A question regarding Special Relativity.

In summary: But, as the rocket accelerates, we would need to know the velocity of the rocket at each instant in time, which would require direct observation.
  • #1
JPBenowitz
144
2
Suppose there is a rocket that is traveling at a constant .9c starting at an initial position x1 and fires a photon every second to observers on earth. (Observers on Earth do not know the velocity of the rocket)

If 1s is the proper time [itex]\Delta[/itex][itex]\tau[/itex] then the time passed on Earth between each photon being fired would be [itex]\gamma[/itex] or 2.29416s. Then dependent on the initial position x1 the time between each measured photon on Earth would be the time for each photon to travel the distance d from each position x at every 2.29416s? From measuring the time between each photon that reaches Earth you could calculate the positions x2...xn. But you would only be able to know the positions of where the rocket was not where it is presently.

Assuming I understand the above, how would someone go about by calculating the positions if the rocket is accelerating?
 
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  • #2
JPBenowitz said:
Suppose there is a rocket that is traveling at a constant .9c starting at an initial position x1 and fires a photon every second to observers on earth. (Observers on Earth do not know the velocity of the rocket)

If 1s is the proper time [itex]\Delta[/itex][itex]\tau[/itex] then the time passed on Earth between each photon being fired would be [itex]\gamma[/itex] or 2.29416s. Then dependent on the initial position x1 the time between each measured photon on Earth would be the time for each photon to travel the distance d from each position x at every 2.29416s? From measuring the time between each photon that reaches Earth you could calculate the positions x2...xn. But you would only be able to know the positions of where the rocket was not where it is presently.

Assuming I understand the above, how would someone go about by calculating the positions if the rocket is accelerating?

Wait a minute, if the rocket is accelerating then [itex]\Delta[/itex]t and [itex]\Delta\tau[/itex] would be changing but the observers on Earth wouldn't be able to determine the relativistic time because it would be an element of the time it takes the photon to travel to earth.
 
  • #3
JPBenowitz said:
But you would only be able to know the positions of where the rocket was not where it is presently.
Yes. That is always true.
 
  • #4
JPBenowitz said:
Suppose there is a rocket that is traveling at a constant .9c starting at an initial position x1 and fires a photon every second to observers on earth. (Observers on Earth do not know the velocity of the rocket)

If 1s is the proper time [itex]\Delta[/itex][itex]\tau[/itex] then the time passed on Earth between each photon being fired would be [itex]\gamma[/itex] or 2.29416s. Then dependent on the initial position x1 the time between each measured photon on Earth would be the time for each photon to travel the distance d from each position x at every 2.29416s? From measuring the time between each photon that reaches Earth you could calculate the positions x2...xn. But you would only be able to know the positions of where the rocket was not where it is presently.

Assuming I understand the above, how would someone go about by calculating the positions if the rocket is accelerating?
You are right, the time observed from Earth of the 1 photon per second coming from the rocket will take even longer than the 2.29416 seconds. It's easy to calculate using the Relativistic Doppler Factor which for comparing intervals for objects moving directly apart from each other is:

D = √[(1+β)/(1-β)]

For β = 0.9
D = √[(1+0.9)/(1-0.9)]
D = √[(1.9)/(0.1)]
D = √19
D = 4.3589

That means as long as the rocket travels at the constant speed of 0.9c away from Earth and continues to send out a photon every second, Earth will receive them every 4.3589 seconds.

Now with a little algebra, you can turn the above formula around to calculate β if you know D:

β = |(1 - D2) / (1 + D2)|

So let's see if this works:

D = 4.3589
β = |(1 - D2) / (1 + D2)|
β = |(1 - 4.35892) / (1 + 4.35892)|
β = |(1 - 19) / (1 + 19)|
β = |-18/20|
β = |-0.9|
β = 0.9

So now we know how to determine the delayed speed of the rocket as it accelerates. If we wanted to keep track of the change in position just by looking at the Doppler Shift Periods, we could integrate the calculated speed measured as each photon was received.
 
Last edited:

1. What is special relativity?

Special relativity is a theory developed by Albert Einstein that explains the relationship between space and time in the absence of gravity. It states that the laws of physics are the same for all observers in uniform motion, and that the speed of light is constant for all observers.

2. What is the difference between special relativity and general relativity?

Special relativity only applies to situations where there is no acceleration or gravity present. General relativity, on the other hand, includes the effects of gravity and explains the relationship between matter and the curvature of space-time.

3. How does special relativity explain the concept of time dilation?

Special relativity predicts that time will appear to move slower for an object in motion compared to an observer at rest. This is known as time dilation and is a result of the constancy of the speed of light.

4. Can special relativity be proven experimentally?

Yes, many experiments have been conducted to test the predictions of special relativity, including the famous Michelson-Morley experiment. These experiments have consistently confirmed the validity of the theory.

5. How does special relativity impact our understanding of the universe?

Special relativity has greatly influenced our understanding of the universe by providing a new framework for understanding the relationship between space and time. It has also led to the development of important concepts such as mass-energy equivalence and the space-time continuum.

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