With COVID-19 putting a damper on social activity, “the gang” doesn’t get together very often, but we still gather occasionally (and carefully). One of the times recently I got into how, even though we’re all sitting essentially motionless in a living room, we’re moving through *time* at the speed of light. I explained why that was, and they found it pretty cool.

Then I ran into someone online who just couldn’t wrap his head around it — just couldn’t accept it (despite explaining in detail and even providing links to some videos). Physics is sometimes challenging to our daily perceptions of reality!

In this case, though, it’s just a matter of some rather simple geometry.

To begin, let me back up and touch on a few basics. (For more details, see my Special Relativity series or any decent textbook or online lecture series.)

What’s important here, firstly, is that things moving relative to other things see spacetime differently — they have different *frames of reference*.

In particular, in a frame we see as moving relative to us, time and distance in that frame appear stretched and compressed. We see the clocks in a moving frame tick slower, and distance along the axis of movement foreshortened.

Note that, because of relativity, observers in that moving frame see *us* as the moving frame, and to them it’s *our* clocks and distances that are affected by motion. Special Relativity insists that both views are equally true. (The seeming paradox is resolved when one frame joins the other.)

The stretching and compressing are governed by the Lorentz transformation. In this post I’m going to show you where it comes from.

As it turns out, it’s also *why*, when we’re not moving through space, we’re moving through time at the speed of light.

In fact, one way or another, we’re *always* moving at the speed of light. The choice is between how much of that velocity goes towards actually moving through space versus through time. The faster we go through space, the less fast we go through time.

If we somehow reach the ultimate speed — light speed — then our motion through time ceases. To *observers*, the clocks of anything moving at light speed appear to stop; time does not pass.

From the point of view of the moving object, the universe is moving past at light speed. Which means the travel path is compressed to zero. To an object moving at light speed, the path it takes appears to have zero length — thus there is no time for time to elapse (and thus their clocks do not change during the journey).

On the other hand, if we’re not moving through space at all, then *all* our velocity is through time.

The bottom line is that our velocity through *spacetime* is always ** c**.

**§**

To unpack this, we need first to understand *Figure 1* (to the right and also at the top of the post).

The diagram depicts the velocity of an observed object.

(It’s not how fast *we’re* going, because we assume we’re always at rest. It’s how fast something we’re *observing* is going.)

The velocity is represented by the *angle* of the red line relative to the vertical, which we’ll consider 0°. Note that we normalize the length of this line to 1.

If the red line is exactly vertical (0°), the velocity is zero. If the line is exactly horizontal (90°), then velocity is maximum (light speed). You can think of the red line as if it were the needle of a speedometer that goes from **0** (straight up) to ** c** (pointing right) in that quarter circle.

The horizontal axis is in terms of ** v/c** — the velocity divided by light speed. Since

**is the maximum possible velocity,**

*c***varies from zero (vertical red line) to one (horizontal red line).**

*v*/*c*The vertical axis is ** gamma** (γ) — the Lorentz factor that determines the compression and stretching. (To be precise,

*gamma*is actually the

*inverse*of the quantity we’ll derive. I’ll get to that.)

**§**

Before we start deriving things, let’s examine two important cases: Standing still and moving at light speed. (The min and max possible, so these are the edge cases.)

*Figure 2* shows the first case, a velocity of zero. The “needle” is standing straight up.

In this case, since ** v=0**, then

**, which is obvious from the diagram.**

*v*/*c*=0Conversely, **1/γ=1**, which means the Lorentz factor is just **1.0** — there is no stretching or compression.

That’s exactly what we should expect. If something isn’t moving relative to us, then it’s in our frame of reference, so its clocks and lengths are “proper” just like ours.

(*Proper time* and *proper length* is what we always experience in our own frame of reference.)

*Figure 3* shows the other edge case, maximum velocity (light speed).

Now, since ** v=c**, then

**, again obvious on the diagram.**

*v*/*c*=1And, likewise, **1/γ=0**, which means that **γ** has to be very large (to make the fraction small). In fact, in order for the fraction to equal zero, gamma has to equal infinity!

As it turns out, that’s exactly what happens.

**§**

Before we dive in, let’s review those edge cases in terms of the Lorentz equation (repeated here for convenience).

