# An "ether-based" introduction into the theories of relativity

Here I will start to develop an introduction of relativity. It differs from all other introductions into relativity because it teaches not only the classical spacetime interpretation of special and general relativity, but also the Lorentz ether interpretation of relativity. The Lorentz ether is well-known only as an interpretation of special relativity, but can be easily extended also to the Einstein equations of general relativity.

One less well-known (in comparison with his work on the Bell inequalities) paper of Bell was "How to teach special relativity", a remarkable paper where he told about his "impression that those with a more classical education, knowing something of the reasoning of Larmor, Lorentz, and Poincare, as well as that of Einstein, have stronger and sounder instincts", and has argued that to teach this classical thinking would be helpful. His point was that "we need not accept Lorentz's philosophy to accept a Lorentzian pedagogy".

Here we will follow this basic idea, and present an introduction into relativity which introduces as the canonical spacetime interpretation, but also the Lorentz ether interpretation. It teaches those parts shared by above interpretations as well as their differences. In the case of general relativity, these differences appear much more important, and the two different interpretations of the Einstein equations become, in fact, different physical theories. What makes them different is that the spacetime interpretation is a background-independent theory, while the ether interpretation defines a theory with a classical Newtonian background.

This background cannot be identified with our measurement devices - this is the effect of relativistic symmetry. But does it follow that this background does not exist? This is an interesting question, and a question worth to be discussed. But most of the physics of relativity, special as well as general, does not depend on the answer to this question. So, one can learn the physics of relativity without making a decision how to answer this question.

# Special Relativity

We start with a simple property of a quite general wave equation, namely that one can construct, for a given solution of the wave equation, other, Doppler-shifted solutions of the same wave equations:

## How to construct Doppler-shifted solutions of a wave equation

The wave equation is the equation $\square u(\vec{x},t) = \left(\frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \nabla^2\right) u(\vec{x},t) = 0$ The operator $$\nabla^2$$ is the Laplace operator, which acts on the spatial variables, so that in the three-dimensional case we have $\nabla^2 = \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right).$ The constant c is the speed of the wave. In relativity, c is the speed of light waves, but such a wave equation can be used also to describe sound waves, or waves on the surface of water.

Let's assume that we have found a solution of this equation u(x,t). Then there exists a surprisingsly simple method to construct other solutions of the same wave equation. We can choose an arbitrary parameter $$|v| < c$$, and define the following coordinates:

$t'=\gamma \left(t-{\frac {v}{c^{2}}}\,x\right),\quad x'=\gamma (x-v\,t),\quad y'=y,\quad \ z'=z, \quad \text{with}\quad\gamma ={\frac{1}{\sqrt{1-v^{2}/c^{2}}}}.$

Then, all we have to do is to replace in the solution $$u(x,y,z,t)$$ all the $$x,y,z,t$$ by $$x',y',z',t'$$, and use the formula above to obtain another, different function $$u_v(x,y,z,t)$$: $u_0(x,y,z,t) \to u_v(x,y,z,t) = u_0(x'(x,y,z,t),y'(x,y,z,t),z'(x,y,z,t),t'(x,y,z,t))$ This function $$u_v(x,y,z,t)$$ is also a solution of the same wave equation. You can simply try it out. $\frac{\partial}{\partial t} u_v = \frac{\partial t'}{\partial t} \frac{\partial}{\partial t'} u_0 + \frac{\partial x'}{\partial t} \frac{\partial}{\partial x'} u_0 = \gamma \left(\frac{\partial}{\partial t'} -v \frac{\partial}{\partial x'}\right) u_0, \qquad \frac{\partial^2}{\partial t^2} u_v = \gamma^2\left( \frac{\partial^2}{\partial t'^2}-2v\frac{\partial}{\partial t'}\frac{\partial}{\partial x'} + v^2 \frac{\partial^2}{\partial x'^2} \right)u_0$ $\frac{\partial}{\partial x} u_v = \frac{\partial t'}{\partial x} \frac{\partial}{\partial t'} u_0 + \frac{\partial x'}{\partial x} \frac{\partial}{\partial x'} u_0 = \gamma \left(-\frac {v}{c^{2}}\frac{\partial}{\partial t'} + \frac{\partial}{\partial x'}\right) u_0, \qquad \frac{\partial^2}{\partial x^2} u_v = \gamma^2\left( \frac {v^2}{c^{4}}\frac{\partial^2}{\partial t'^2}-2\frac {v}{c^{2}}\frac{\partial}{\partial t'}\frac{\partial}{\partial x'} + \frac{\partial^2}{\partial x'^2} \right)u_0$ $\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} \right)u_v = \gamma^2 \left(\left(1 - \frac {v^2}{c^{2}}\right) \frac{1}{c^2} \frac{\partial^2}{\partial t'^2} + \frac{v^2 -c^2}{c^2} \frac{\partial^2}{\partial x'^2} \right) u_0 = \left(\frac{1}{c^2}\frac{\partial^2}{\partial t'^2} - \frac{\partial^2}{\partial x'^2} \right)u_0 = 0.$

