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