When ** v**=

**0**, then

**=**

*v*/*c***0**, and its square is also zero. Subtracting that from the

**1**gives us

**1**, and the square root of

**1**=

**1**, so the bottom part of the fraction is

**1**, and the whole thing is

**1**/

**1**or just

**1.0**, which is what we saw above.

When ** v**=

**, then**

*c***=**

*v*/*c***1**, and its square is also one. Subtracting that from

**1**gives us

**0**, and the square root of

**0**=

**0**, so now the bottom part of the fraction is zero. Which is a problem, because division by zero is undefined. (Some claim it’s equal to infinity, but this is mathematically incorrect.)

What we see here is one reason why something with mass can never accelerate all the way to ** c**. For an object with mass, it involves undefined math.

We can also see why objects can’t move *faster* than ** c**. If

**>**

*v***, then that fraction is greater than**

*c***1.0**, and the subtraction results in a value less than zero. Square roots of negative numbers take us into the imaginary number regime!

**§**

To finally bring this home, returning to the diagram, we invoke the Pythagorean theorem.

Note how the red line, combined with the green and blue lines, give us three sides of a triangle.

(Imagine either moving the green line down, or the blue line left, to form the actual triangle. What we care about are the lengths of those lines.)

We’ve normalized the red line to 1 to make calculation easier. It means we can use a simpler version of Pythagoras’s rule:

For us, remember, ** x** is (

**) and**

*v*/*c***is (**

*y***1/γ**).

We know our velocity (** v/c**), so we can solve for

**:**

*y*And then take the square root of both sides:

Which should look familiar. It’s the bottom part of the Lorentz equation’s fraction. Just replace ** x** with (

**).**

*v*/*c*That basically is all there is to it — the geometric version of the Lorentz transformation. (Special Relativity is a geometric theory.)

**§**

A last detail involves why we take the *inverse* of the square root of one minus the square of the velocity ratio. (The Lorentz equation is one *over* the square root of etc.)

All that’s happening is that we’re *dividing* length or time by the calculated factor. (Or multiplying by the inverse, which is the same thing.)

For example, if ** v** is one-half

**, then**

*c***=**

*v*/*c***0.5**, and the square of that is

**0.25**. Then

**1**–

**0.25**=

**0.75**, and the square root of that is

**0.866…**(the inverse is

**1.1547…**). It’s an irrational number — it’s precisely

**√3/2**.

Effectively that means we see time difference in a frame moving at **0.5 c** as moving at a rate of

**(which is the same as multiplying**

*t*/0.866…**by**

*t***1.1547…**). Thus time appears to move slower in the moving frame. One second for for us appears as only

**0.866**seconds on the moving clock. Alternately,

**1.154**seconds for us is just one second on the moving clock.

Taking the inverse also produces both the extreme behavior at near light speed as well as the impossibility of accelerating to light speed. As the lower part of the fraction approaches zero, the factor increases towards infinity (but, of course, can never reach it).

**§**

To wrap it up, the red line (the “speedometer needle”) is always the same length (one), which reflects that every object is always moving through spacetime at ** c**, the maximum velocity.

What changes is how much of that speed goes towards moving through the space part of spacetime. The faster an object moves through space, the slower it moves through time.

On the other hand, not moving through space at all means all our velocity is entirely through the time part of spacetime.

Stay speedy, my friends! Go forth and spread beauty and light.

∇

December 9th, 2020 at 12:10 pm

If you’re wondering about the post’s title, it’s the sixth of the Extras posts in my Special Relativity series. (Technically the seventh, but the first was part of the series itself.)

You can find a lot more detail about the Lorentz transformation in the Wiki page, including the mathematical derivation. The Lorentz Factor page has a nice table and chart showing Lorentz factors at different speeds. It also has a diagram showing the quarter circle relationship.

December 9th, 2020 at 12:12 pm

To be honest, it’s probably more correct to say the circle diagram derives from the Lorentz equation, which is why it’s so easy to get back to the equation from the diagram.

But it’s a great illustration of why and how we’re always moving at light speed through spacetime.

December 9th, 2020 at 12:30 pm

It’s also another illustration of why we can’t go faster than light speed — the “needle” would go beyond the diagram. It “pegs” at

.cDecember 9th, 2020 at 12:20 pm

December 9th, 2020 at 12:20 pm

December 9th, 2020 at 12:22 pm

One more:

December 9th, 2020 at 2:54 pm

This PBS SpaceTime video from 2015 goes into more detail about where the Lorentz transformation comes from:

It’s weird to see Matt five years younger! 🙂

December 9th, 2020 at 3:36 pm

This PBS SpaceTime video from 2017 gets more into the Lorentz transformation and includes good bits on the spacetime interval and how we define simultaneity. It also nicely shows the geometric aspects of SR.