The transformation of the coordinates which we have used here to create the new solution is, in the case of c being the speed of light, named Lorentz transformation. For other wave equations, like sound waves or water waves, the name "Lorentz transformation" is not used, and the Lorentz transformations are seldom used. But, nonetheless, the mathematics of the Lorentz transformation works in the same way for these equations too.

This new solution of the same wave equation is known as the Doppler-shifted solution. It has a well-defined physical meaning: If the source of the wave is, in the original solution, at rest, then the Doppler-shifted solution is the solution where the same source is moving with the velocity v.

Note that above solutions are clearly physically different. The source is, in the first solution, at rest, while it moves in the second solution. And if the first solution has spherical symmetry. The second solution does no longer have such a symmetry: If the train moves in your direction, you hear a different sound than if the train moves away. Despite these differences, some properties remain unchanged an symmetric - namely the speed of the wave.

## Relativistic denotations

It seems time to introduce a lot of relativistic conventions, conventions which allow to write memorize quite short formulas for otherwise quite long and boring sets of equations.

The first idea is to handle time - an a priori something qualitatively quite different from spatial coordinates - like a spatial coordinate. This coordinate is denoted $$x^0=ct$$. The introduction of the factor c into the definition of $$x^0$$ allows to get rid of a lot of factors c in the formulas, and to recover them correctly all one has to remember is that formula $$x^0=ct$$. The spatial coordinates obtain indices from 1 to 3. A typical relativistic formula contains the space and time coordinates in an equivalent way, thus, is a formula for some "four-dimensional spacetime". But there will be, often enough, also formulas which are not fully relativistic, formulas which handle spatial indices separately. Now, there is a convention which handles this: Whenever an index runs from 0 to 3, thus, is a spacetime index, one uses a greek letter to denote it. If the index is, instead, a spatial index only, one uses latin indices for it. So, one can write a function which depends on all four coordinates as $$f(x^\mu)$$, or as $$f(x^0, x^i)$$.

A nice convention if one considers transformations of coordinates, like the Lorentz tranformation, is to use the same index for the new coordinates, only primed. So, one writes the quite short formula $$x^{\mu'} = x^{\mu'}(x^\mu)$$ to describe the four new coordinates, as functions depending on the four old coordinates. The rule how to transform partial derivatives between different coordinates are, obviously, the following: $\frac{\partial}{\partial x^{\mu}} = \sum_{\mu'=0}^3 \frac{\partial x^{\mu'}}{\partial x^{\mu}}\frac{\partial}{\partial x^{\mu'}}, \qquad \frac{\partial}{\partial x^{\mu'}} = \sum_{\mu=0}^3 \frac{\partial x^{\mu}}{\partial x^{\mu'}}\frac{\partial}{\partial x^{\mu}}.$

The next simplification is to write partial derivatives $$\frac{\partial}{\partial x^\mu}$$ simply as $$\partial_\mu$$. This gives: $\partial_{\mu} = \sum_{\mu'=0}^3 \frac{\partial x^{\mu'}}{\partial x^{\mu}}\partial_{\mu'}, \qquad \partial_{\mu'} = \sum_{\mu=0}^3 \frac{\partial x^{\mu}}{\partial x^{\mu'}}\partial_{\mu}.$

