changes average mass-energy density and changes universe shape. If space is finite, as it contracts, it stays finite. Contracting space changes average mass-energy density and changes universe shape.
universe maximum density at origin
Perhaps, universe started with maximum mass, minimum volume, and maximum mass-energy density.
expansion or contraction with no equilibrium
Even if gravity exactly balances universe space expansion, so space neither expands nor contracts at that time, space cannot stay in that state. Because particles always travel at light speed through space-time, system always has perturbations, and perturbations decrease or increase gravity and space expansion. Because decreased gravity makes more expansion and decreases gravity more, and increased gravity makes less expansion and increases gravity more, non-equilibrium states always continue to expand or contract. Therefore, universe must always expand or contract. There is no steady state or equilibrium point.
Star masses make universe gravitational field, which is an absolute reference frame for accelerated motion, including rotational motion. Water in spinning buckets is concave because it rotates with respect to universe, not with respect to bucket {bucket argument}.
Newton imagined a water bucket {bucket experiment} [1689]. On Earth, bucket hangs on a rope and spins. At first, bucket rotates, but water does not, and water surface is flat. Then water rotates, and water surface becomes concave. If bucket slows and stops, water first rotates faster than bucket but then becomes less concave, and then becomes flat. What will happen if bucket rotates in outer space? What will happen if bucket rotates in empty space?
Universe absolute curved space-time shape can be a 4-cylinder {hypercylinder}, with time as cylinder axis and space as cylinder three-dimensional cross-section.
Space-time surfaces and hypersurfaces have a path {geodesic} between two space-time points (events) that has shortest separation {space-time separation}. For no-curvature space-times (planes and hyperplanes), geodesics are straight lines. For no-curvature space-times, separation has shortest distance and shortest time. On spheres and saddles, shortest space distance between two points is great-circle arc. See Figure 1.
spheres
Convex, positive-curvature space-times include spherical surfaces, which have two dimensions, have centers, and have same constant curvature for both coordinates. Starting from nearby points, parallel geodesics converge. Geodesics have shortest-distance and longest-time trajectory.
saddles
Concave, negative-curvature space-times include saddle surfaces, which have two dimensions and have no center or two centers. Coordinates have constant opposite curvature. Starting from nearby points, parallel geodesics diverge. Geodesics have longest-distance and shortest-time trajectory.
geodesics
Experiments show that particles and objects always travel at light speed through space-time, along shortest-separation trajectory (geodesic) between two space-time points, whether or not matter and/or energy are present. Masses free fall along space-time geodesics. Observers and objects traveling along geodesics feel no tidal forces.
object mass
All objects and particles follow the same geodesics. Because inertial mass and gravitational mass are the same, object mass does not affect trajectory. Gravity is not a force but a space-time curvature field.
In a metric field with isometry, vector fields {Killing vector field} can preserve distances. In relativity, translations, rotations, and boosts preserve space-time separation.
Convex surfaces have two points {conjugate point} through which many geodesics have same distance, so geodesics are not unique. For example, Earth North Pole and South Pole have many equivalent geodesics (longitudes).
Curved space-time can have discontinuities {singularity, relativity}|, when geodesics are not continuous and/or points do not have neighborhoods. Those space-time events have no past or no future points, and so start or stop world-lines.
gravity
If gravity is high enough to prevent light from exiting a space region, space-time curvature becomes so great, with curvature radius equal Planck distance, that space closes on itself. The space region has a surface from which nothing can escape. As orthogonal light rays converge, spatial surface {trapped surface} has decreasing area. Space-time geodesics do not continue infinitely in space-time but stop at space boundary.
causes
Stellar and galactic-center collapse can make singularities, such as black holes.
Perhaps, Big Bang, white holes, Big Crunch, and/or black hole are space-like or light-like singularities. Perhaps, universe beginning was a singularity and began time. For black holes and Big Crunch, tidal distortions can be large. For Big Bang, at low entropy, tidal distortions (described by Weyl curvature tensor) are small. Perhaps, white holes violate the second thermodynamics law.
physical law
At space-time singularities, all physical laws break down, so field equations do not hold. Because space-time has high curvature, singularities violate CPT symmetry. Space-time-curvature radius is approximately Planck length, so space-time separations are approximately zero.
physical law: quantum mechanics
Quantum-mechanical-system states develop in unitary, deterministic, local, linear, and time-symmetric evolution in Hilbert configuration space. By Liouville's theorem, phase-space volumes are constant. However, "reduction of state vector" is asymmetric in time, and "collapse of wave function" adds phases and information, so phase-space volumes are not constant, and past and future have different boundary conditions, just as singularities have discontinuities between space-time pasts and futures. Quantum-mechanics measurements cause wave-function collapse.
