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 c is the maximum possible velocity, v/c varies from zero (vertical red line) to one (horizontal red line).
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 v/c=0, which is obvious from the diagram.
Conversely, 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.
Figure 3 shows the other edge case, maximum velocity (light speed).
Now, since v=c, then v/c=1, again obvious on the diagram.
And, 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=c, then 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>c, then that fraction is greater than 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 (v/c) and y is (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 c, then 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 0.5c as moving at a rate of t/0.866… (which is the same as multiplying t by 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.