Euclidean Relativity

July 9,2021

Many texts on relativity often present it in a way that makes it seem unintuitive and difficult to visualize . However, we can easily formulate relativity in terms of ordinary Euclidean geometry.

An easy way to see this is by starting with the Minskowski approach and switching the roles of time and proper time to get $$t^2 = \tau^2 + x^2$$ (using natural units $$c=1$$). This means we now treat proper time as the 4th coordinate rather than common time. This approach is also sometimes referred to as Proper Time Geometry or Circular Function Geometry .

This approach has several advantages, the first being that on a diagram, time is now conveyed by the length of an actual line, rather than by an arbitrary parameter, making it easier to visualize. It also allows us to extend our usual geometric intuition to explain some otherwise difficult relativistic concepts.

The goal of this article is to explore this idea in some depth and see how we can utilize this Euclidean approach to better gain insight into certain aspects of relativity. I will start by first looking at the postulates, then show how we can use this geometry to understand various ideas in relativity. The aim is to build an intuitive picture of spacetime.

The postulates and geometry

To build this intuitive picture of spacetime, We can start by first seeing how Einstein's two postulates can be related directly to statements in Euclidean geometry.

Einstein's first postulate says that the laws of physics in one inertial frame are the same in any other inertial frame. This means that the motion of any objects described in one inertial frame can be equally so described by any other inertial.

In particular, this means that for any object there is always some reference frame in which the object can be regraded as stationary . We call such a frame the proper frame of the object. For example: As we are moving along with the earth, we can say we are in its proper frame.

We will identify proper frames with vertical lines in our coordinate system, and thus proper time will be measured by the length of such lines (in appropriate units).

Einstein's second postulate says that there is a maximum speed limit that is the same for all inertial frames. This speed is called the speed of light , as light was the first thing we discovered that could travel at this speed.

This postulate implies two things. Firstly it gives us a way to measure distance. As the speed of light is a constant independent of any motion, we can simply infer distances from the time taken for light to travel between two points.

Secondly this also suggests a way for us to synchronize two distant clocks.We can use the fact that both clocks already agree on the speed of light.If two clocks A and B exchange light signals, then we can say that they are synchronized if their time values are set such that the time from A to B is the same as the time from B to A. This method of synchronizing clocks is called the Poincaré-Einstein method.

Diagram of Euclidean Spacetime

On a Euclidean diagram we draw lines representing the trajectory an object moves from different points of view with respect to an inertial reference frame of reference.

Let's say we have a rocket moving at a certain velocity away from Earth. We want to compare time values measured from the point of view of Earth with those measured inside the rocket itself. We place clock A on the surface of the earth and clock B inside the rocket.

To construct our diagram we draw lines representing the motion of the rocket relative to each clock. Since Clock B is inside the rocket, there is no motion relative to the rocket so we represent its path by a vertical line. Clock A,on the other hand, is on earth, so the rocket is moving away from it. We represent this path as a diagonal line.

Furthermore, we assume we can compare times by connecting the paths via horizontal lines. This assumption is justified by the fact that we can use light beams to synchronize two clocks.

4D Euclidean Vectors

The Euclidean approach allows us to formalize the intuitive idea that there is a trade off between speed through space and speed through time , i.e. the faster you move through space the slower you must move through time. We can see this by first defining the velocities. $$v$$ is velocity through space and $$\alpha$$ is the velocity through time .

$$v = \frac{dx}{dt}$$ $$\alpha = \frac{d\tau}{dt}$$
We can easily show that
$$\left( \frac{dx}{dt} \right) ^2 + \left( \frac{d\tau}{dt} \right) ^2 = 1$$
and thus
$$v^2 + \alpha ^ 2 = 1$$

We can define $$\left( \alpha, v \right)$$ as the Euclidean 4-velocity an equivalent to the usual 4-velocity of Minkowski space, but with a positive definite $$\left( ++++ \right)$$ metric, as opposed to the usual $$\left( -+++ \right)$$ metric of Minkowski space.

We can similarly define Euclidean 4-vectors for other quantities.

A Trigonometric Approach

We can define an angle $$\Phi$$ such that the velocity $$v = \sin \Phi = 0$$. We can call such an angle the aberration angle . Because as it turns out this angle does have a physical interpretation in stellar aberration. As an object changes velocity by an amount $$v$$, the angular position of a distant star in the night sky of the object changes by precisely this angle.

On the diagram this represents the angle by which an object's motion is tilted towards the x axis. $$\Phi = 0^\circ$$ for stationary objects and $$\Phi = 90^\circ$$ for objects traveling at the speed of light.

From the definition we can see the velocity through time $$\alpha = \cos \Phi$$. We can also define another kind of velocity $$u = \tan \Phi$$. This is usually called the proper velocity . The proper velocity doesn't actually represent the velocity of anything. It is more useful to think of it as a ratio between the two velocities. It is useful for constructing other important quantities like energy and momentum.