Above formulas are sufficiently easy to remember, and have something in common: The index of the summation appears two times - once as an upper index, once as a lower index. This rule applies so often, that it can be considered as a general rule: Whenever there is a summation, it has to be over one upper and one lower index. And whenever there is an index used twice, as an upper index as well as a lower index, there is summation over this index. The second part became simply a convention, so that the summation sign can be omitted if the index is used twice, as an upper and a lower index. So, the formula becomes: $\partial_{\mu} = \frac{\partial x^{\mu'}}{\partial x^{\mu}}\partial_{\mu'}, \qquad \partial_{\mu'} = \frac{\partial x^{\mu}}{\partial x^{\mu'}}\partial_{\mu}.$

In relativistic physics, there are a lot of objects with various indices, and for each index one has to remember how it has to be transformed if the coordinates are transformed. The formula above decribes two possibilities - one which uses the partial derivatives of the new coordinates as functions depending on the old coordinates, and one which uses the partial derivatives of the old coordinates as functions of the new coordinates. Above four times four matrices are inverse to each other. To make remembering easy, there has been a simple convention: the upper vs. the lower position of the index defines which is the transformation rule. And which rule has to be used corresponds to the rule above: the sum has to be about one upper and one lower index. So, a lower index transforms in the same way as the partial derivatives. Say, for some $$a_\mu$$ we obtain the transformation rule: $a_{\mu} = \frac{\partial x^{\mu'}}{\partial x^{\mu}}a_{\mu'}, \qquad a_{\mu'} = \frac{\partial x^{\mu}}{\partial x^{\mu'}}a_{\mu}.$ Instead, if we have some upper index, we have to use the other, inverse rule: $a^{\mu} = \frac{\partial x^{\mu}}{\partial x^{\mu'}}a^{\mu'}, \qquad a^{\mu'} = \frac{\partial x^{\mu'}}{\partial x^{\mu}}a^{\mu}.$

## The Lorentz group

With these denotations, let's rewrite now the operator $$\square$$ for the wave equation: $\square = \partial_0^2 - \sum_i \partial_i^2 = \eta^{\mu\nu}\partial_\mu \partial_\nu$ where $$\eta^{\mu\nu}$$ is a $$4\times 4$$ matrix where only the diagonal entries are nonzero, and have the values $$\eta^{00}=1, \eta^{ii}=-1$$. Now, let's assume we have a general coordinate transformation which is linear in the coordinates, thus, $$x^{\mu'} = a^{\mu'}_\mu x^\mu$$. This gives $$\partial_{\mu} = \frac{\partial x^{\mu'}}{\partial x^{\mu}}\partial_{\mu'} = a^{\mu'}_\mu \partial_{\mu'}$$ and $\square = \eta^{\mu\nu}\partial_\mu \partial_\nu = \left(\eta^{\mu\nu} a^{\mu'}_\mu a^{\nu'}_\nu \right) \partial_{\mu'}\partial_{\nu'} = \eta^{\mu'\nu'}\partial_{\mu'}\partial_{\nu'}.$ Thus, the property of the Lorentz transformation we need to obtain another solution of the same wave equation is $\eta^{\mu\nu} a^{\mu'}_\mu a^{\nu'}_\nu = \eta^{\mu'\nu'}.$ These special coordinate transformations form a group: We can apply two such transformations, one after the other, and the result would be the same, yet another solution of the wave equation. And we can invert it, to get the original solution back - and the inverted transformation would be also, yet, another example of such a transformation. This group is named the Lorentz group.

## An important difference: Active vs. passive coordinate transformations

The mathematics of coordinate transformations can be used in two from a physical point of view very different ways.