Perhaps, quantum-mechanics measurements and wave-function collapse relate to general-relativity singularity space-time points and their formation. Perhaps, general relativity disrupts, or makes unstable superpositions of, quantum states and breaks equilibrium at measured states (objective reduction). General relativity has non-local negative-gravity potential energy and has positive-energy gravity waves, while state-vector-reduction time depends on inverse diameter and energy.
Singularities {naked singularity} can have high density but not enough gravity to form event horizons. Space-time paths that go through time can enter and leave naked singularities (but cannot leave other singularities). For example, spindle-shaped singularities have spindle ends that are naked singularities. Objects with spin faster than mass-determined rate are naked singularities. Objects with electric charge higher than mass-determined rate are naked singularities.
Perhaps, some or all singularities {thunderbolt} go to infinity and have no confinement, thus removing their space-time points from space-time.
General relativity is about gravity {gravitation, relativity}| and accelerations. For small gravity, observers calculate that gravitation and acceleration have the same local effects on space-time curvature. Because gravitational field strength varies inversely with distance, observers calculate that gravitation and acceleration have different global effects on space-time curvature.
time
Because stationary observers calculate that gravity rotates space-time time and radial-space coordinates toward each other, clocks in gravitational fields, or undergoing accelerations, run slower. People age slightly more quickly on Moon than on Earth, because Moon has smaller gravitational field. People age more slowly on accelerating rockets than on Earth.
object length
Observers calculate that accelerating massive objects decrease length. After accelerating finishes, observers calculate that length returns to previous amount.
object mass
Observers calculate that accelerating massive objects increase mass. However, mass increase increases inertia and resists further acceleration. After accelerating finishes, observers calculate that mass returns to previous amount.
energy
Gravity depends on mass, directed potential energy, directed kinetic energy, and random-energy temperature. Mass and random energy are always positive. Gravity fields cannot cancel, because they are only positive. Because gravity is infinite and is only positive, gravity can have unlimited energy amounts.
sources
General-relativity stress-energy tensor has ten independent gravitational-field sources and ten independent internal-stress sources. Sources all conserve energy and momenta. Field equations are d'Alembert potential equations.
physical-law invariance
Gravitation and acceleration curve space-time, so non-locally physical laws vary under coordinate transformations.
uncertainty principle
Gravitational-field values correspond to position. Field-value-change rates correspond to momenta. Therefore, uncertainty principle applies to gravitational-field values and value-change rates.
black holes
Because gravity is unlimited, gravity can become strong enough to overcome all object accelerations, so even light cannot escape the space region. Outgoing geodesics converge. Space curves so much that it closes on itself, forming a region separate from space-time, not observable from outside. Only gravity can cause space-time singularities, because it is never negative.
gravitational entropy
Spaces have entropy that depends on topology (Euler number). Gravity curves space-time and creates different topologies, so gravity has entropy. Because only gravity is always positive, only gravity has entropy. Other forces cannot curve space-time, because they are not infinite and/or are both positive and negative.
gravitational entropy: black hole
Because gravity has entropy and forms black holes, black holes trap entropy. Black-hole trapping amount depends on event-horizon radius, so black-hole entropy depends on event-horizon spatial area.
Because black holes have entropy, they have surface temperature at event horizon. At event horizon, virtual-particle creation can allow one virtual-pair member to tunnel through event horizon to space, causing black hole to lose matter and eventually dissipate. Entropy decreases, rather than always increasing. Black holes disrupt quantum-state deterministic development and mix states {mixed quantum state}.
repulsion
Perhaps, gravity can temporarily repulse, and cause universe origin. Exotic particles can have negative pressure, causing repulsion. Larger spaces have more repulsion because pressure is in space, not in ordinary particles.
Objects with mass have gravitational forces {gravitational pressure} on top, bottom, middle, and sides, all pointing toward mass center. See Figure 1.
Imagine that object is fluid. Gravitation pulls all points straight down toward mass center. Pull is directly proportional to mass m and inversely proportional to distance r squared: m / r^2. Pull is least at farthest points least and most at nearest points.
