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.
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Date Modified: 2022.0225