Lorentz, waves, Einstein and bodies: transformations +/- gamma

In summary, the conversation discusses the use of Lorentz transformations in relation to a waveform and the effect of the gamma factor on the transformations. The speaker concludes that the transformations without the gamma factor are sufficient for transforming the waveform, but the gamma factor is necessary for other applications. The determinant of the transformation matrix is also mentioned as not being equal to 1.
  • #1
ANvH
54
0
I am using a wikipedia page, Derivation of the Lorentz transformations and a lot of historical papers. To follow through I came up with my own transformations that do not contain the gamma factor:

##x^{'}=x-\beta ct##​
##t^{'}=t-\beta \frac{x}{c}##​

When applying them to a waveform

##\omega t^{'}-kx^{'}=(1+\beta)\omega t-(1+\beta)kx##​

The speed of the wave is then
##u=\frac{\omega}{k}\frac{1+\beta}{1+\beta}=c##​

So for a waveform the above transformations suffice given the speed of the Doppler shifted wave is equal to the non Doppler shifted wave. However, using the wikipedia page: http://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations, the transformations need the gamma factor when I am following through with the above equations. For the waveform this will increase the Doppler shifts:

##\omega t^{'}-kx^{'}=\gamma (1+\beta)\omega t-\gamma(1+\beta)kx##​

The speed of the wave is then
##u=\frac{\omega}{k}\frac{\gamma(1+\beta)}{\gamma(1+\beta)}=c##​

I would conclude that the transformations without the gamma factor is sufficient to transform the waveform. The transformations with the gamma factor is apparently necessary when the waveform is not utilized to test the validity of the transformations, but comparing the above with the reasoning in Wikipedia seems to be confusing.

What am I thinking wrong?
 
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  • #2
Note that the determinant of your transformation is not equal to 1.
 
  • #3
Lacking the gamma, your transformation includes a scale transformation. It preserves null vectors but does not preserve the length of spacelike or timelike four-vectors. Especially it does not preserve the rest mass of a particle.
 
  • #4
robphy said:
Note that the determinant of your transformation is not equal to 1.

Ok, the Lorentz transformation matrix

\begin{align}
A=\gamma\begin{pmatrix}1 & -\beta/c\\ -\beta c & 1\end{pmatrix}
\end{align}

gives a determinant

det##A=\gamma(1-\beta^{2})=\frac{(1-\beta)(1+\beta)}{\sqrt{(1-\beta)(1+\beta)}}=\sqrt{(1-\beta)(1+\beta)}##
that does not lead to unity either. Please enlighten.
 
  • #6
Bill_K said:
det##A=\gamma^2(1-\beta^{2})##

gosh..., thanks.
 

1. What is the Lorentz Transformation and how does it relate to Einstein's theory of relativity?

The Lorentz Transformation is a mathematical formula used to describe how measurements of space and time change for different observers moving at constant velocities. It is a key concept in Einstein's theory of relativity, as it allows for the consistency of physical laws regardless of the observer's frame of reference.

2. What are waves and how are they related to the concept of energy?

Waves are a type of disturbance that travels through a medium or space, carrying energy with it. In physics, energy is defined as the ability to do work, and waves have the ability to transfer energy from one place to another. This makes them an important concept in understanding the behavior of matter and energy.

3. How does Einstein's famous equation E=mc² relate to the concept of mass-energy equivalence?

Einstein's equation, E=mc², states that energy (E) and mass (m) are equivalent and can be converted into each other. This means that mass can be thought of as a form of energy, and vice versa. This concept is known as mass-energy equivalence and is a fundamental principle of modern physics.

4. What is the significance of the gamma factor in the Lorentz Transformation?

The gamma factor, represented by the symbol γ, is a mathematical term used in the Lorentz Transformation to describe the time dilation and length contraction effects predicted by Einstein's theory of relativity. It is a measure of how much an object's perceived time and length change as it moves at high speeds relative to an observer.

5. How do the Lorentz Transformation and gamma factor impact our understanding of the behavior of bodies in motion?

The Lorentz Transformation and gamma factor have major implications for our understanding of the behavior of bodies in motion, especially at high speeds. They help explain phenomena such as time dilation, length contraction, and the constancy of the speed of light. These concepts have been confirmed by numerous experiments and are essential for understanding the structure of the universe.

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