LightClock

What is a Light Clock?
Following Einstein we define time to be what is measured with a clock, and the ideal time measurement device is a Light Clock.
A Light Clock uses a light pulse sent from one location to a mirror at another location, which reflects the pulse back. The distance to the mirror is held constant.
The completion of each roundtrip increments the clock by one 'tick' and initiates the next cycle. Counting only completed roundtrips complies with the "all physics is local" rule.
Whichever observer you choose to be, the tick count of your clock is in the bottom right corner of the window; it shows your proper time progressing at a constant rate unaffected by the Lorentz factor (γ).
The animation uses four mirrors for the Light Clock so the length contraction in the direction of motion can be seen clearly. The tick count (proper time) of the Light Clock is centered between the mirrors.
The Light Clock is always stationary in the Rocket, Shell, and Far frames. For non-zero values of v, the Light Clock is in motion relative to the Lab and Rain frames.
Tip: Click the t time button, then click the 'mini light clocks' button to see how light clocks look in the Lab or Rain frames.

What is the scale of the animation?
The scale is assumed to be appropriate for the animation to depict an inertial reference frame (flat spacetime).
For the Lab and Rocket animations of Special Relativity the scale can be from the sub-microscopic to the very large, but smaller than cosmological distances where expanding space would be a factor.
For the Rain, Shell and Far animations of General Relativity the scale is small enough for gravitational effects (e.g. tidal distortions) to be insignificant.
Lengths are expressed as dimensionless fractions of the proper distance D between the mirrors, with no spatial scale specified.
Similarly, time durations are expressed as dimensionless fractions of the proper period T, with no time scale specified.
Tip:Note that times are shown as decimal fractions of the clock period (e.g., t/T=0.750 has the S clock hand in the "9 o'clock" position).

What do the colors signify?
Colors are used to indicate different reference frames.
Lime represents the S inertial reference frame of a stationary observer (laboratory frame) in Special Relativity.
S is also the frame of a Free-Falling observer in General Relativity (local rain frame).
Cyan represents the S' inertial reference frame of the moving unaccelerated Light Clock (rocket frame) in Special Relativity.
S' is also the frame of a stationary fiducial observer in General Relativity (local shell frame).
Orange represents the S" reference frame of a far-away bookkeeper in General Relativity (global Schwarzschild frame).
S" is an idealized global coordinate system which extends far-away flat spacetime to radius r as if there where no gravitating mass M.
Note: mass M in Schwarzschild geometry is assumed to be a non-rotating, uncharged, point mass.

You can select whether the animation shows Special or General Relativity, and which reference frame is yours:

S	Lab	Laboratory Frame	Special Relativity	You are a local observer with the Light Clock moving to the right (left) at velocity v (-v) through motionless space.
S'	Rocket	Rocket Frame	Special Relativity	You are a local observer with the Light Clock stationary in your frame; space is motionless.
S	Rain	Rain Frame	General Relativity	You are a local observer free-falling past the Light Clock from far away with zero initial velocity. The Light Clock is at a constant distance r = \|Δx\| from mass M as determined by a far away bookkeeper. You are comoving with space toward mass M at speed [2GM/r]^1/2 relative to the Shell frame.
S'	Shell	Shell Frame	General Relativity	You are a local observer with the Light Clock stationary in your frame. The Light Clock is at a constant distance r = \|Δx\| from mass M as determined by a far away bookkeeper. Space is moving past you toward mass M at speed [2GM/r]^1/2 relative to your frame.
S"	Far	Far-Away Frame	General Relativity	You are a global bookkeeper far away from mass M (your location is at r = ∞). You determine the Light Clock to be a constant distance r = \|Δx\| from mass M which is at r = 0. You are unaware (or choose to ignore) that space is moving toward mass M at speed γ^-2[2GM/r]^1/2 relative to your frame.

