If you move away from me at 0.6 c, and you measure something moving away from you in the same direction at 0.4 c, then I see it moving away from me at 0.806 c, rather than 0.6 c + 0.4 c = c.

On the other hand, if you see something moving away from (or toward) you at velocity w = c (or "V" in Einstein's notation) then I also get detect U = c.

That solves our receding / approaching flashlight problem. We always agree on things that move at the speed of light.

Once Einstein understood how to translate between inertial frames, he quickly revised our common-sense rules for combining velocities.

1) You move relative to me with constant velocity v

2) You see something moving in that direction with velocity w

3) I measure its velocity as U, given by:

In Einstein's presentation of these “Lorentz transformations," our relative speed appears as v, and the speed of light is denoted V.

(Nowadays we universally use c for the speed of light.)

This way of moving back and forth between "where and when" descriptions by two inertial observer insures that they both detect light propagating at the same, constant speed.

Why does this work? It's a secret symmetry hidden in Maxwell's equations for electrodynamics. It was there waiting to be found!

I actually wrote a nice long post about this the other day! How do we reconcile two different descriptions of where and when things happen, in a way that preserves important notions like causality?

In the image above, my "where and when" description t, x, y, and z for an event appears on the right-hand side of these equations. The coordinates I called t', x', y', and z' in this linked thread are the symbols on the left-hand side.

https://mastodon.social/@mcnees/111103830608682584

As a result, we begin to disagree on things that we previously took for granted. For example, I may regard two events as simultaneous, while you insist that one happened before the other.

Two observers moving relative to each other will generally give different descriptions of where and when things happen.

To reconcile their descriptions we must accept that their notions of space and time get mixed up. Einstein explains the translation between reference frames:

Your everyday experience has led to some common-sense rules for determining the relative velocity of two objects. And in everyday situations, those rules work just fine. But in this case, those rules are noticeably wrong. To fix this, we have to make some big changes.

To preserve this idea that you and I both see light as propagating at speed c, even when we are moving relative to one another, we have to toss the Newtonian idea of absolute time out the window.

Neither one of us is accelerating, so we both agree that Newton's laws describe mechanical phenomena. If you are an inertial observer then so am I. According to Einstein we therefore agree on the same laws of electrodynamics, as well.

Since we agree on the same laws of electrodynamics, we must agree on the speed of light they predict. The light from my flashlight follows these laws. You must conclude that it is catches up with you at c, rather than 0.4 c.

Now apply that same reasoning to light.

Suppose you are moving away from me at 60% of c (the speed of light). You see me receding at 0.6 c when I shine a flashlight at you. Your everyday experience with things like cars and baseballs and sound waves suggests that the light from my flashlight should “catch up” with you at 40% of c.

According to Einstein, that's *not* what happens.

Again, compare that with the propagation of sound waves in air. Once they are released from the source, their speed depends on the properties of the medium. So if a moving source emits sound waves, they still travel at the speed of sound.

On the other hand, if *you* are moving towards the source or away from it, you detect those waves as moving at a different relative velocity. Basically the speed of sound plus or minus the speed with which you are approaching or receding from the source.

The speed of light is a consequence of the laws of electrodynamics.

It had been assumed, up to that point, that these laws hold only in a special frame of reference defined by the aether. But Einstein asserts that *all* inertial observers agree on these laws.

Therefore, all inertial observers must agree on the speed at which light propagates, even if they are in motion relative to the source.

(Sorry, one last parenthetical comment:

If I am in an inertial reference frame, and you and I are moving relative to each other with constant velocity – at constant speed along a straight line – then your reference frame is also inertial.)

(What exactly is meant by an “inertial reference frame?”

A reference frame is basically the clock and set of rulers that a particular observer uses to record their measurements of where and when things happen.

If the measurements made with this clock and rulers agree with Newton's laws of motion – that is, if objects only accelerate in proportion to the net force acting on them, without the need for any fictitious forces – then the reference frame is said to be "inertial.")

This conjecture does more than obviate the aether. Once we assume that the laws of electrodynamics hold in all "inertial" reference frames, we reach some confusing conclusions that force us to revise our most basic ideas about space and time.

But by 1905, scientists had been looking for the aether for nearly 20 years with no success.

Einstein proposes a beautifully simple explanation: There is no special frame of reference for light. No motion through the aether had been detected because there is no aether!

It had long been assumed that light waves propagate through a medium, like mechanical waves.

This medium – the luminiferous aether – would establish a special frame of reference. Someone moving through the aether would detect a different speed of light than someone at rest.

Compare this with the Doppler shift for sound waves. There are genuinely different explanations for the change in pitch based on whether the listener or source is moving. Why? Because there's a medium (the air) to refer to.

Einstein addresses these issues (and revolutionizes physics) with a simple conjecture that has radical consequences.

He postulates:

“...the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good."

If the phenomena truly depend only on relative motion, why should the explanations be so qualitatively different depending on which point of view one adopts?

Second, Einstein reminds the reader that attempts to detect the motion of Earth through the “luminiferous ether” -- a hypothetical medium assumed to be necessary for the propagation of electromagnetic waves – had failed.

Let's go through some of his arguments!

At the beginning of the paper, Einstein presents two reasons why our understanding of electrodynamics (the rules governing charges, currents, and electromagnetic fields) must be incomplete.

First, he points out odd asymmetries in explanations of electrodynamic phenomena associated with relative motion between two objects. The explanations were often very different depending on the point of view: Object A at rest and Object B moving, or vice-versa.

Happy 118th birthday to special relativity!

Albert Einstein introduced special relativity in the paper "On The Electrodynamics Of Moving Bodies," published #OTD in 1905 in the journal Annalen der Physik.

Manuscript: https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.19053221004
English Translation: https://www.fourmilab.ch/etexts/einstein/specrel/www/

The speed of light is, to the best of our knowledge, the upper limit on the speed with which any causal influence can propagate in Nature.

Looking at the stars, we are not seeing things as they are. Even looking at a person standing in front of you, there is a short delay between light leaving them and reaching your eye.

We see everything as it was and nothing as it is. Every story the Universe tells is a history.