The first way is named passive or alias coordinate transformations. In this case, one and the same physical solution is described in different ways, using different coordinates. This is nothing but an application of pure mathematics, without any physical importance. If the mathematics are used correctly, it does not matter at all which system of coordinates you use to describe the solution. Your choice of coordinates is arbitrary. You can check this, by trying to compute something measurable using different coordinates. The final result should be the same. If not, you have made a mathematical error. But you are not obliged to do such things at all. You can, as well, choose one system of coordinates, and refuse even to look at any other one, but you are nonetheless able to compute everything physical using only that single system of coordinates.

The second way is named active or alibi coordinate transformation. In this case, the coordinates remain unchanged. What changes is the solution. This is what we have used here to obtain a new, Doppler-shifted solution of the wave equation out of a given one.

Note the difference: We could have used any other coordinate transformation to get another description of the same solution. If that other coordinate transformation would not be a Lorentz transformation, we would have to rewrite the same wave equation in these other coordinates. It would have been a different-looking equation, but the different-looking solution of the different-looking equation would have been simply another, equally valid, description of the same solution of the same equation.

Instead, a different coordinate transformation would not have allowed us to construct a new, different solution of the same equation. For this trick, it was essential that the transformed equation looked, by accident, like the original equation. In general, the tranformation would have given us some solution of some other equation, something in no way useful to study the solutions of the wave equation. But in our case, the Lorentz transformation has given us, by happy accident, another, new solution, with different properties, on the same wave equation.

Could we have used the Lorentz transformations to describe the same wave equation in other coordinates? Yes, of course. But these other coordinates would have been a quite unnatural, strange choice. Of course, one is free to use whatever coordinates one likes. But, in the case of water waves, would be the point of using a "time coordinate" which have nothing to do with real time?

So, once we have understood this important difference between active (alibi) and passive (alias) transformations, we have found a useful application of the Lorentz transformations as active transformations, transformations which create new, different solutions of the wave equation.

## Other equations where Lorentz transformations allow to find new solutions

Once the Lorentz transformations allow to create new solutions of the wave equation, another question appears: Are there other interesting equations with the same property, so that we can use Lorentz transformations to find new, different solutions? The answer is positive. There are a lot of other, more complex but very interesting equations where the Lorentz transformations allow to find new solutions.

### Mass terms

First of all, one can add a mass term. This gives a Klein-Gordon equation, which describes a massive scalar particle: $\frac{\partial^2}{\partial t^2} u(\vec{x},t) - c^2 \nabla^2 u(\vec{x},t) + \left(mc^2\right)^2 u(\vec{x},t) = 0.$

In the more economic relativistic denotations, with the units taken in such a way that $$c=1$$, this equation looks like $\square u(x^\mu) + m^2 u(x^\mu) = 0.$ This equation is also linear. The solutions of this equation are waves, but their velocity is already lower than c.

### Interaction terms of different fields

The field itself is not obliged to be a simple real field, it can have many components. And these many components can interact in quite complex ways. All one needs to preserve the property that the Lorentz transformation of a solution gives another solution is that the interaction term does not contain any spatial derivatives. Thus, the general form of the equation would be the following: Some number of fields $$u^\alpha(x^\mu)$$ and the following equation: $\square u^\alpha(x^\mu) + V^\alpha(u^\beta(x^\mu)) = 0.$ These interaction terms make the system of equations itself nonlinear. This makes it usually difficult, or even impossible, to find exact solutions. But, despite this, the basic property of the Lorentz transformation remains - if we have an exact solution, the Lorentz transformation creates a new, different solution of the same equation.

### The electromagnetic field

Another very interesting example is the electromagnetic field. The simplest way to describe it, and to see how the Lorentz transformation can be used, is to consider the equation for the electromagnetic potentials $$\phi(x,t), \vec{A}(x,t)$$ in a gauge condition $\frac{1}{c}\frac{\partial}{\partial t} \phi + \frac{\partial}{\partial x} A^x +\frac{\partial}{\partial y} A^y + \frac{\partial}{\partial z} A^z = 0,$ which is named Lorenz gauge. Note that it is not "Lorentz gauge", because it is not named after Hendrik Lorentz, but after another physicist, Ludvig Lorenz.

