# Lorentz Transformation: Wave Equation vs. Interval Invariance

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• jk22
In summary, the Lorentz Transformation is a set of equations used in special relativity to relate measurements of space and time between different frames of reference. It can be used to derive the wave equation for electromagnetic waves and maintains interval invariance, a fundamental principle in special relativity. This allows for the prediction of phenomena such as time dilation and length contraction. However, the Lorentz Transformation is only applicable to systems with constant velocities and may not accurately describe other physical laws.
jk22
To find the Lorentz transformation, should it start with the invariance of the wave-equation ?

If so, then it gives 5 equations, 2 of them being wave-equations again.

If however the invariance of the space-time interval is demanded only 3 quadratic equations come out.

Which way should be taken to start relativity ?

I don't believe Lorentz started with either of these approaches. Further, can you provide more detail on what you mean by each alternative. The way I would do either approach, I do not get the number and type of equations you claim.

jk22 said:
To find the Lorentz transformation, should it start with the invariance of the wave-equation ?

If so, then it gives 5 equations, 2 of them being wave-equations again.

If however the invariance of the space-time interval is demanded only 3 quadratic equations come out.

Where are you getting this from? Can you give references?

I mean in formulas the first way were :

$$\frac{\partial^2 f}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2 f}{\partial t^2}=0\\=\frac{\partial^2 f}{\partial x'^2}\left[\left(\frac{\partial x'}{\partial x}\right)^2-\frac{1}{c^2}\left(\frac{\partial t'}{\partial t}\right)^2 \right]\\-\frac{1}{c^2}\frac{\partial^2 f}{\partial t'^2}\left[-c^2\left(\frac{\partial t'}{\partial x}\right)^2+\left(\frac{\partial t'}{\partial t}\right)^2 \right]\\+2\frac{\partial^2 f}{\partial x'\partial t'}\left[\frac{\partial x'}{\partial x}\frac{\partial t'}{\partial x}-\frac{1}{c^2}\frac{\partial x'}{\partial t}\frac{\partial t'}{\partial t}\right]\\+\frac{\partial f}{\partial x'}\left[\frac{\partial^2 x'}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2 x'}{\partial t^2}\right]\\+\frac{\partial f}{\partial t'}\left[\frac{\partial^2 t'}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2 t'}{\partial t^2}\right]$$

as it can be seen the two last equations are wave equations for the change of coordinates.

For example the two last equation would imply :

$$x'=ax+bt+f(x-ct)+g(x+ct)\\ t'=dx+et+h(x-ct)+k(x+ct)$$

So is there any hope that the coordinate transformation would allow to know what happens at the speed of light in vacuum $c$, as it was questioned by Einstein at his epoch (I remember having read that but I could not find where again, because Lorentz transformation are diverging at that speed) ?

jk22 said:
I just start from vague laws claimed by physicists during history

"Vague laws" are not a good starting point. Also Wikipedia is not a good source, you need to be looking at textbooks or peer-reviewed papers that specifically talk about how the Lorentz transformations can be derived and from what axioms. Also that Wikipedia article is very long and I don't see anything in it that corresponds to the claims you made in your OP.

jk22 said:
I mean in formulas the first way were

Where are you getting all this from?

From the invariance of the wave-equation, by using the chain rule.

jk22 said:
From the invariance of the wave-equation, by using the chain rule.

In other words, you don't have five equations. You just have one. "Five equations" would mean five independent equations, none of which can be derived from any of the others.

You seem to have a fundamental confusion about how to count "equations" and what it means to "derive" the Lorentz transformations. Searching PF should turn up some good past discussions on this topic. I would strongly recommend checking them out and also looking at the literature on different ways of deriving the Lorentz transformations from particular sets of axioms. That way you will be able to start a new thread with a better basis for discussion.

In the meantime, this thread is closed.

## 1. What is the Lorentz transformation?

The Lorentz transformation is a mathematical equation that describes how the measurements of space and time change between two reference frames that are moving at a constant velocity relative to each other. It was developed by Dutch physicist Hendrik Lorentz in the late 19th century and later refined by Albert Einstein in his theory of special relativity.

## 2. How does the Lorentz transformation relate to the wave equation?

The Lorentz transformation is closely related to the wave equation, which describes how waves propagate through space and time. The Lorentz transformation is used to calculate the effect of relative motion on the properties of a wave, such as its frequency and wavelength. This is important in understanding the behavior of electromagnetic waves, which are governed by the wave equation.

## 3. What is the significance of interval invariance in the Lorentz transformation?

Interval invariance is a fundamental concept in the Lorentz transformation. It means that the interval between two events in space and time remains the same for all observers, regardless of their relative motion. This is a key principle of special relativity and is essential in maintaining the consistency of physical laws across different reference frames.

## 4. How does the Lorentz transformation impact our understanding of space and time?

The Lorentz transformation has had a profound impact on our understanding of space and time. It showed that the traditional concepts of absolute space and time, as described by Isaac Newton, were not accurate and that the laws of physics are the same for all observers in uniform motion. This led to a major shift in our understanding of the nature of space and time and paved the way for the development of modern physics.

## 5. Can the Lorentz transformation be applied to all types of motion?

Yes, the Lorentz transformation can be applied to all types of motion, including linear, rotational, and accelerated motion. However, it is most commonly used to describe the effects of relative motion between two objects moving at a constant velocity. For more complex motions, the equations of general relativity, which were also developed by Einstein, are needed to accurately describe the effects of gravity and acceleration.

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