Volume reduction changes mass density and energy density, and so changes pressure. Gravity tends to reduce volume, and increase pressure, until outward pressure force per area balances inward gravitation force per area.
Gravitational-field accelerations make waves {gravity wave}| {gravitational wave}. Gravitational waves make space-time curvature oscillate in two dimensions.
speed
Gravitational waves travel at light speed.
frequency
Gravitational-wave frequencies are about 1000 Hz.
medium
Gravitational waves oscillate gravitational-field surfaces. Gravity waves need no other medium.
quadrupoles
Gravity waves have two orthogonal linear-polarization states, at 45-degree angle, making field surfaces (not just lines). Gravity waves are quadrupole radiation. Because mass can only be positive (unlike electromagnetic positive and negative charges), no mass-dipole or gravitational-dipole radiation can exist.
At peaks, potential energy is maximum, and kinetic energy is minimum. As they pass, gravitational waves stretch and compress (vibrate) objects with mass.
spin
Gravitational waves can rotate. Primordial gravitational waves have different spin {polarization, gravity} than current ones.
graviton
Gravitational-force exchange particles are gravitons and have no mass. Gravitons have spin 2, which is invariant under 180-degree rotation around motion direction.
sources
Gravity waves come from oscillating and/or accelerating masses, such as pulsating stars, irregularly rotating stars, collapsing stars, exploding stars, or interacting star clusters.
superposition
Because masses are always positive, gravitational fields cannot cancel each other. However, locally, accelerations and/or decelerations can cancel gravitational fields. Because gravitational waves are non-local and have components in more than one direction, and accelerations are in only one direction, accelerations and/or decelerations cannot cancel gravitational waves.
comparison with electromagnetic waves
Gravitational fields have advanced and retarded solutions and their equations are similar to those for electromagnetic waves.
renormalization
Gravitational waves are infinite and require renormalization for gravitational-wave calculations.
Pressure measures momentum exchange. System external pressure puts force per area on system-boundary surfaces. It is due to kinetic energy, which increases with temperature.
internal pressure
System internal pressure {internal pressure}| puts force per area on system particles. It measures system potential energy changes as system expands or contracts while keeping temperature constant. Internal pressure is positive for attractive forces and negative for repulsive forces.
Vacuum has no forces, so its internal pressure is zero. Particles have no internal forces, so their internal pressure is zero. Solids have attractive forces, but particle distances do not change at constant temperature, so internal pressure is zero.
positive internal pressure
Gas particles slightly attract, and system volume can change at constant temperature, so particle distances can change at constant temperature, and gases can have positive internal pressure. Hotter gases push particles farther apart against attractive forces, increasing positive potential energy, so hotter gases have more internal pressure than cooler gases. Photons have radiation pressure that pushes against electromagnetic forces, increasing positive potential energy, so photon "gases" have positive internal pressure.
negative internal pressure
Systems that have internal repulsive (negative) forces have negative potential energy and negative internal pressure. For example, if external force compresses rubber membranes, rubber has repulsive forces that tend to push particles apart. The internal restoring force is negative, so internal potential energy is negative, with negative internal pressure.
gravity
At space-time points, gravity G depends on mass-energy density M and on internal pressure P: G ~ M + 3 * P. Hotter gas has more positive internal pressure than cooler gas and so more positive gravity. Photon "gas" has positive internal pressure that is one-third of energy density, so gravity doubles: M + 3 * (M/3) = 2 * M.
Quantum vacuum has negative (repulsive) force that expands space, increasing negative potential energy (dark energy) by subtracting universe positive kinetic energy, and so cooling the universe. Quantum vacuum has negative internal pressure between one-third and one of mass-energy density, so repulsive antigravity is between zero and negative two times mass-energy density: M + 3 * -(M/3) = 0 and M + 3 * (-M) = -2*M.
Gravitational fields have different strengths at different distances from mass-energy. In gravitational fields, objects have different forces {tidal force} on side nearest to mass-energy, side farthest from mass-energy, and middle. Tidal distortions depend on gravitational-field strengths at different space points.
Gravity varies inversely with distance squared {inverse square law}, so tidal effects vary inversely with distance cubed (by integration). Therefore, tidal effects can measure gravitational-field strength.