In relativistic denotations, the electric scalar potential $$\psi$$ and the magnetic vector potential $$\vec{A}$$ are combined into an electromagnetic four-potential $$A^\mu$$, with $$A^0=\phi$$. Then, the Lorenz gauge obtains the much simpler form $\partial_\mu A^\mu = 0.$ Let's first check that the Lorenz gauge remains the Lorenz gauge condition We also have to check what happens with the Lorenz gauge during the Lorentz transformation: $\partial_{\mu'} A^{\mu'} = \left(\frac{\partial x^{\mu}}{\partial x^{\mu'}}\partial_\mu\right) \left(\frac{\partial x^{\mu'}}{\partial x^{\mu}} A^\mu\right) = \left(\frac{\partial x^{\mu}}{\partial x^{\mu'}}\right) \left(\frac{\partial x^{\mu'}}{\partial x^{\mu}}\right) \partial_\mu A^\mu = \partial_\mu A^\mu.$ Here we have used that the coefficients $$\frac{\partial x^{\mu'}}{\partial x^{\mu}}$$ are constants, so can be taken out of the partial derivative, and that $$\frac{\partial x^{\mu'}}{\partial x^{\mu}}$$ is the inverse matrix of $$\frac{\partial x^{\mu}}{\partial x^{\mu'}}$$. In the Lorenz gauge, the Maxwell equations become simply wave equations for the electromagnetic potential, so that we have: $\square A^\mu(x^\nu) = 0.$ This is already a good starting point, given that we already know that the operator $$\square$$ behaves as necessary. But the Lorentz transformation of this equation also has to do a transformation of the vector index. So, the result of the Lorentz transformation will be a solution of the follwing equation: $\square A^{\mu'} = \square \left(\frac{\partial x^{\mu'}}{\partial x^{\mu}} A^\mu\right) = \left(\frac{\partial x^{\mu'}}{\partial x^{\mu}}\right) \square A^\mu(x^\nu) = 0.$ This is, fortunately, also equivalent to the original equation. Again, we can take the constant coefficients out of the differential operator. And the resulting system of equations is equivalent to the original one, because the matrix $$\frac{\partial x^{\mu'}}{\partial x^{\mu}}$$ is invertible, with $$\frac{\partial x^{\mu}}{\partial x^{\mu'}}$$ being the inverse matrix. So, the system is equivalent to the original system of equations $$\square A^{\mu} = 0$$.

The Maxwell equations have been, in fact, the equations where the first equations where the Lorentz transformations have been found.

### The Dirac equation

The Dirac equation is another important equation in modern physics. It is the equation used to describe fermionic fields. It is an equation for a system of four complex (or, equivalently, eight real) fields. The equation was obtained by Dirac in an attempt to take a square root out of the Klein-Gordon equation. But the Klein-Gordon equation is already among the equations for which the Lorentz transformation allows to construct new solutions. So, it is not really a surprise that this works for the Dirac equation too.

In relativistic notations, the equation looks like $i \gamma^\mu \partial_\mu \psi(x^\mu) = m \psi(x^\mu).$ The square of the operator $$\gamma^\mu \partial_\mu$$ is, by construction of the matrices $$\gamma^\mu$$, the Laplace operator $$\square$$. So, it follows from this equation that $$-\square \psi = m^2 \psi$$, thus, the equation is a square root of the Klein-Gorden equation.

It also follows immediately that the Lorentz-transformed equation fulfills the same basic property, namely that its square gives the Klein-Gordon equation. Unfortunately, this is nonetheless not exactly the same equation, but a different representation. Fortunately, all these different representations are equivalent, thus, one can find a transformation U so that $$\psi'(x^\mu) = U \psi(x^\mu)$$ is already again a solution of the Dirac equation in its original representation.

The subtle point is that this operator U is not uniquely defined. The operator -U would do it too.

### And, again, various interaction terms

What has been said about pointwise interaction terms for the scalar wave equation holds also for all the other equations considered - it is possible to add various pointwise interaction terms. The freedom of choice is somewhat restricted, not completely without any restrictions (except for containing no derivatives) as in the scalar case - as the EM field, as the Dirac operator follow some transformation laws, and these transformation laws have to fit each other.

