# I The velocity of a moving frame of reference

#### RossBlenkinsop

Summary
train carriages in a MFR
In the attached fig 1 you have a moving frame of reference (MFR) moving in the direction V at velocity V

Looking from above in the MFR there are 2 identical train carriages (car 1 and car 2), Car 1 is on a long set of tracks. Car 2 cannot move but is on train tracks.

There are also 2 Identical clocks, clock 1 and clock 2 which are attached to the train tracks. Clock 2 can receive electrical signals from car 2 thru the tracks . Clock 1 can receive electrical signals from car 1 thru the tracks

Car 1 can move along the train tracks in the direction v1 at a constant velocity v1

Fig 2 shows inside one of the train carriages, in it is a light tube of length L. The light tube has a light and detector at one and and a mirror at the other. The light tube has a wire that is connected to the train tracks . When photons are emitted by the light source in the light tube, an electric pulse is sent down the wire to the train tracks. The electric pulse flows along the train track back to the clock which starts the clock running .

When an electric pulse is sent it makes a mark on the train track, so the position of a train at the time the electric pulse was sent is known..

Back at the light tube, the photons are reflected by the mirror, in the light tube, back to the detector, in the light tube. When the photons arrive at the detector, in the light tube, a second electric pulse is sent down the wire, along the train tracks and back to the clock, which stops the clock, thus measuring the time it takes for light to traverse the length L. the length of the light tube

Both train carriages are identical in every way and both train carriages are set up in an identical fashion. They are identical in every way except one can move along the tracks, the other cant.

if the MFR is travelling in direction V at velocity V. When the velocity of Car 1 ( velocity V1) is equal to the velocity of the MFR (V) and when the direction of car 1 (direction of V1) is exactly opposite direction of the MFR (direction of V), the time to traverse the length of the light tube, in Car 1, will be the shortest.

Is that correct ?

From the above the velocity and direction of the MFR can be calculated

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#### jbriggs444

Homework Helper
There are a number of errors with the analysis of this experiment. One is that the "moving frame of reference" is never used. Another is that there is no accounting for signal delays in the wire in the train car or along the train tracks.

#### RossBlenkinsop

Im not sure using another terminology changes the overall result ?

the relevant time is the duration between pulse 1 and the second pulse. The delay will be the same for both pulses moving in the same direction along the same track, so the time difference between pulse 1 and pulse 2 will remain constant

#### RossBlenkinsop

the wires, train track, wheels etc etc are all at constant velocity, constant length, constant direction , constant ... constant constant

the delay for pulse 1 will be identical to the delay of pulse 2 , so the time difference between pulse 1 and pulse 2 will also be constant

any delay as a result of the train being slightly further away with pulse 2 can be accounted for by measuring the marks on the track and the speed of electricity thru the train track ....that was the purpose of the marks on the track as I anticipated that question

#### RossBlenkinsop

the speed of electricity thru the train track would be measured using a clock at rest wrt the train track

#### Ibix

Your description is over-complex, but you appear to have a light clock on a cart and to be measuring its tick rate in the track frame (once you correct for signal delay). This will just be one tick per $1/\sqrt{1-v_1^2/c^2}$ seconds, where $v_1$ is the velocity of the cart with respect to the track.

I fail to see how you think you can deduce anything about another frame of reference (your MFR). You do not appear to be making any measurements using it.

#### RossBlenkinsop

In order to clarify my logic

Referring to fig 99

There is a moving frame of reference ( MFR ) moving to the right at velocity MFRV

In the time between T1 and T2 the MFR moves a distance to the right of mfrD see to the left of fig 99

Inside the MFR there are three light clocks a green light clock (GLC), a red light clock (RLC) and a blue light clock (BLC)

The BLC is at rest wrt the MFR

The GLC is able to move to the left at a constant velocity GV

The RLC is able to move to the right at a constant velocity RV

RV = GV

In the time between T1 and T2 GV and RV move a distance of d see the small square top right

At time To the GLC and RLC are both at position X see the top of fig 99. There is a strobe light at position X . A To the strobe goes off

As it is difficult to represent diagrammatically I have drawn the MFR at time 0 at the top, T1 below that and T2 below that again.

