Lorentz transformation of multiple events into one frame of observation

[Oranginayo]

Homework Statement

Object A travels past a point B at 0.5c and the distance is 1.3 light seconds to arrive at point C. When A passes point B a microwave is sent at point B towards point C, where an observer D who is travelling ahead of object A arrives at point C at the same time as the microwave.
Clocks at point B one at point C are synchronised.

Lorentz transform three events into into object A's frame and draw diagrams showing the three events from object A's frame.

Relevant Equations

x' = /gamma(x-Vt), t' = /gamma(t - Vx/c^2)

The three events;
t = 0s, x=0
t=1.3s, x=1.3 light seconds
t=2.6s, 1.3 light seconds

 

[PeroK]

Perhaps you ought to label those events $1, 2, 3$.

The question is to transform those events into A's frame?

(I'm not I understand what D has to do with this. Is that for another part of the question?)

 

[Oranginayo]

So are the events correct? (in B, C system)

Yes transform into A's frame.

 

[PeroK]

So are the events correct? (in B, C system)

Yes. Those events being: 1) A passes B (and microwave is emitted from B); 2) Microwave (and D) arrives at C; 3) A arrives at C.

 

[Oranginayo]

So the transformation of the events I have..
1. t' = 0, x'=0
2. t' = 0, x' = 0
3. x' = 0, t' = 2.25s

Are these correct?

 

[PeroK]

So the transformation of the events I have..
1. t' = 0, x'=0
2. t' = 0, x' = 0
3. x' = 0, t' = 2.25s

Are these correct?

The second one looks suspicious, don't you think?

 

[Oranginayo]

Yes, I'm not sure how to express in a division of zero in the lorentz factor.

 

[PeroK]

Yes, I'm not sure how to express in a division of zero in the lorentz factor.

Division by zero is a problem that implies a mistake somewhere.

 

[Oranginayo]

These are my thoughts. The microwave is traveling at speed c which I've used in the lorentz factor to be undefined.

 

[PeroK]

These are my thoughts. The microwave is traveling at speed c which I've used in the lorentz factor to be undefined.

That would be the frame of the microwave. Which is undefined in SR. You want the event in the frame of A.

 

[Oranginayo]

Unless they are t=0.75s and x=0.75??

 

[PeroK]

Unless they are t=0.75s and x=0.75??

They are indeed.

 

[Oranginayo]

Thank you. So when observer D is traveling at 0.5c and has a clock in sync with object A. According to observer D what is the time on the B-C clock and observer D's clock when the microwave arrives?
Is this just delta T0 = 1.3 sqrt(1-0.5^2) for the B-C clock and T0 = 0.75 sqrt(1-0.5^2)?

 

[PeroK]

Thank you. So when observer D is traveling at 0.5c and has a clock in sync with object A. According to observer D what is the time on the B-C clock and observer D's clock when the microwave arrives?
Is this just delta T0 = 1.3 sqrt(1-0.5^2) for the B-C clock and T0 = 0.75 sqrt(1-0.5^2)?

If A and D are moving at the same velocity in the B-C frame, then A and D have effectively the same rest frame. Assuming they synchroise their clocks and agree on a common spatial origin.

Fundamentally, A and D are local observers representing their shared frame. If we take A to be at the origin, then A's clock measures the time of events at the origin in the A-D frame; and, D's clock measures the time of events at $x' = 0.75$ in that frame.

Similarly, C's clock measures the time of events at location $x = 1.3$ in the B-C frame.

With that in mind, do you want to reconsider your answers?

 

[Oranginayo]

Does that mean $\Delta T = 1.3/ \sqrt(1-0.5^2)$ for C's clock and $\Delta T = 0.75/ \sqrt(1-0.5^2)$ for D's clock ??

 

[PeroK]

Does that mean $\Delta T = 1.3/ \sqrt(1-0.5^2)$ for C's clock and $\Delta T = 0.75/ \sqrt(1-0.5^2)$ for D's clock ??

What is $\Delta T$?

 

[Oranginayo]

The time interval for time dilation

 

[PeroK]

The time interval for time dilation

What time interval?

 

[Oranginayo]

The time interval as seen from observer D or is this incorrect?

 

[PeroK]

The time interval as seen from observer D or is this incorrect?

The time interval between what two events?

 

[Oranginayo]

There is just one event - the arrival of the microwave so does that mean; Clock C = 1.3s, Clock D = 0.75s

 

[PeroK]

There is just one event - the arrival of the microwave so does that mean; Clock C = 1.3s, Clock D = 0.75s

For a time interval you need two events. For a single event, you have a single time coordinate.

That's the same in classical physics.

 

[Oranginayo]

So the clocks read the same? Both must read 1.2s then

 

[PeroK]

So the clocks read the same? Both must read 1.2s then

Can you please just tell me what the question is? Not the answer. What is the question?

 

[Oranginayo]

Observer D is traveling ahead of object A at 0.5c and has a clock in sync with object A. According to observer D what is the time on the C clock and observer D's clock when the microwave arrives?

 

[PeroK]

Observer D is traveling ahead of object A at 0.5c and has a clock in sync with object A. According to observer D what is the time on the C clock and observer D's clock when the microwave arrives?

So, that's not a time interval. That's a time on each clock. We need to go back to this post:

If A and D are moving at the same velocity in the B-C frame, then A and D have effectively the same rest frame. Assuming they synchroise their clocks and agree on a common spatial origin.

Fundamentally, A and D are local observers representing their shared frame. If we take A to be at the origin, then A's clock measures the time of events at the origin in the A-D frame; and, D's clock measures the time of events at $x' = 0.75$ in that frame.

Similarly, C's clock measures the time of events at location $x = 1.3$ in the B-C frame.

With that in mind, do you want to reconsider your answers?

Let me help you out, because I think you are confused.

The microwave arriving at $C$ is an event (the second event). The time of that event in the B-C frame is $1.3s$. Therefore, C's clock must read $1.3s$.

Likewise, in the A-D frame that event takes place local to D. So, the time on D's clock is $0.75s$, as you have already calculated.

 

[Oranginayo]

Thanks. So what are the times on object A's clock and on C's clock when object A arrives at C?

 

[PeroK]

Thanks. So what are the times on object A's clock and on C's clock when object A arrives at C?

I'll let you work that out.