See Figure 1. The larger object is denser and has much more mass than smaller object. The smaller object is fluid. The objects are not far apart.
near and far
Gravitation pulls smaller-object nearer side, farther side, and middle straight toward larger-mass center. Nearer side feels strongest gravity, and its particles accelerate most. Middle feels intermediate gravity, and its particles accelerate intermediate amount. Farther side feels weakest gravity, and its particles accelerate least. Along vertical, small object tends to stretch out from middle, keeping same volume.
left and right
Gravitation pulls left and right sides toward larger-mass center diagonally, straight down along vertical component and across inward along horizontal component. Left and right sides feel slightly less gravity than middle, because they are slightly farther away from larger-mass center. Those particles accelerate downward slightly less than middle does. Left and right sides also accelerate small amount horizontally toward smaller-mass center. This pushes other molecules equally up and down and contributes to vertical stretching out.
waves
Changing gravity changes tidal forces and can cause mass oscillations. Mass accelerations make gravitational waves.
Rotating objects with mass pull space-time around {frame dragging}| {Lense-Thirring effect} {gravitomagnetism}. An analogy is rotating masses drag viscous fluid around them. For particles orbiting around rotating masses, relativity causes orbit-plane precession, because rotation and angular momentum couple.
Objects can move forward and backward in space, and physical laws have no preferred space direction. Objects cannot move forward and backward in time, though physical laws have no preferred time direction. In space-time, can objects move forward and backward in time {time travel}|?
If time travel is possible, people can deduce what has happened from knowledge of the future. Past-time observation affects past. Because contradictions violate causation, nothing can communicate or transport backward through time. The meaning of space and movement prevents moving forward and backward in time. Relative velocity is moving through space over time. Movement is always in space-time spatial dimension. Moving forward and backward in time cannot separate from moving forward and backward in space.
Space paths must not reverse time, so nothing can happen at two times. Between a past point and future points reachable from the past point, along geodesic, all space-time points must be reachable {hyperbolicity} {hyperbolic space-time, relativity}. If geodesics exist, space-time has no singularity.
One twin stays on Earth. The other twin takes a high-speed trip, traveling to a space point and then back to Earth. Second twin must accelerate to leave Earth and travel in space, must accelerate to round point in space, and must decelerate to land on Earth. Traveling twin's clocks appear to run slower to Earth observer. Second twin is younger than first twin on return to Earth. Traveling twin ages more slowly than Earth twin {twin paradox}|.
length contraction
Traveling at almost light speed, people can cross universe in 86 years of their time, because universe lengths contract greatly. People on Earth age 13 billion years during that time.
space-time graph
Space-time graphs {Minkowski diagram} can show travel effects. The diagram assumes first twin is stationary and is observer. First twin has vertical world-line on space-time graph. Second twin has angle to right, away from Earth as twin leaves Earth, and angle to left, toward Earth as twin returns to Earth.
At beginning, twin accelerates to leave Earth and has curved world-line, with greater angles to time axis. At turning point in space, twin changes direction and has curved world-line, with lesser angles to time axis, reaches vertical, then has curved world-line, with greater angles to time axis. At landing, twin decelerates to stop on Earth and has curved world-line, with lesser angles to time axis. See Figure 1.
space-time trajectory
The shortest path is the longest time. Traveling twin has longer path and shorter time.
universe
If second twin is observer, twin on Earth travels, relative to second twin, with same motions and accelerations as described above. However, first twin does not undergo acceleration relative to universe masses, as second twin does. To second twin, universe masses have same speeds and accelerations as first twin. During acceleration relative to universe masses, time slows, because mass curves space-time. Curved space-time makes longer path and shorter time.
Permanent aging happens only during accelerations and decelerations. Uniform-velocity time dilations are symmetric between observers, are momentary, and are reversible.
Masses have gravitational fields, which have energy, and energy has mass and makes gravitational fields. Masses interact with their gravitational fields to make energy {self-energy}| {matter field}. Because mass is only positive, gravity has interaction energy greater than zero.
renormalization
Mathematical renormalization adjusts values to prevent infinities.
electromagnetism
Perhaps, if charge moves (and external electric field is zero), charge gains velocity by interacting with its electric field, because energy in point-charge field is infinite (by Maxwell's equations or quantum electrodynamics). However, because positive and negative charges can induce each other, electromagnetism has no interaction energy.