Local reference frames with the same color have the same relativistic effects (Lab = Rain, Rocket = Shell). In special relativity the effects are due to relative motion through space (motionless space, flat spacetime),
while in general relativity the effects are due to relative motion of space (moving space, warped spacetime). This is how the Special and General Theories are linked.
Tip: Toggle between the Lab frame and the Rain frame to see that there is no difference in space or time effects. The same is true for toggling between the Rocket frame and the Shell frame.
The Far frame is not like any of the other frames; the extra space and time distortions (spacetime warping) you see are due to not accounting for the motion of space.

Why does the Light Clock stay contracted and its time dilated when 'Center' and Lab are selected?
Centering the Light Clock with Lab selected automatically drags the scene to the left (right) at constant speed -v (+v), with the Light Clock artificially held in the center of the window.
It is an animation trick for the convenience of you the Lab observer. It is not the same as viewing the Light Clock in its own rest frame.
From the point of view of the Light Clock's Rocket frame you are moving and it is your lengths and clocks that are distorted.
With 'Center' selected you are not moving at the same speed as the Rocket frame; it just looks that way. By definition, you and your space coordinate grid have no relative motion, despite how it appears when the Light Clock is centered.
Centering the Light Clock with Shell selected accurately depicts the stationary Light Clock in General Relativity. Check xy to see the length contracted Rain frame moving downward toward M.
Tip: Set v to a high positive value. Select Lab and 'Center'. Check the xy space grid to see it being dragged to the left. You can tell the scene is artificial because the grid is not contracted in the x direction.
Select 'Uncenter' to see a more intuitive depiction of the physics, without the illusion that your Lab frame is in motion.

Is the Lab frame always stationary and the Rocket frame always in motion?
No, either frame can be stationary or in motion, as required by the Principle of Relativity.
Perhaps the traditional Special Relativity terms Lab and Rocket are misleading. Lab implies stationary and Rocket implies motion.
Terms from an earlier era like Station Platform and Railroad Car had the same problem.
Also, Rocket could imply acceleration if the engines are firing, which in this animation they never are.
Tip: Consider if alternative names might be used that are neutral as to which frame is moving. Perhaps Green and Blue, or Alice and Bob.

How are the spatial coordinate axes oriented in the animation?
The x, x', and x" axes are collinear and horizontal, increasing to the right. The y, y', and y" axes are vertical, increasing upward. The positive z, z', and z" axes extend out of the screen toward you. At time t = 0 the origin of the xyz coordinate system is at the center of the light clock. As time changes the xyz system moves relative to the x'y'z' and x"y"z" coordinate systems which have their origins always at the center of the light clock. Only the xy, x'y', and x"y" planes (z = z' = z" = 0) are shown in the animation so each circular wave is a 2D slice through the center of a 3D spherical wavefront. Lorentz contraction is limited to the x-axes, because all motion is in the x direction. For the General Relativity cases the gravitating mass M is on the x-axis, at a fixed distance r from the stationary light clock, as measured in the Far frame (r is the radial coordinate of spherical Schwarschild geometry).
Tip: Select a GR frame and turn on the xy, x'y', and x"y" grids. Set v to a non-zero value. Notice the xy grid moving relative to the x'y' and x"y" grids. The horizontal grid lines stay colinear while the vertical grid lines for the three frames have different spacing.

What is Time Dilation?
Time Dilation is the lengthening of time durations of a moving clock, as measured by a stationary clock.
A clock measures time differences between two events at the same position within a reference frame: S clock has Δx = 0, S' clock has Δx' = 0
The slower ticking S' moving clock measures less elapsed time Δt' than the elapsed time Δt of the stationary clock.
The ratio of measured durations between the two frames is the Lorentz factor: Δt = γΔt'
Tip: Select the Lab reference frame. Set γ to 2 and compare clock ticks. The S' clock runs two times slower than your S clock, or equivalently, your S clock runs twice as fast as the S' clock.