The interaction terms which are important in the standard model of particle physics are:

1.) Non-abelian gauge fields: This is a generalization of the EM field, and formally looks like several such EM fields which additionally interact with each other.

2.) The interaction of these gauge fields with Dirac fermions. The interaction term has a quite special form, namely a replacement of the partial derivative by an additional term: $i \partial_\mu \to i \partial_\mu + g A_\mu,$ with the charge of this fermion field being g.

### The equation of the Standard Model of particle physics

So, we have found a lot of wave equations which all share the same nice property: If we have a solution of these equations, we can, using a Lorentz transformation, create new, different solutions of these equations.

How important is this class of wave-like equations? The surprising news is that all the fundamental fields, all the fields used in the Standard Model (SM) of modern particle physics, are described by such equations. And they all share the same constant c - the speed of light. In principle, this SM can be considered as a single big equation, let's name it the SM equation, containing many different parts, which interact with each other. But all these parts, and all the interaction terms, fit into the list of equations above. So that the SM equation is also of this type. If we have one solution of the SM equation, we can apply a Lorentz transformation, and will obtain another, different solution of the SM equation, a solution which will be, in comparison with the original solution, Doppler-shifted.

## Relativistic effects

What is the consequence of the fact that all fields, the whole SM, follow an equation where we can apply a Lorentz transformation to obtain a new solution of the same equation? Some surprising results about the behavior of clocks follow.

### Time dilation

Let's construct, out of what we have, namely of of things described by the SM, a clock. This clock will be described by some trajectory, with some numbers on it which denote the time measured at this point. Let's simply assume, as an example, that this trajectory is, for the initial solution with the clock at rest, a line with the spatial coordinates (0,0,0) and with a result 0 at time 0 and result 1 at time 1.

What happens now with this solution, if we apply some Lorentz transformation? We obtain another solution. This other solution describes a clock of the same construction, but moving. The Lorentz transformation is linear, thus the point $$(x^\mu)=(0,0,0,0)$$ remains the same. But the point where the clock shows 1, which was originally at $$(x^\mu)=(1,0,0,0)$$, will now be in $$(x^{\mu'})=(\gamma,-\frac{v}{c}\gamma,0,0)$$. The line remains a line, so that the clock is moving with the speed v (which is $$\frac{v}{c}$$ in terms of the time coordinate $$x^0=ct$$. But the clock shows the result 1 at real time $$\gamma$$ instead of real time 1. But we have $\gamma = \sqrt{1-\frac{v^2}{c^2}}^{-1} > 1$ for every $$0 < |v| < c$$. That means, the moving clock shows clock time 1 only at real time $$\gamma > 1$$, thus, is dilated.

### Length contraction

What happens with a ruler? The ruler at rest is some solution of the SM, but we cannot idealize it as a single point, we need two points of it. The begin we put, again, at $$(x^\mu)=(0,0,0,0)$$, the mark of the ruler with length 1 will, if we measure the x-direction, be at $$(x^\mu)=(0,1,0,0)$$. At rest, the begin moves toward $$(x^\mu)=(1,0,0,0)$$ and the mark with the 1 toward $$(x^\mu)=(1,1,0,0)$$.

We apply the same procedure and obtain a solution of the same type, thus, also a ruler, but moving. The four resulting points will be the following: The begin will be, like for the clock, moving along the line from $$(x^{\mu'})=(0,0,0,0)$$ to $$(x^{\mu'})=(\gamma,-\frac{v}{c} \gamma,0,0)$$. The point with mark 1 will, instead, move from $$(x^{\mu'})=(-\gamma \frac{v}{c},\gamma,0,0)$$ to $$(x^{\mu'})=(\gamma (1-\frac{v}{c}),\gamma(1-\frac{v}{c}),0,0)$$. Where is the mark 1 of this moving ruler at $$t=0$$? We have to compute where the line between the two points intersects the line $$t'=0$$. The result is $$(x^{\mu'})=(0,\gamma(1-\frac{v^2}{c^2}),0,0)$$. Thus, the mark 1 of the moving ruler will be, at time $$t=0$$, at a distance $$\gamma^{-1} < 1$$ from the origin. The moving ruler is shorter.

