Why must the Lorentz Transformations Be Linear?

In summary, the requirement for Lorentz Transformations to be linear stems from the fact that it must preserve the nature of timelike lines, which represent inertial particles. If the transformation were not linear, then an object would appear to be accelerating in one inertial frame but moving at a constant velocity in another, violating the assumption of relativity. This also explains the use of F=dp/dt instead of F=ma in Special Relativity, as mass can vary at high speeds to maintain a consistent momentum.
  • #1
emob2p
56
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All derivations of the Lorentz Transformations I've seen assume a linear transformation between coordinates. Why must this be the case? Thanks.
 
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  • #2
A transformation is linear if and only if it keeps the origin fixed, and it maps lines to lines.
 
  • #3
In particular, a Lorentz transformation must preserve the nature of timelike lines, which represent inertial particles.
 
  • #4
So if the transformation were not linear, then an object would appear to be accelerating in one inertial frame but moving at a constant velocity in another. This would mean F=ma doesn't hold in both frames, a violation of a relativity assumption. Does that argument sound good to you guys?
 
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  • #5
linearity of the LET

emob2p said:
All derivations of the Lorentz Transformations I've seen assume a linear transformation between coordinates. Why must this be the case? Thanks.
I used to say to my students that the LET should be linear because to a pair of space-time coordinates in one inertial reference frame should correspond a single pair of space-time coordinates in an other one.
 
  • #6
LET? Oh, you mean Lorentz-Einstein transforms, not Lorentz Ether Theory.

a pair of space-time coordinates in one inertial reference frame should correspond a single pair of space-time coordinates in an other one.
That happens with any 1-1 transformation...
 
  • #7
linearity

Hurkyl said:
LET? Oh, you mean Lorentz-Einstein transforms, not Lorentz Ether Theory.


That happens with any 1-1 transformation...
of couse! in any consistent theory.
sine ira et studio:smile: :
 
  • #8
emob2p said:
So if the transformation were not linear, then an object would appear to be accelerating in one inertial frame but moving at a constant velocity in another. This would mean F=ma doesn't hold in both frames, a violation of a relativity assumption. Does that argument sound good to you guys?
Yes, the LT is linear so that if there is no accelartion in one LF, there will no acceleration in any other LF. But, don't talk about F=ma in SR.
 
  • #9
why no F=ma in SR?
 
  • #10
I guess it's more appropriate to say F=dp/dt.
 
  • #11
a and dp/dt are related by:
[itex]\frac{d\bf p}{dt}=\frac{d}{dt}(m{\bf v}\gamma)
= m\frac{d}{dt}\left[\frac{\bf v}
{\sqrt{1-{\bf v}^2}}\right]
=m\gamma^3[{\bf a}+{\bf v\times(v\times a)}].[/itex]
 
  • #12
emob2p said:
So if the transformation were not linear, then an object would appear to be accelerating in one inertial frame but moving at a constant velocity in another. This would mean F=ma doesn't hold in both frames, a violation of a relativity assumption. Does that argument sound good to you guys?

What? How do you figure that a frame can be accelerated to one inertial frame but not accelerated to another inertial frame. This makes no sense. F=ma holds in all inertial frames. However, as mentioned above, mass can vary. If mass did not vary then, due to the speed limit of c, a given force in an inertial frame @ .9999c would obviously yield a much smaller acceleration than that same force at .00001c, and if only acceleration changed due to a given force then the postulate of special relativity would be violated. But, at very high speeds the lack of acceleration is made up for in a large increase in mass, so regardless of speed, a given force yields the same momentum.
 
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1. Why do we need the Lorentz Transformations to be linear?

The Lorentz Transformations are a set of mathematical equations that relate space and time coordinates between two reference frames in special relativity. They must be linear in order to accurately describe the behavior of objects moving at high speeds, as non-linear transformations would lead to inconsistencies and paradoxes in our understanding of space and time.

2. What does it mean for the Lorentz Transformations to be linear?

Linearity in mathematics refers to a relationship between two variables in which a change in one variable leads to a proportional change in the other variable. In the context of the Lorentz Transformations, this means that the equations must maintain their form and obey the laws of physics regardless of the reference frame used to measure them.

3. How do the Lorentz Transformations ensure consistency in our understanding of space and time?

The Lorentz Transformations were derived from Einstein's theory of special relativity, which states that the laws of physics must be the same for all observers in uniform motion. By being linear, the transformations allow for the consistent measurement of space and time between two reference frames, ensuring that the laws of physics hold true for all observers.

4. Can the Lorentz Transformations be non-linear?

No, the Lorentz Transformations must be linear in order to accurately describe the behavior of objects moving at high speeds. If they were non-linear, it would lead to contradictions and inconsistencies in our understanding of space and time, violating the principles of special relativity.

5. How are the Lorentz Transformations derived and validated?

The Lorentz Transformations were first derived mathematically by Hendrik Lorentz and Joseph Larmor in the late 19th century. They were later refined and incorporated into Einstein's theory of special relativity. These transformations have been extensively tested and validated through experiments and observations, such as the famous Michelson-Morley experiment, which confirmed the constancy of the speed of light and the validity of the Lorentz Transformations.

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