Relative Velocity in Lorentz Transform: Agree or Disagree?

Click For Summary

Discussion Overview

The discussion revolves around whether two inertial observers in relative motion agree on their relative velocity, particularly in the context of the Lorentz transform. It explores the implications of differing measurements of distance and time between observers and how these affect the perceived relative velocity.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants propose that since velocity is defined as distance per unit time, and observers do not agree on distance or elapsed time, they may not agree on their relative velocity.
  • One participant suggests that if the apparent distance in one frame is shorter and the elapsed time is longer, the apparent relative velocity must be smaller in that frame.
  • Another participant questions which observer measures the greater velocity, raising the issue of perspective and the relativity of simultaneity, noting that each observer perceives the other's clock as running slow and lengths as contracted.
  • There is mention of deriving the inverse Lorentz transform by solving for coordinates in terms of each other, indicating a mathematical approach to the problem.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether observers agree on their relative velocity, with multiple competing views and uncertainties expressed throughout the discussion.

Contextual Notes

The discussion highlights limitations related to assumptions about measurements of time and distance, as well as the dependence on the definitions of the frames involved. There are unresolved questions regarding the implications of the Lorentz transform on perceived velocities.

exmarine
Messages
241
Reaction score
11
Do two inertial observers in relative motion agree on their relative velocity? Velocity is distance per unit time and they don’t agree on the distance or the elapsed time. If the apparent distance in the prime system is shorter and the elapsed time is longer, then it seems that the apparent relative velocity in the prime system must be smaller. The inverse transform gets a little strange if one can’t assume the exact same relative velocity.
 
Physics news on Phys.org
exmarine said:
Do two inertial observers in relative motion agree on their relative velocity? Velocity is distance per unit time and they don’t agree on the distance or the elapsed time. If the apparent distance in the prime system is shorter and the elapsed time is longer, then it seems that the apparent relative velocity in the prime system must be smaller. The inverse transform gets a little strange if one can’t assume the exact same relative velocity.

Why don't you check this yourself? By definition, the Lorentz transform is constructed so that the world line with ##x' = 0## has ##x = vt##. Can you figure out how to measure the velocity of the spatial origin of the unprimed frame in the primed one?
 
exmarine said:
Do two inertial observers in relative motion agree on their relative velocity? Velocity is distance per unit time and they don’t agree on the distance or the elapsed time. If the apparent distance in the prime system is shorter and the elapsed time is longer, then it seems that the apparent relative velocity in the prime system must be smaller. The inverse transform gets a little strange if one can’t assume the exact same relative velocity.

Suppose two inertial observers are in relative motion. If they don't measure the same relative velocity, then which one measures the greater velocity? The one moving left? Or the one moving right? Or, the one you choose to call O'? Of the one you choose to call O (without a prime)?

Is it the one whose clock is running slow? But, to each, it's the other's clock that is running slow and whose lengths are contracted.
 
Of course, ultimately, you do not need to know this to derive the inverse formula. Just solve for ##x## and ##t## in terms of ##x'## and ##t'##. It is simply taking the inverse of a 2x2 matrix.
 

Similar threads

  • · Replies 14 ·
Replies
14
Views
3K
  • · Replies 9 ·
Replies
9
Views
2K
  • · Replies 5 ·
Replies
5
Views
2K
  • · Replies 57 ·
2
Replies
57
Views
3K
  • · Replies 34 ·
2
Replies
34
Views
4K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 13 ·
Replies
13
Views
2K
  • · Replies 13 ·
Replies
13
Views
2K
  • · Replies 31 ·
2
Replies
31
Views
3K