What is Lorentz Contraction?
Lorentz Contraction is the shortening of the length of a moving object in the direction of its motion as measured by a stationary observer,
which requires the distance between two positions be measured at the same time by the stationary observer: S length Δ x has Δt = 0.
The ratio of measured lengths between the two frames is the Lorentz factor: Δx' = γ(Δx-vΔt)=γΔx.
In this animation two pairs of mirrors are used to demonstrate that length contraction happens only in the direction of motion, not perpendicular to the motion.
Shortened separation between mirrors in the direction of motion is necessary to keep the clock ticking by ensuring all light pulses return to the source at the same time.
Tip: Pause the animation to see a snapshot with all events having the same time in your reference frame, e.g. in the S frame (Δt = 0 for all event pairs).

What is a Spacetime Interval?
The Spacetime Interval is defined to be the combination of space and time coordinates for two events such that same value results in all inertial reference frames:

Invariant Spacetime Interval in SR
≡ (Δx)² + (Δy)² + (Δz)² - (cΔt)² = (Δx')² + (Δy')² + (Δz')² - (cΔt')²
Invariant Spacetime Interval in GR (small spacetime region)
ds² ≡ (dx)² + (dy)² + (dz)² - (dt)² = (dx')² + (dy')² + (dz')² - (dt')² = (γdx")² + (dy")² + (dz")² - (cdt"/γ)²

The Lorentz factor γ in the dx" and dt" terms represent the warping of spacetime (flowing space).
For a sufficiently small spacetime region the Spacetime Intervals of SR and GR are identical (Δx = dx, ...)

Types of Spacetime Intervals: negative is "timelike", zero is "lightlike", positive is "spacelike".
Relative motion in only the x, x' or x" direction means the Interval is (Δx)² - (cΔt)² = (Δx')² - (cΔt')² = (γΔx")² - (cΔt"/γ)² because Δy = Δy' = Δy" and Δz = Δz' = Δz".
All events in each Lab or Rain animation frame have the same t, therefore the Interval is (Δx)² = (Δx')² - (cΔt')² = (γΔx")² - (cΔt"/γ)² for all event pairs in an animation frame.
All events in each Rocket or Shell animation frame have the same t', therefore the Interval is (Δx)² - (cΔt)² = (Δx')² = (γΔx")² - (cΔt"/γ)² for all event pairs in an animation frame.
All events in each Far animation frame have the same t", therefore the Interval is (Δx)² - (cΔt)² = (Δx')² - (cΔt')² = (Δx")² for all event pairs in an animation frame.
Tip: Check Retarded Waves. Pause and Reset the animation. Check Events and Set Event A at the center of the Light Clock. Step the animation forward a few steps to expand the wave without reflection.
Set Event B on the wave and notice the Interval is lightlike (= 0). Drag Event B inside the wave for timelike (< 0), or outside the wave for spacelike (> 0).

What is Relativity of Simultaneity?
By definition an observer's simultaneous events happen at the same time, so they have the same time coordinate: Δt = 0 within S, Δt' = 0 within S', Δt" = 0 within S".
However, two reference frames that are moving relative to each another will generally assign different times to two events, hence simultaneity is relative to the reference frame.
The only case where both frames assign the same time is when the relative motion is perpendicular to the line connecting the locations of the two events.
Consider, for example, frames S and S' with two events that have times Δt = Δt' = 0, therefore Δx² = Δx'². With Δx = Δx'/γ, this means γ = 1 or Δx = Δx' = 0.
If γ is not 1 and two S' clocks have non-zero Δx' separation then they do not measure the same time as observed by S.
Note that all events depicted in an animation frame are simultaneous from the selected observer's perspective, all the observer's clocks within an animation frame are synchronized with each other.
Tip: Select the Lab frame. Set γ to 2 and show t' time. Notice the clocks are only synchronized where Δx = Δx' = 0. Check the t time to see that all the S clocks are synchronized with each other.

