Time difference of events when moving at relativistic speeds

In summary, the distance between Planet A and Planet B is 8.3 light minutes and event A occurs at t=0 while event B occurs at t=2 minutes. An observer traveling from Planet A to B at 0.8c would observe event B to occur 464 seconds before event A due to a combination of time dilation and the fact that Planet B is not sitting at the origin. This can be calculated using Lorentz transformations for time and distance.
  • #1
Whistlekins
21
0

Homework Statement


Lets say that Planet A and Planet B are moving in the the same inertial reference frame. The distance between them is 8.3 light minutes. Event A occurs on Planet A at t=0, and Event B occurs on Planet B at t=2 minutes. If an observer is traveling from Planet A to B at 0.8c, what is the time difference between the two events?


Homework Equations




The Attempt at a Solution



I want to say that we just need to use the Lorentz transformation, plugging in 2 minutes as the proper time, but it's probably not that simple.
 
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  • #2
probably not.
event A occurs at tA'=0, xA'=0 in planet frame (taking A as the origin).
event B occurs at tB'=2 min, xB'=8.3 light minutes in planet frame.

find out when these are in the moving frame tA, tB.

you get the piece due to time dilation, but also a piece due to the fact
that planet B is not sitting at the origin.
 
  • #3
So if I use Lorentz transformations and say tA = γ(t'A - v/c2 xA), similarly for tB, I get tA = 0 obviously, and tB = -464s, meaning that Event B occurs 464s before Event A in the reference frame of the observer? I hope I'm understanding this properly.
 
  • #4
More help to Clarify

This should just be a straightforward application of time dilation.

Assume your ship's origin lines up with A's origin. x=0, t=0, x'=0, t'=0.

Now B is at rest with respect to A, so they are in the same frame, call this the ground frame.

Your event is going to take place in the ground's frame at (x=8.3 light-mins, t = 2 min).

Thus, use x'=γ(x-vt) where you plug in the ground's frame x and t from above. The x' that pops out is the coordinate where the ship observes the event to take place.

Use the next Lorentz equation: t'=γ(t-vx/c^2) where you plug in the ground frame's x,t from above. This will give the t' that the ship observes. If you get an overall minus on the t', that's okay, that just means it happened before the orgiins lined up in the ship's frame.
 
  • #5


I would like to clarify some details before providing a response. When you say "moving in the same inertial reference frame," do you mean that both planets are stationary relative to each other? Also, when you mention an observer traveling from Planet A to B at 0.8c, do you mean that they are traveling at a constant velocity of 0.8 times the speed of light? Assuming these conditions, I will provide a response to the question.

In special relativity, the time difference between two events can be calculated using the Lorentz transformation. However, this transformation takes into account the relative velocities between the two frames of reference. In this case, we have two reference frames: the stationary frame of Planet A and the moving frame of the observer traveling from Planet A to B.

To calculate the time difference between Event A and Event B, we need to first determine the time in the moving frame of the observer. This can be done using the time dilation formula, which states that the time in a moving frame is dilated (slower) compared to the time in a stationary frame. In this case, the time dilation factor is given by γ = 1/√(1-v^2/c^2), where v is the velocity of the observer (0.8c in this case) and c is the speed of light. Plugging in these values, we get γ = 1/√(1-0.8^2) = 1.67.

Next, we can use the Lorentz transformation to calculate the time in the moving frame of the observer. This transformation is given by t' = γ(t-vx/c^2), where t is the time in the stationary frame, v is the velocity of the observer, and x is the distance between the two events (8.3 light minutes in this case). Plugging in the values, we get t' = 1.67(2-0.8*8.3/c^2) = 1.67(0.4) = 0.67 minutes.

Therefore, the time difference between Event A and Event B as measured by the moving observer is 0.67 minutes. This is shorter than the time difference of 2 minutes measured in the stationary frame, due to the time dilation effect. This result is in agreement with the predictions of special relativity.
 

1. What is the theory of relativity?

The theory of relativity is a fundamental theory in physics that explains how objects behave in relation to one another. It is comprised of two main theories: special relativity and general relativity. Special relativity deals with objects moving at constant speeds, while general relativity deals with objects in accelerated motion or in the presence of gravity.

2. How does the theory of relativity affect the concept of time?

The theory of relativity states that time is not absolute, but rather it is relative to the observer's frame of reference. This means that time can appear to pass differently for two observers depending on their relative speeds and positions in space.

3. What is time dilation in the context of relativity?

Time dilation refers to the phenomenon where time appears to pass slower for objects that are moving at high speeds. This is due to the fact that as an object's speed approaches the speed of light, time slows down for that object relative to a stationary observer.

4. Can time travel be achieved by moving at relativistic speeds?

While time dilation does allow for time to pass differently for objects moving at high speeds, it does not necessarily mean time travel is possible. Time travel would require the ability to move through time as well as space, which has not been proven to be possible according to the current understanding of physics.

5. How does the time difference between events change at relativistic speeds?

At relativistic speeds, the time difference between events can change significantly. This is due to the time dilation effect, where time appears to pass slower for objects moving at high speeds. As a result, events that may seem simultaneous to an observer at rest may appear to occur at different times for an observer in motion.

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