Wheeler-Feynman theory
In Wheeler-Feynman theory, universe particles absorb moving-charge electric field, so field at large distances is zero, and system has no advanced solutions and no infinities. However, this theory is not correct.
perfect absorption
In perfect absorption, electric field is relativistically invariant, so all force-induced fields, including reaction forces, form other particles using photon exchanges and go to zero. Perfect absorption has only retarded solutions, because advanced solutions are improbable by thermodynamic laws. In expanding universes, absorption happens at low frequency for retarded solutions and at high frequency for advanced solutions. However, this theory is not correct.
Time-like space-time paths have points where space-time curvature and path curvature are not the same, and net gravity is zero {general energy condition}.
For classical matter, energy density is greater than or equal to zero in all reference frames {weak energy condition}. However, weak energy condition is false for quantum-mechanical scales.
For classical matter for long enough distances, energy density is greater than or equal to zero for all time-like paths {strong energy condition}. However, strong energy condition is false for quantum-mechanical scales.
Geodesics have space-time separation {geodesic deviation} along (straight) line perpendicular to geodesics. An equation {equation of geodesic deviation} calculates separation: (D^2)r / Ds^2 + G * r, where D^2 is second partial derivative, r is curvature radius, D is first partial derivative, s is space coordinate, and G is Gaussian curvature. In empty space-time, geodesics are parallel straight lines. Empty space-time has no curvature, so r is zero, and geodesic deviation is zero.
Geodesics converge along tangent vector to hypersurface path. Geodesic-convergence rate relates to shear and gravitation {Newman-Penrose equation} {Raychaudhuri equation}.
In special and general relativity, field equations {d'Alembert equation, relativity} describe how masses, their gravitational fields, and space-time gravitation potentials determine object motions.
Similar field equations describe how charges, their electrostatic fields, and space-time electrostatic potentials determine object motions. Such equations {electrodynamics} are similar to curved-space-time special-relativity equations.
Equations {Einstein field equation} describe how mass-energy affects space-time geometry, and how space-time geometry affects mass-energy motions. Local-space-time average curvature tensor G {Einstein tensor} is proportional to mass-energy tensor T {stress-energy tensor}: G = 8 * pi * T.
Einstein tensor has six components for tide-producing acceleration: particle position, particle velocity, field amplitude, field-change rate, geometry, and geometry-change rate. Einstein tensor has four components for space-time coordinates.
Stress-energy tensor has components for stresses, momentum densities, and mass-energy density.
Einstein tensor G relates to local-space-time curvature tensor R (Riemann curvature tensor): G = R - gamma * R/2. Stress-energy tensor T relates to Riemann curvature tensor R: R - gamma * R/2 = 8 * pi * T. Riemann-curvature tensor has 20 components. In empty space-time, stress-energy-tensor gradient is zero, so Einstein-tensor gradient equals zero, and Riemann curvature tensor is zero.
Surfaces have Gaussian curvature. Tensors {Riemann curvature tensor} represent space-time curvature using geodesic separation. Riemann curvature tensor represents total curvature. It adds tidal distortions (Weyl curvature tensor) and volume changes (Ricci curvature tensor).
Two-dimensional space requires one curvature component, curvature radius. Three-dimensional space requires six curvature components, three for each dimension's curvature and three for how dimensions curve in relation to each other. Four-dimensional space requires 20 curvature components, four for each dimension's curvature, twelve for how pairs of dimensions curve in relation to each other, and four for how triples of dimensions curve in relation to each other.
invariance
Curvature is invariant over linear space-time-coordinate transformations.
electromagnetism
Like gravity, electromagnetism exerts force that decreases with distance squared {Lorentz force equation}. Lorentz force equation and Riemann curvature tensor are equivalent. At low velocity, because relativistic effects are negligible, only the nine Lorentz-equation electric-field components, and the corresponding Riemann-curvature-tensor mass components, are significant.
Curvature tensors {Ricci curvature tensor} can describe space volume changes, which is local curvature caused by local matter.
Perhaps, at one second after universe origin, thermal variations in Ricci curvature tensor formed particles and black holes.
Curvature tensors {Weyl curvature tensor} can describe tidal distortions, which is non-local curvature caused by non-local matter.
At Big Bang, quantum fluctuations and damping cause small variations. At Big Crunch, variations have no damping and can be large. Perhaps, this asymmetry causes time to have direction. Alternatively, past and future singularities can be different.