What is Relativistic Causality?
If event A is said to cause event B, then relativity requires B to be on or inside the future light cone of A (A must lie on or inside the past light cone of B).
Event A can cause event B (A and B are said to be causally connected) only if a signal can pass from A to B at a speed less than or equal to the speed of light.
Two causally connected (timelike or lightlike separated) events always appear in the same time order as viewed from any frame of reference.
Two causally disconnected (spacelike separated) events, however, can appear in any time order by choosing to view them from different frames of reference. [For three or more events see this reference].
Tip: Select the Lab frame. Set γ to 2. Check Retarded Waves. Pause and Reset the animation. Check Events and Set Event A at the center of the Light Clock. Event A has Lab frame time t = 0.
Step the animation forward a few steps to advance the time. Set Event B somewhere inside the circular wavefront. Event B has Lab frame time t > 0. Event A always precedes Event B in the Lab frame.
Drag Event B to other positions inside the wavefront (timelike separation) and notice how its time changes in the Rocket frame, with Event B always later than Event A in that frame.
Drag Event B to positions outside the wavefront (spacelike separation) and notice that Event B's spatial position determines whether its Rocket frame time is earlier than, simultaneous with, or later than Event A's Rocket frame time.

Why do waves and photons reflect off a moving mirror differently than off a stationary mirror?
The apparent angle of a rotated mirror is due to length contraction along the direction of motion.
The waves obey the Huygens Principle with each wavelet centered on a different spot in space due to the mirror's motion.
The photons obey the principle of least time. [see references]
Tip: Set gamma to 2, Pause, and Reset. Check Retarded Waves and Full Arcs. Check Events and Set Event A at center of Light Clock, Resume.
Pause when the center of the Full Arc of the reflected wave for the right mirror first appears. Set Event B there. Notice that Events A and B have the same t' time.

Why are the light waves elliptical when the Far frame option is selected?
Far depicts how a far-away bookkeeper represents the events in the vicinity of the light clock. It is not what the bookkeeper observes directly; he is not a local observer of those events.
The free-falling observer (Rain frame) and the stationary observer (Shell) frame are local observers, therefore they see events consistent with special relativity.
The Far distortions are the result of extrapolating from far-away flat spacetime to the vicinity of the light clock, without compensationg for the effects of the downward flowing space.
These distortions (apparent spacetime warping) include contracted radial lengths (Δx" = Δx = Δx'/γ) and slowed clocks (Δt" = γ²Δt = γΔt'). This explains the Far view of Shell frame length contractions (e.g. elliptical appearence of the light waves, contracted appearance of the Light Clock even though it is motionless in the Far frame), time dilations (e.g. slowed clocks), and the extra slow inward speed (v/γ²) of the free-falling Rain frame.
Tip: Set the speed to zero to see the distortions disappear: same effects are seen by the far-away bookkeeper, free-falling observer, and stationary shell observer.
This is because zero relative speed between the stationary shell observer and the free-falling observer happens only at r → ∞ (or M=0), which means in this special case the far-away bookkeeper is a local observer.

What is an Inertial Reference Frame?
An Inertial Reference Frame is defined as a frame of reference in which Newton's first law holds (constant motion with no forces). An inertial observer does not detect acceleration, or forces, or gravity.
We represent an inertial frame as a uniform spatial coordinate grid with identical synchronized clocks at each position in the grid, to locally measure the position and time of spacetime events. Clocks are synchronized using Einstein's method of round-trip light signals that are assumed to travel at the same speed c in all directions (light travel time A-to-B is the same as B-to-A).
Check any of the xy x'y' x"y" space grids to see the spatial coordinates. Check any of the t t' t" times to see the time coordinates depicted as clocks.
Tip: Watch the four photons travel round-trip from S' clock tick to the next tick. Notice the mirror reflection events are all at half-tick t' times, even with the mirrors rotated.

Is the Shell frame an Inertial Reference Frame?
Locally, Yes. Globally, No.

Yes: If the scale of the spacetime region is sufficiently small to exclude gravitational effects (see next FAQ) then the Shell frame is "locally inertial", but not "globally inertial". The animation does not depict space as accelerating toward the gravitating mass; it assumes the radial extent of the Shell frame is small enough that the speed of downward moving space is constant across the window width. The Shell frame is depicted as a stationary inertial frame while the inertial Rain frame (comoving with space) moves downward. Effectively, the spacetime region has been magnified enough to appear flat and only Special Relativity applies.