Writing these out we have:

$$v = \sin \Phi$$ $$\alpha = \cos \Phi$$ $$u = \tan \Phi$$

Dynamics: The Energy-Dispersion Relation

In natural units it becomes clear that the important dynamical quantities mass, energy and momentum, all have the same units in relativity. This means we can sometimes interchange these concepts. Historically, this has lead to a wide variety of definitions like relativistic mass, transverse mass, proper momentum etc. However in recent times it has become increasingly clear that a better approach is to give a simple suitable definition for each quantity, and find a useful relation between them. This relation is usually called the Energy Dispersion Relation or the Energy Momentum Relation .

The simplest way to define mass is as an invariant quantity having no dependence at all on motion. It can be defined exclusively from within the proper frame of an object. This fits nicely with our intuition of mass being a quantity of matter.

Since we can always measure mass from within the proper frame of an object, we can represent it with vertical lines on a Euclidean diagram, in much the same way we did for proper time.

Our second dynamical quantity is momentum. We can define momentum as a quantity of motion, that does not exist for objects that are at rest. It can only be increased by applying forces to an object. Secondly we will require that momentum be conserved in all interactions.

The usual Newtonian definition of momentum satisfies the first condition, but fails to be conserved within the theory of relativity. For example: consider the case of lightbeam striking against a mirror...

If we replace velocity with proper velocity, we get a quantity that is conserved. Thus we will define momentum as $$p = mu$$. We can also write this in trigonometric form $$p = m\tan\Phi$$ .

Our third dynamical quantity energy, can be defined by two requirements. Firstly it must be conserved. Secondly it must satisfy the Work-Energy theorem. The Work-Energy theorem is the requirement that in an interaction, the amount of energy exchanged (usually called work), is proportional to the Force times the distance travelled. More generally, if we push an object along a continuous path. We can say the work done is the integral along that path.

And since force is just the rate of change of momentum. We can write the Work-Energy Theorem as follows.

$$E = \frac{dp}{dt} dx$$

To derive a formula for energy, I will make use of the trigonometric formulas derived early. From this, the Energy-Dispersion relation will emerge by means of a simple trigonometric identity.

First let's substitute the following into the Work-Energy formula:

$$dp = d \left(m \tan\Phi \right) = d\sec^2\Phi d\Phi$$ $$dx = \sin\Phi dt$$

We should get

$$E = m\int \sec^2 \Phi \sin \Phi d \Phi$$
Simplify
$$E = m\int \sec \Phi \tan \Phi d\Phi$$

We can solve this by simply recognizing the expression in the integral is just the derivative of $$\sec \Phi$$. Thus we obtain

$$E = m\sec \Phi d\Phi$$

And this is our formula for Energy. We can now derive the Energy dispersion relation by using the trig identity $$\sec^2 \Phi = 1 + \tan^2 \Phi$$.

$$m^2\sec^2 \Phi = m^2 + m^2\tan^2 \Phi$$ $$E^2 = m^2 + p^2$$

This is the Energy dispersion relation. A geometric relation between the three important dynamical quantities. From the relation we can see the special cases. A particle at rest will have an energy equal to its mass $$E = m$$. This is Einstein's mass Energy equivalence.

De Broglie Matter Waves

If we take seriously the idea that not only does all matter travel through spacetime at the same speed of light, but that all matter behaves in exactly the same way as light, we can understand intuitively one of the most important ideas in the history of science. According to De Broglie's theory, all matter behave like waves, in the same way light does.

De Broglie's matter waves are deeply connected with the theory of relativity, and in particular,with the Energy-dispersion relations derived earlier.

Einstein had already worked out, in 1905, the idea that light comes in discrete energy quanta. We now call these photons. This was based on Planck's theory of radiation. The Energy of this light is proportional to the frequency. $$E = \hbar \omega$$. Here $$\omega$$ is the frequency in radians per second, and $$\hbar$$ is Planck's reduced constant. Planck's original constant $$h$$ was defined for frequencies measured in Hertz.

De Broglie realized that the same could also hold for particles. A stationary particle with mass $$m$$ has energy.

Acceleration and Non-inertial frames

In an accelerating frame of reference things become a bit more tricky, as objects are constantly changing inertial frames. This means the postulates we derived do not always apply. In general, we need to assume at least one of the clocks are in an inertial frame for comparison to work.

Uniform circular motion

The simplest case study of acceleration is that of an object in uniform circular motion. If we consider the proper time axis on diagram, this motion is described by a helix through spacetime. From the point of view of the object's own proper frame however, the object is stationary and described by a vertical line. Without any calculations, we can see that this helix is longer than the line. This means a clock placed at the center of the circular motion, would observe a longer time interval, than a clock in circular motion.

To calculate precisely how much time dilation is experienced by the object, we would simply calculate the length of the helix, and divide it by the proper time. The formula for the length of a helix is given by the formula

$$t = \sqrt{c^2 + h^2}$$

Where $$c$$ is the circumference and $$h$$ is the height of the helix. In this case $$h = \tau$$ and $$c = \omega^2 r$$, where $$r$$ is the radius, and $$\omega$$ is the angular velocity measured in the proper frame of the object.

$$t = \sqrt{\tau^2 + (\omega^2r)^2}$$