Space-time theories {Kaluza-Klein theory} can use four space dimensions and one time dimension. Fourth space dimension is only several Planck lengths long, has curvature so high that it makes a circle, and is unobservable. A small fourth space dimension allows the vacuum to have higher energy density than three space dimensions have.
Time relativistic theories {kinematic relativity theory} describe finite expanding universes.
Gravitation theories can use metrics {metric theory} or be non-metric. A ten-parameter general metric theory {parametrized post-Newtonian formalism} can model all metric gravitation theories, which then differ only in parameter values.
parameters
The ten parameters model: How mass causes space curvature. How gravity-field superposition is non-linear. If space has preferred reference frame, or all spatial directions are equivalent. If all four space-time components have momentum conservation. If distant galaxies affect local interactions. If general metric theory does or does not include gravitational-radiation effects or other gravity-strength changes.
types
Metric theories include general theory of relativity, scalar-tensor theories, vector-tensor theories, tensor-tensor theories, conformally flat theories, stratified theories, and quasi-linear theories.
non-metric
Non-metric gravitation theories violate completeness, consistency, relativity, and/or Newtonian limit.
Abstract spaces {superspace} can have approximate three-dimensional space by tetrahedron skeletons and have tetrahedral edge lengths. They can have space dynamics, change over time, and represent different geometries.
Riemann surfaces are Riemann sphere, torus, and pretzel-shaped surface. Their angles are the same as in Euclidean space. Riemann surfaces can define field theories {conformal field theory} that pair with string theory.
General-relativity dynamics {geometrodynamics}| is three-dimensional Riemann-space dynamics, using a method {ADM formalism} {canonical quantization} developed by Paul Dirac and later Richard Arnowitt, Stanley Deser, and Charles Misner.
Geometric optics {geometric optics}| models plane waves in flat space-time. Geometric optics applies if wave-packet wavelengths are much less than wave-front space-time curvature radius. Wave photons have same momentum and polarization. Photon number determines ray amplitude. Like adiabatic flow, photon number conserves. Light rays are null geodesics. Polarization vector is perpendicular to rays and propagates along rays.
Quantum general-relativity gravitation theories {relational quantum theory} have different observers whose calculations are the same at corresponding space-time points.
In one renormalization, electric field is relativistically invariant, so all force-induced fields, including reaction forces, form other particles using photon exchanges and go to zero {perfect absorption}. Perfect absorption has only retarded solutions, because advanced solutions are improbable by thermodynamic laws. In expanding universes, absorption happens at low frequency for retarded solutions and at high frequency for advanced solutions. However, this theory is not correct.
In a renormalization theory {Wheeler-Feynman theory}, universe particles absorb moving-charge electric field, so field at large distances is zero, and system has no advanced solutions and no infinities. However, this theory is not correct.
If general relativity has canonical quantization, Wheeler-DeWitt equation has no time coordinate {frozen time problem} {problem of frozen time} {problem of time} {time problem}.
In empty space, space-time can have many equivalent reference frames {covariance, relationalism} {covariance, relativity} {general covariance, relationalism}.
In relativity theories {relationalism}, mass-energy determines space-time curvature and shape, and space and time are not absolute or real but differ for different observers.
Perhaps, space and time are real and absolute {substantivalism, relativity}, and mass-energy alone does not determine space-time curvature and shape.
Vectors can equal another vector plus a scalar term {gauge, relativity}|. Scalar gauges can change with position. For example, space-time curvature can change with position, and gauges can represent linear curvature changes with position.
Using linear transformations {gauge transformation}, gauges can relate vectors expressed in different coordinate systems. Gravitation, electromagnetism, and chromodynamics use gauge transformations to model infinitesimal, finite, scalar-coordinate transformations. For local space-time regions, general relativity is invariant under finite coordinate transformations, and a generalized gauge transformation represents general relativity. Using gauge scalars can simplify differential equations.
Because derivatives of scalars equal zero, gauge changes do not affect physical measurements, motion differential equations do not change, and gauge transformations preserve invariants.
In gravitational fields so weak that space-time has negligible curvature, gravity does not move gravitational-field-source masses and does no work on them, so masses have no self-energy. For this case, theories {linearized theory of gravity} represent space-time-coordinate changes as infinitesimal gauge changes, which change space-time-metric coefficients.
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Date Modified: 2022.0225