Special Relativity: Time Intervals & Colocated Origins

[Whatifitaint]

I have a simple question about special relativity time intervals.

Say when the origins of 2 frames are colocated, they sync their clocks to 0.

At time t1, frame 1 emits a light pulse. Then at time t2>t1 frame 1 emits a second light pulse. That is a time interval I in frame 1.

In frame 2, it receives these light pulses at times t1a' and t2a'. Then frame 2 does this calculation. At unknown time t1', the origin of frame 1 emits the light pulse. We have v*t1' = d1' as the distance the frame 1's origin traveled at the instant in frame 2's time at the 1st light wads emitted.

The travel time of the light pulse to reach the primed origin is d1'/c. So, t1a' = t1' + d1'/c.

Then, t1'a = t1' + t1'(v/c). Therefore, unknown t1' = t1'a/(1+(v/c)).

Also, t2' = t2'a/(1+(v/c)). Name this interval I' for t1' to t2'.

Question 1. Is the algebra right under special relativity?

Question 2. Does this mean time interval I occurs in frame 1 if and only if time interval I' occurs in frame 2?

Thanks

 

[PAllen]

It looks ok to me - you are doing all measurements and calculations in frame 2 coordinates, so it is consistent. You are computing Doppler redshift in terms of frame 2 times - so time dilation does nat factor in.

As to the relation between your I and I', these would be the same only for an equivalent set up: emitter and receiver moving directly away from each other, and all measurements taken in coordinates based on the receiver.

 

[Whatifitaint]

It looks ok to me - you are doing all measurements and calculations in frame 2 coordinates, so it is consistent. You are computing Doppler redshift in terms of frame 2 times - so time dilation does nat factor in.

As to the relation between your I and I', these would be the same only for an equivalent set up: emitter and receiver moving directly away from each other, and all measurements taken in coordinates based on the receiver.

Yes, they have identical clocks when at rest with each other. Also, all coordinate distance and time are in the context of frame 2 when measuring the light pulses from frame 1.

So, you agree with the OP conclusions assuming these things?

 

[PAllen]

Yes, they have identical clocks when at rest with each other. Also, all coordinate distance and time are in the context of frame 2 when measuring the light pulses from frame 1.

So, you agree with the OP conclusions assuming these things?

Yes, it looks fine.

 

[Whatifitaint]

Yes, it looks fine.

I don't mean to offend you, but I would like some consensus on this. A while ago, I was banned from some other website and I don't want to provide names for claiming this was true. I went through it over and over and could not see an error.

So, I just want to make sure.

 

[Dale]

At time t1, frame 1 emits a light pulse.

Frames are coordinate systems, mathematical objects. I assume that you mean that an emitter at rest at the spatial origin of frame 1 emits a light pulse.

In frame 2, it receives these light pulses at times t1a' and t2a'.

Again, I assume that you mean a receiver at rest at the spatial origin of frame 2 receives the light pulse.

Then, t1'a = t1' + t1'(v/c). Therefore, unknown t1' = t1'a/(1+(v/c)).

Also, t2' = t2'a/(1+(v/c)). Name this interval I' for t1' to t2'.

Question 1. Is the algebra right under special relativity?

Yes.

Question 2. Does this mean time interval I occurs in frame 1 if and only if time interval I' occurs in frame 2?

I don't think so. I could think of ways to get I' through different emission patterns and vice versa. For example, consider clocks that are not at rest at the respective origins.

 

[Whatifitaint]

Frames are coordinate systems, mathematical objects. I assume that you mean that an emitter at rest at the spatial origin of frame 1 emits a light pulse.

Again, I assume that you mean a receiver at rest at the spatial origin of frame 2 receives the light pulse.

Yes.

I don't think so. I could think of ways to get I' through different emission patterns and vice versa. For example, consider clocks that are not at rest at the respective origins.

All of your assumptions above are correct.

For your last statement, the receiver is the origin of frame 2. So, the origin of I determined the times of the light flashes and the origin of the I' is the receiver of these light flashes.

So, does time interval I associate with I' under these conditions?

 

[Dale]

Then it is an if relationship, but not an if and only if relation.

In other words, this is a correct inference: If the emitter is at rest at the origin of the unprimed frame and the receiver is at rest at the origin of the primed frame and nothing interferes with the signal between emission and reception and the interval between emission is I then the interval between reception is I'.

But this is not a correct inference: If the interval between reception is I' then the emitter is at rest at the origin of the unprimed frame and the receiver is at rest at the origin of the primed frame and nothing interferes with the signal between emission and reception and the interval between emission is I.

There are just too many conditions to the initial "if" to make it easily into an "if and only if".

There probably is a correct inference that: If the interval between reception is I' and the emitter is at rest at the origin of the unprimed frame and the receiver is at rest at the origin of the primed frame and nothing interferes with the signal between emission and reception then the interval between emission is I.

That may be more what you had intended, but I don't know how to describe that one. It isn't an if and only if relationship on the emission and reception, nor on the more specific statement.

 

[Whatifitaint]

Then it is an if relationship, but not an if and only if relation.

In other words, this is a correct inference: If the emitter is at rest at the origin of the unprimed frame and the receiver is at rest at the origin of the primed frame and nothing interferes with the signal between emission and reception and the interval between emission is I then the interval between reception is I'.

But this is not a correct inference: If the interval between reception is I' then the emitter is at rest at the origin of the unprimed frame and the receiver is at rest at the origin of the primed frame and nothing interferes with the signal between emission and reception and the interval between emission is I.

There are just too many conditions to the initial "if" to make it easily into an "if and only if".

There probably is a correct inference that: If the interval between reception is I' and the emitter is at rest at the origin of the unprimed frame and the receiver is at rest at the origin of the primed frame and nothing interferes with the signal between emission and reception then the interval between emission is I.

That may be more what you had intended, but I don't know how to describe that one. It isn't an if and only if relationship on the emission and reception, nor on the more specific statement.

Yes, I was not thinking about I'->I.

You have completely answered my question.

Have a nice day.