Deriving the Lorentz Transformation from the Homogeneity of Spacetime

In summary, this source/solution provides an explanation of how the result that ##f = f_0## emerges from the properties of a 1-parameter Lie group. However, the solution does not seem to be completely accurate, and I am not sure how I could provide any help.
  • #1
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Homework Statement


Show that the isotropy and homogeneity of space-time and equivalence of different inertial frames (first postulate of relativity) require that the most general transformation between the space-time coordinates (x, y, z, t) and (x', y', z', t') is the linear transformation,
x'=f(v2)x-vf(v2)t; t'=g(v2)t-vh(v2)x; y'=y; z'=z
and its inverse,
x=f(v2)x'+vf(v2)t'; t=g(v2)t'+vh(v2)x'; y=y'; z'=z'

Homework Equations

The Attempt at a Solution


Now, I know that homogeneity implies that the transformation must be linear in x and t and that the isotropy of space implies that the coefficients can only be functions of the magnitude of the velocity (not the direction) at most. Therefore, I am stuck at the following:
x'=f(v2)x-vf0(v2)t; t'=g(v2)t-vh(v2)x; y'=y; z'=z
However, I am having trouble proving that f = f0. The solution I am looking at says that this follows from the homogeneity of space-time but I am having trouble using that fact to prove it.
 
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  • #2
Hmm. Which source/solution are you looking at? (Can you post a reference or link?)

In my derivation of this stuff, the result that ##f = f_0## emerges from the properties of any 1-parameter Lie group (which is what you're deriving here). I.e., 2 successive transformations (in the same direction) with different parameters ##v,v'## must commute. This imposes some constraints on the various functions.
 
  • #3
strangerep said:
Hmm. Which source/solution are you looking at? (Can you post a reference or link?)

In my derivation of this stuff, the result that ##f = f_0## emerges from the properties of any 1-parameter Lie group (which is what you're deriving here). I.e., 2 successive transformations (in the same direction) with different parameters ##v,v'## must commute. This imposes some constraints on the various functions.
http://faculty.uml.edu/cbaird/all_homework_solutions/Jackson_11_1_Homework_Solution.pdf
 
  • #4
Hmm. (Sigh.) Well, I will say that I think there are some unnecessary fudges in that solution. So I'm not sure how I can usefully help you. I could (possibly) show you (a version of) my derivation, but it would be rather different from the solution you've been given.
 

1. What is the concept of the Lorentz Transformation?

The Lorentz Transformation is a mathematical formula used in special relativity to describe how measurements of space and time are affected by an observer's motion relative to an object or event. It was developed by Dutch physicist Hendrik Lorentz and later modified by Albert Einstein.

2. How does the Homogeneity of Spacetime relate to the Lorentz Transformation?

The Homogeneity of Spacetime refers to the idea that the laws of physics are the same for all observers in uniform motion. This means that the basic principles of physics, such as the speed of light, are constant and do not change based on an observer's perspective. The Lorentz Transformation is derived from this concept, as it allows for measurements to be consistent for all observers.

3. Can you explain the mathematical derivation of the Lorentz Transformation?

The mathematical derivation of the Lorentz Transformation involves using the Pythagorean theorem and the principle of relativity to derive a set of equations that describe how measurements of time and space are affected by an observer's relative motion. This includes the famous equation E=mc^2, which relates energy and mass.

4. Why is the Lorentz Transformation important in physics?

The Lorentz Transformation is important in physics because it allows for the prediction and understanding of how objects and events behave in space and time. It is a fundamental concept in special relativity and has been used in numerous experiments and theories, including Einstein's theory of general relativity.

5. Are there any real-world applications of the Lorentz Transformation?

Yes, the Lorentz Transformation has many real-world applications, particularly in fields such as astrophysics and particle physics. It is used to calculate the effects of time dilation and length contraction in high-speed objects, and has been tested and confirmed through experiments such as the famous Michelson-Morley experiment.

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