### Impossibility to identify global contemporaneity

A similar fate waits for devices which allow to measure absolute contemporaneity. Such a device would tell us that the two events $$(x^\mu)=(0,0,0,0)$$ and $$(x^\mu)=(0,1,0,0)$$ have happened at the same moment of time. But now we apply a Lorentz transformation to this device, and what is the result? The same device, only in a moving state, claims that $$(x^\mu)=(0,0,0,0)$$ and $$(x^{\mu})=(-\gamma \frac{v}{c},\gamma,0,0)$$ have happened at the same time.

### Impossibility to measure absolute rest

And now it is already clear what happens if we try to measure, with some physical device, what is absolute rest. Suppose we have such a device. This device measures that the line from $$(x^\mu)=(0,0,0,0)$$ to $$(x^\mu)=(1,0,0,0)$$ is at absolute rest. We do the same trick again, and apply the Lorentz transformation to this measurement of absolute rest. What do we obtain? A solution which describes the same measurement device, which claims that the line from $$(x^\mu)=(0,0,0,0)$$ to $$(x^\mu)=(\gamma,-\frac{v}{c}\gamma,0,0)$$ is at absolute rest. Which is, of course, wrong.

The consequence is the relativity principle: Once all our physical equations allow the application of Lorentz transformations to obtain a new solution of our equations, only moving relative to the original solution, there cannot exist a physical device which identifies the own state of movement.

## The Lorentz ether vs. Minkowski spacetime

What follows from the relativity principle?

### The Lorentz ether

In the Lorentz ether interpretation, nothing follows. Wave equations appear, most natural, as a sort of sound wave equations of some material. Once we observe something nicely described by wave equations, the reasonable hypothesis is that they are all similar to sound waves of some ether.

The relativity principle is, then, only an unfortunate consequence of the fact that all what we can actually observe are sound waves of the same ether. For usual condensed matter, there exist other things, like light rays, which are not sound waves of this condensed matter. With these additional possibilities accessible to use, the relativity principle would become invalid, and we would be able to measure as absolute rest, as absolute contemporaneity, and measure absolute lengths and time. Without them, we are simply unable to distinguish things which, in reality, are different.

In the Lorentz ether, time dilation and length contraction are distortions of our measurement devices caused by their motion relative to the ether. True distances and true time could be measured - if our measurement devices would be, by accident, really at rest. Once we cannot know if they are really at rest, one cannot be sure that our measurements are undistorted by the ether.

### The Minkowski spacetime

There is another interpretation of special relativity - the spacetime interpretation proposed by Minkowski. It is the one accepted by the mainstream of physics. At this place I have to admit that, given that I'm not a proponent of the spacetime interpretation, the argumentation in its favor will be suboptimal.

From point of view of the spacetime interpretation, once there is no possibility, by any measurement, to distinguish if we are at rest, then no such animal like "absolute rest" exists in reality. The same for absolute contemporaneity: Once there is no possibility to establish it, uniquely, by measurement, there is no absolute contemporaneity in our real world.

This is, of course, in sharp contradiction with classical common sense. Common sense has no problem with the mathematical possibility of a spacetime. But this would be simply a collection of the states of space, in various moments of absolute time. It would not be a description of what exists, but of all what has existed in the past, exists now, and will exist in the future. All this would have to be taken together, and given a common status of existence, without any reference of when it existed. This would be, in fact, a new concept of timeless existence. There is no more any difference between events which have happened and those which will happen, they all simply exist.

In this four-dimensional world there would be no present at all. What is the present which we experience? It can be only something particular, derivative, restricted to the particular trajectory in this spacetime which describes our own lifestream, our worldline, which contains all the events of our own life, from our birth to our death.

# General Relativity

To be continued ...