At the bottom of fig 99 I have simply shown the position of the various clocks at To, T1 and T2, relative to X

At time T1 the GLC will be a distance of (mfrD – d) from X (the original position of the light source)

At time T1 the RLC will be a distance of (mfrD + d) from X (the original position of the light source)

At time T2 the GLC will be a distance of (mfrD + mfrD - d - d) from X (the original position of the light source)

At time T2 the RLC will be a distance of (mfrD + mfrD + d + d) from X (the original position of the light source)

The positions of the various clocks wrt the position X is shown at the bottom of fig 99

At the bottom of fig 99 I have shown the path taken by the photons in the RLC and GLC

At To the RLC and GLC will be at X
At T1 the GLC will be at GT1
At T1 the RLC will be at RT1
At T2 the GLC will be at GT2
At T2 the RLC will be at RT2

Any problems thus far ?

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#### jbriggs444

Homework Helper
In order to clarify my logic

Referring to fig 99

There is a moving frame of reference ( MFR ) moving to the right at velocity MFRV.
As I understand it, you have a set of tracks that are at rest relative to this frame. And you have a car (car 2) which is also at rest relative to this frame.
In the time between T1 and T2 the MFR moves a distance to the right of mfrD see to the left of fig 99
In what frame of reference are T1 and T2 measured? The above statement appears to pre-suppose that simultaneity is absolute.

#### RossBlenkinsop

appears there are no problems

as in the previous example the GLC and RLC are on train tracks and an electric pulse is transferred from the GLC and RLC to the train tracks for each tick of the RLC and GLC. The electric pulses flow back along the train tracks to the BLC which measures the duration between the pulses.

The BLC has a constant tick rate as it is tied to the MFR

will the tick rates of the GLC and RLC be the same or different ?

#### RossBlenkinsop

As I understand it, you have a set of tracks that are at rest relative to this frame. And you have a car (car 2) which is also at rest relative to this frame.

In what frame of reference are T1 and T2 measured? The above statement appears to pre-suppose that simultaneity is absolute.
T1 is the time that the light pulse strikes the bottom of the GLC and the RLC. The electric pulse then moves back along the train track to the BLC.

T1 is not actually measured I have just used To,T1 and T2 to demonstrate the chronology of events for the purposes of making my logic understandable .

The only thing that is "measured" is the duration between electric pulse 1 and electric pulse 2 off the train tracks by the BLC, ie how fast the RLC and GLC are ticking relative to the BLC

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#### RossBlenkinsop

also in fig 99 I now have three cars sorry my fault that is confusing

there is a car at rest wrt MFR . That car houses the BLC

there is another car, not at rest wrt the MFR, it houses the GLC

there is another car, not at rest wrt the MFR, it houses the RLC

The RLC is moving at a constant velocity along train tracks to the right hand edge of fig 99

The GLC is moving at a constant velocity along train tracks to the left of fig 99

the velocity of the CLC is the same as the velocity of the RLC

I have omitted the cars tracks wheels etc etc as it make the picture too confusing I have just shown the moving clocks

#### RossBlenkinsop

sorry the other thing that is measured is the speed of electricity thru the train tracks, the marks on the tracks and the velocity and direction of the various train carriages moving wrt to the MFR

#### RossBlenkinsop

the purpose of the experiment is to adjust the velocity of the various trains, moving wrt the MFR, so that the clock tick rate, in the moving train carriage, is at its fastest as measured by the BLC

at a given velocity the tick rate should peak and then if the velocity is increased or decreased, from that specific value, the tick rate should slow

#### PeterDonis

Mentor
the purpose of the experiment is to adjust the velocity of the various trains, moving wrt the MFR, so that the clock tick rate, in the moving train carriage, is at its fastest as measured by the BLC
The clock rate of a train, as measured by the BLC, will be fastest if it is at rest relative to the BLC. This is just a special case of the general result that, relative to any clock, a second clock will tick fastest if it is at rest relative to the first clock. There is no need for any complicated mathematical machinery to reach this result; it's a simple consequence of the relativistic time dilation formula.

#### RossBlenkinsop

in order to keep thing simple lets analyse only the 2D version

place the above models in a 2d Cartesian coord system

lets say when the strobe goes off it is at position x1,y1

assuming the entire experiment is moving, the entire experiment is moving away from that original position X1, Y1, at some rate V

appears to me if the train that can move is moving in the right direction and at the right rate so that it remains at position x1,y1

that will maximize the tick rate of that clock , in fact that will be the maximum tick rate for any clock in that frame

?

#### jbriggs444

Homework Helper
in fact that will be the maximum tick rate for any clock in that frame
Which is what @PeterDonis just finished saying. It is a simple consequence of the time dilation formula.

