Based on a photo titled “Albert Einstein by Doris Ulmann”. Source: Wikimedia Commons. Public Domain.

Einstein’s Theory of Relativity

Panda the Red
Nov 11, 2019 · 18 min read

“Everything should be made as simple as possible, but no simpler.”

The special theory of relativity is without question one of the most important discoveries in the history of science and second only to Newton’s discovery of the laws of mechanics in its importance to physics. In spite of this, special relativity is poorly understood and there is an abundance of misinformation on the internet and in the media on the subject. This is not helped by a mostly undeserved reputation for being too difficult for most people to understand.

Reference frames, covariance, and Galilean relativity

The basic idea of relativity is that two different observers who are in motion relative to each other must agree on the laws of physics. When two different observers are in relative motion, they are said to be in different reference frames, and when their relative velocity is constant, those reference frames are said to be inertial. When all observers in all inertial reference frames agree on a physical theory, then that theory is said to be covariant. We will only consider inertial frames.

Source: Wikimedia Commons. Public domain.
  • Time is absolute: ΔT=ΔT′
We are ignoring the y′ and z′ coordinates for simplicity.
  • There is only one special reference frame in which Maxwell’s equations are true, namely the rest frame of the so-called luminiferous aether.
  • The Galilean transform is wrong, and therefore so are the underlying assumptions about space and time.

The Lorentz Transform and Einstein’s theory of relativity

In 1892, Hendrik Lorentz published a paper in which he showed that the transformation under which Maxwell’s equations are covariant is:

  • The speed of light has the same value in all inertial reference frames, that is, it is invariant.

Time Dilation

Let S′ be the rest frame of a train that is in standard configuration with S, the rest frame of someone standing on the platform. An experiment is carried out on the train in which some physical process takes place over a time interval Δt′. We will show that the observer on the platform will see that same physical process take place over a time interval Δt, where Δt and Δt′ are related by:

Not to scale.

Demonstration: Muon decay

A muon is a subatomic particle that is identical to an electron in every way except its mass: A muon is about 207 times heavier. The weak interaction (one of the four fundamental forces) causes a muon to decay into an electron and two other particles called an electron antineutrino and a muon neutrino:

Length Contraction

An observer at rest in frame S sees a particle with velocity Vx pass a post at point A at time t=0, and then at time Δt she sees the particle pass a post at point B, with both points on the x-axis separated by length L. The rest frame of the particle is S′, and in S′ the particle is stationary and the two posts, separated by length L′, are approaching the particle with velocity -Vx. The first post passes the particle at time t′=0 and the second passes the particle at time Δt′=Δt/γ. Since L=VΔt and L′=VΔt′, we see that L′=L/γ. This means that length is contracted in the moving frame.

The Lorentz Transform

Now we can prove that the Lorentz transform relates the coordinate systems of the two reference frames in standard configuration. We will show that:

Be aware that other conventions exist for the minus signs.

Demonstration: The classical limit

The Lorentz transform looks very different from the Galilean transform. How could physicists have been this incorrect for such a long time?

Spacetime

It cannot be emphasized strongly enough that time dilation and length contraction are properties and space and time themselves. They are not the result of forces that cause clocks to run slower depending on who’s looking at them, nor does moving at relativistic speed induce forces that stretch or compress objects. It is also not the result of a measurement error or optical illusion that causes observers in different frames to misjudge the length of an object or the rate that a clock ticks. When observers in different frames report different lengths for measuring rods or different frequencies for ticking clocks, they are all correct because lengths and time intervals are not invariant, and that’s just how space and time work.

Demonstration: Mass-energy equivalence, E=mc²

One of the most famous consequences of special relativity is that rest mass is equivalent to energy. The rest mass of a particle is its mass as measured in the frame in which the particle is not moving. This section is meant to provide a justification, though not a formal proof, for this claim.

Conclusion and copyright stuff

Any images that have not been given a citation are my own original work. Some of the examples that I used are based on examples covered in the textbook Modern Physics for Scientists and Engineers, 2nd edition by Taylor, Dubson, and Zafiratos.

Cantor’s Paradise

Medium’s #1 Math Publication!

Panda the Red

Written by

Index gymnast, operator quantizer, and lover of all things nanoscale. I am eternally trapped in the library, someone please send help.

Cantor’s Paradise

Medium’s #1 Math Publication!

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