Matt’s definitely has a new look two years later. 😀

December 10th, 2020 at 7:58 am

It’s interesting how much the Pythagorean theorem comes up in working out these equations. I was surprised to learn that it’s involved in working out the Born rule. I guess I shouldn’t be that surprised since it’s a fundamental theorem.

December 10th, 2020 at 10:39 am

In something like |c|^2 = |a|^2 + |b|^2? Yeah, it’s a fundamental truth about any geometric system. In this case, it even works with complex numbers.

Just last night I was watching a math lecture where the teacher said the Pythagorean Rule ought to be named after the Chinese guy who discovered it 900 years before Pythagoras. 🙂

December 10th, 2020 at 2:32 pm

The Greeks do tend to get credit for a lot of stuff they learned from other cultures. There was a blogger a while back who looked at all the concepts that show up in Greek philosophy a century or two after they’re attested in India. The Ionian coast was on the edge of the Persian empire, an empire that I think also projected into the western part of the Indian subcontinent.

Of course, China is a lot further away, so it’s conceivable Pythagoras came up with it independently. But it’s also possible it diffused through Eurasia.

All that said, the written evidence in these cultures often doesn’t go back nearly as far as in the west. So there are probably some cases where a tradition of a concept arising in their culture before the Greeks is just that, a tradition.

December 10th, 2020 at 4:43 pm

We have, as you say, no way of knowing in this case, but the Pythagorean relationship is so fundamental, I can easily believe the Greeks figured it out — it’s very much the sort of thing they were figuring out at the time. (Many math fundamentals are documented to have been co-discovered — it’s one argument that math is inevitable.)

As you also note, the Chinese (and Japanese), Indians, and Persians, were all serious mathematicians — some of what they contributed (especially the Persians) did find its way into Western maths (“algorithm” is one of my faves).

December 10th, 2020 at 5:13 pm

One thing Morris Kline pointed out in his book, is that much of the mathematics the Greeks inherited from the Egyptians and Babylonians were built on explicitly empirical foundations. Those cultures didn’t feel the need to develop an a priori justification for those observed relations. (Or if they did, they didn’t leave evidence of it.) The Greeks may have been the first to be relentless about actual proofs, enabling them to take it to a new level.

The Pythagoreans actually made mathematics their religion, which seems nuts. But it might have been what was needed to start such a new form of thought.

December 10th, 2020 at 5:49 pm

Ha, yeah, the Egyptians and Babylonians — those guys, too!

I think it’s in Steven Strogatz’s book about calculus that he talks about how the Greeks created a mental block in math that wasn’t really cleared until Newton and Leibniz came up with calculus.

The Greeks, as you say, got really serious about their math, but they tended to stay on the rational side — the square root of two was really upsetting to them. Xeno’s paradoxes reflect their inability to wrap their head around certain notions of change (let alone

ratesof change — change that changes).They also tended to treat their geometry as the source — curves and lines were givens, but once drawn you could prove interesting things about them. That, as you say, was one place they really shone. Geometry students have been studying those proofs ever since.

It wasn’t until calculus that mathematicians saw a whole new world of

functions. The line was no longer a given, it came from something that could be manipulated.As an example, if you took calc, you may remember that f'(x), the derivative of f(x), is a function that gives the slope of f(x) for any x. The Greeks found geometric ways to calculate the slope of some

givencurve at somegivenpoint, but the idea of deriving f'(x) from f(x) was completely outside their mental conceptions.December 10th, 2020 at 8:30 pm

I saw or read something about a Greek mathematician who had to solve a problem involving digging a tunnel through a mountain. It was something where calculus would have been the right tool, but since he didn’t have it, he just brute forced his way through it.

I also find it interesting that they didn’t have anything like our notation. Apparently things like the Pythagorean theorem were described in language, which sounds awful. (Even more so since ancient writing didn’t have other niceties like spaces between words or paragraphs. Life was just a lot harder back then.)

December 10th, 2020 at 11:53 pm

Really torturous language, too. It’ll give you a whole new appreciation for the word problems from math in school.

January 25th, 2023 at 9:22 am

[…] For that matter, Special Relativity, which is entirely geometric, is yet another example of the Pythagorean theorem, but that’s another story. (One I’ve already told. See: SR #X6: Moving at Light Speed) […]