That does not help you determine how fast the "moving frame of reference" is moving in any absolute sense.

#### RossBlenkinsop

I disagree that is what he said

I understand he said if the moving clock (GLC or RLC) is at rest wrt the BLC then that will be the maximum tick rate as measured by the BLC

I am saying if the RLC or GLC is at rest wrt to the origin of the light source, at point X1,Y1, then that will be the max tick rate

#### RossBlenkinsop

I refer to the attached image

u have a clock that is at rest wrt the source of the photons; clock 1
u have a second clock (clock 2) that is not at rest wrt the source of the photons

clock 2 simply must tick slower otherwise it defies physics

or am I missing something

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#### Ibix

clock 2 simply must tick slower otherwise it defies physics
It must tick slower in the frame where the source is at rest. But in the frame where clock2 is at rest and the source and clock1 are in motion, clock1 will tick slower than clock2.

#### PeterDonis

Mentor
I am saying if the RLC or GLC is at rest wrt to the origin of the light source, at point X1,Y1, then that will be the max tick rate
Max tick rate relative to which clock?

You were asking for the max tick rate relative to the BLC. That is what I answered.

If you want the max tick rate relative to some other clock, that will be a different answer. What state of motion gives the max tick rate depends on the state of motion of the clock you are measuring the tick rate relative to--the max tick rate relative to that clock will be for an object at rest relative to that clock. Which is the more general version of what I said.

#### PeterDonis

Mentor
clock 2 simply must tick slower
Relative to clock 1, yes. "Tick rate" is relative.

or am I missing something
You appear to be missing that tick rate is relative.

#### Ibix

You appear to be missing that tick rate is relative.
...and/or that there's nothing special about the frame where the light emitter is at rest, at least as far as this experiment is concerned.

#### Janus

Staff Emeritus
Gold Member
I refer to the attached image

u have a clock that is at rest wrt the source of the photons; clock 1
u have a second clock (clock 2) that is not at rest wrt the source of the photons

clock 2 simply must tick slower otherwise it defies physics

or am I missing something
What you seem to be missing is that the speed of light is invariant. It it is the same(c) relative to either frame as measured from that frame.

So, for example, if we start with both clocks at the same spot and the source (white dot) is at rest with respect to clock 1 (blue bar), then this is what happens according to the rest frame of clock 1.

The light expands outward at c from the source as clock 2(yellow bar) moves off to the right. The edge of the expanding light reaches the end of clock 1 before it reaches the end of clock 2.

However, in the rest frame of clock 2, this is what is happening:

The light expands outward from the point where it was emitted relative to clock 2, which at that moment coincides with the position of the light source and the top of clock 2. It expands outward at c from that point while both Clock 1 and the source move off to the left. You can't have the center of the expanding circle moving to the left along with the source, because then the left edge of the expanding light would have to be moving at more than c relative to clock 2 ( the right edge at less than c). Light must move at c in all frames.

The expanding light reaches the bottom of clock 2 before it reaches the bottom of clock 1.

#### RossBlenkinsop

I think ppl might be confusing what I refer to as a light source and being at rest wrt the light source.

see attached

there is a mfr with a light in the roof. At t2 the light briefly strobes at the point x1,y1. I am saying the moving clock in the MFR move in such a way that it remains at rest wrt the point X1,Y1

I have shown the photons at successive times t3 t4 t5 t6 as they spread out . There is a Cartesian coor system in dark blue and there is a light blue line to the point on the x axis X1

as it is difficult to draw I have shown the position of the MFR at various time with t2 draw below t1 to show the rough chronology of events

I am saying if the moving train inside the MFR moves in such a way (in the right direction at the right rate) that it remains at rest wrt the point X1 Y1 then the clock within that train carriage must tick faster than any other clock in that frame. Those fast ticks will be transferred to the rail tracks and measured at the BLC. And the BLC will at all times be ticking slower than the clock at rest wrt the point X1Y1

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#### RossBlenkinsop

or another way

The train carriage inside the MFR moves in the right direction and at the right rate so that it remains directly below (remains on a vertical line drawn thru the point X1) and at rest wrt the point X1Y1. I am saying if that is the case the tick rate of the clock in the moving train carriage will peak and that fast ticking will be transferred to the train tracks.

As the BLC is not at rest wrt the point X1 Y1 it must tick slower. Further as it is moving at some angle that is not perpendicular to a tangent of the photon wave fronts, it must tick slower.

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