Applying the lorentz transformation

Click For Summary

Discussion Overview

The discussion revolves around the application of the Lorentz transformation to understand how a circle appears in a moving reference frame. Participants explore the implications of relativistic effects on the shape of objects as observed from different frames, particularly focusing on the transformation of a circle into an ellipse and the concept of length contraction.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions whether the Lorentz transformation can be directly applied to the equation of a circle, seeking clarification on how it would appear in a moving frame.
  • Another participant asserts that the circle will appear as an ellipse with a major axis of 1 and a minor axis of 1/γ, suggesting a need to relate parameters to complete the analysis.
  • A participant expresses confusion about the necessity of relating parameters u and t, initially believing that transforming x coordinates alone suffices.
  • One participant notes that they initially observed the circle stretching rather than contracting, indicating a misunderstanding in their application of the transformation.
  • Another participant explains that all observed lengths are less than or equal to the proper length, referencing the length contraction formula L = L₀/γ.
  • A later reply challenges the assertion that the circle simply transforms into an ellipse, emphasizing the role of simultaneity in measurements across different frames and the complications introduced by light travel time.
  • One participant warns against relying solely on the length contraction equation, suggesting that using the complete Lorentz transformation can help avoid common mistakes.

Areas of Agreement / Disagreement

Participants express differing views on the transformation of the circle and the implications of simultaneity in relativity. While some agree on the general outcome of the transformation, there is no consensus on the specifics of how to apply the Lorentz transformation correctly or the interpretation of the results.

Contextual Notes

Participants highlight the importance of accounting for relative simultaneity when transforming shapes between frames, indicating that misunderstandings may arise from neglecting this aspect. There are also unresolved questions regarding the relationship between parameters in the transformation process.

cephas
Messages
10
Reaction score
0
I am trying to understand how to use the lorentz transformation to calculate things and I have a few questions about it.

Suppose I have a circle of radius 1 in frame K which is at rest. Frame k' is moving at velocity v relative to frame k. What would the circle look like to someone in frame k' as they pass by it?

Can I just apply lorentz equation straight to the equation of the circle?

Rest frame k

x(u)=Cos(u)
y(u)=Sin(u)

moving frame k'

x'(u)=(cos(u)-v*t)*gamma
y'(u)=Sin(u)

then if I graph the above parametric equation for frame k' would that be what the circle would like to the person as they move by?

I really want to nail this stuff down. I don't understand why this reference frame stuff confuses me so much. It should be simple. Thanks for the help in advance
 
Physics news on Phys.org
It will just look like an ellipse with major axis 1 and minor axis [itex]1/\gamma[/itex].

If you want to parameterize the ellipse, you'll need to relate u to t somehow to be able to finish the analysis.

- Warren
 
Why would I need to relate u and t? x=Cos(u) just means I am transforming all x coordinates from 0<=u<=2Pi to x' I don't see why they need to be related.

Also, when I tried doing this, the circle stretched. I thought is was supposed to shrink. I have to be doing something wrong.
 
Ok, now I got it so it contracts like it is supposed to. I don't understand why i have to take the reciprocal of gamma though. I will try studying my books some more.
 
All observed lengths are less than or equal to the proper length. Gamma is always greater than or equal to one. Ergo

[tex]L = L_0/\gamma[/tex]

- Warren
 
chroot said:
It will just look like an ellipse with major axis 1 and minor axis [itex]1/\gamma[/itex].- Warren

That's not quite true is it?

Relativity says that if you MEASURE the length of a moving object by determining the locations of its ends simultaneously (in the rest frame) that the result will be less than that for the same object when its stationary. But what you SEE is not quite the same, because the light takes longer to get from the more distant parts of the object. This isn't an issue for a one dimensional object moving along its axis, but with a circle it comes into play.

Just thought I'd mention it.
 
cephas said:
Ok, now I got it so it contracts like it is supposed to. I don't understand why i have to take the reciprocal of gamma though. I will try studying my books some more.

It sounds to me like you're making a very common mistake, which is the result of using the length contraction equation instead of the compelte LT for length. Until you're VERY confident with this stuff, stick with the LTs, it's much harder to go wrong with them, because you can see which frame the simulataneous measurements have to be in.

I don't know if this helps or not.
 
cephas said:
I am trying to understand how to use the lorentz transformation to calculate things and I have a few questions about it.

Suppose I have a circle of radius 1 in frame K which is at rest. Frame k' is moving at velocity v relative to frame k. What would the circle look like to someone in frame k' as they pass by it?

Can I just apply lorentz equation straight to the equation of the circle?

Rest frame k

x(u)=Cos(u)
y(u)=Sin(u)

moving frame k'

x'(u)=(cos(u)-v*t)*gamma
y'(u)=Sin(u)

then if I graph the above parametric equation for frame k' would that be what the circle would like to the person as they move by?

I really want to nail this stuff down. I don't understand why this reference frame stuff confuses me so much. It should be simple. Thanks for the help in advance

Actually you didn't transform what a metal loop looks like from one frame to another. You transformed where a set of events occur from one frame to the other. If you apply the time transformation as well you will see that if they happen at the same time according to your initial frame that they occur at different times according to the second frame. If these events happen at the same time along a solid ring according to a frame at which it is at rest then they can not describe the shape of the ring according to any other frame because the ends of the ring have moved between the different times that the events occur. Most paradoxs or problems people seem to have in understanding relativity seem to be tracable to unaccounted for relative simultaneity.
 
Last edited:

Similar threads

Undergrad Lorentz transformation on mode functions
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Derive the Lorentz transformation in Minkowski four-dimensional spacetime spacetime
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Question about full Lorentz transformation
  • · Replies 101 ·
4
Replies
101
Views
8K
Undergrad Why is the Lorentz Force always perpendicular to velocity?
  • · Replies 55 ·
2
Replies
55
Views
6K
High School Reciprocal time dilation
  • · Replies 15 ·
Replies
15
Views
3K
Undergrad Deriving Lorentz Transformations: Hyperbolic Functions
  • · Replies 33 ·
2
Replies
33
Views
3K
High School Confusion in notation of Lorentz Transformations
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Can we choose different functional forms for Lorentz transformation?
  • · Replies 10 ·
Replies
10
Views
2K
Graduate Show Lagrangian is invariant under a Lorentz transformation without using generators
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Another interpretation of Lorentz transformations
  • · Replies 53 ·
2
Replies
53
Views
4K