No: If the scale of the spacetime region is not sufficiently small then gravitational effects arise and the Shell frame is neither "locally inertial" nor "globally inertial". Globally, a Shell frame must be accelerated to maintain a fixed position at a constant Schwarzschild r-coordinate. If accelerations were included then gravity would appear, because space accelerating downward past the Shell observer is equivalent to the Shell observer accelerating upward through space, which Einstein tells us is equivalent to the Shell observer being motionless in a gravitational field.
Tip: Set v to zero and select the Rain frame. Notice how the Light Clock, the waves, the xy spatial grid, and the t clocks look inertial. Change v to a non-zero value and select the Shell frame to see the same (locally) inertial view. The vertical x'=0 line through the center of the Light Clock is correct. The "locally inertial" approximations become less correct as the distance from x'=0 increases.

Where are the gravitational effects of General Relativity?
For simplicity, the animation is limited to only the kinematic effects of Special and General Relativity. This means no accelerations are depicted. The Rocket frame moves at constant speed relative to the Lab frame, and the Rain frame falls at constant speed relative to the Shell and Far frames. Without accelerations we can focus on the effects of radial length contraction and time dilation which are identical for both Special and General Relativity. In the GR cases we see the effects of space moving radially downward toward the gravitating mass, but the scale is assumed to be small enough so only the SR effects are significant.

At sufficiently small scale (as depicted in the animation) GR effects are neglible:
a) the region of spacetime has negligible curvature (locally flat spacetime)
b) the accelerating downward flow of the Rain frame past the Shell frame is neglible (no increasing speed as it falls)
c) the radial stretching and transverse shrinking of falling objects is neglible (no tidal distortions)
d) the spherical symmetry of the spatial grid is not noticeable (uniform rectangular spatial grid is sufficient)
e) the gradient of radial flow velocity and its effect on light frequency is negligible (no gravitational redshift or blueshift)
f) the combined time and space gradient effects are neglible (no bending of light or perihelion advance)
g) the expansion of the universe has virtually no effect (no cosmological redshift)
See this video and this web site for animations that include accelerating downward flow of space.
Tip: To compare relativistic effects at multiple radial distances from M, open two (or more) windows and position them to be vertically adjacent. Start an animation in each, select Rain frame and t time in each window. Set rc²/GM to progressively lower values from rightmost to leftmost window (v > 0). Notice radial length contraction and time dilation effects are greater closer to M. This gradient is equivalent to gravity.

What are Retarded and Advanced waves?
A Retarded wave propagates forward in time, producing effects which are retarded from (later than) the originating event at the wave source. An Advanced wave propagates backward in time, producing effects which are in advance of (earlier than) the originating event at the wave source. A Retarded wave from a source is the top half of a lightcone with the source event at the vertex; an Advanced wave from that event is the bottom half of that light cone.
When viewed forward-in-time a Retarded wave expands outward from its source, and an Advanced wave contracts inward toward its source. When viewed backward-in-time a Retarded wave contracts inward toward its source, and an Advanced wave expands outward from its source.
Notice that a Retarded wave from an emission event overlaps at only one point with the Advanced wave from the corresponding absorption event. This results in particle-like behavior from waves: point-like superposition moves at light speed in a straight line from emission to absorption, tracking exactly the photon path(s).
Add many iterations forward-and-backward in time, plus a π phase shift at the end points, and one arrives at the Transactional Interpretation of Quantum Mechanics (TIQM). [see references, especially John Cramer's book and video]
Tip: Consider whether combining TIQM with the flowing-space model of General Relativity might be a path toward Quantum Gravity:
if (QM is time-symmetric waves in 3D space) and (GR is flowing 3D space) then (QG is time-symmetric waves in flowing 3D space)

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