A post for general readers:
Einstein’s relativity. Everybody’s heard of it, many have read about it, a few have learned some of it. Journalists love to write about it. It’s part of our culture; it’s always in the air, and has been for over a century.
Most of what’s in the air, though, is in the form of sound bites, partly true but often misleading. Since Einstein’s view of relativity (even more than Galileo’s earlier one) is inherently confusing, the sound bites turn a maze into a muddled morass.
For example, take the famous quip: “Nothing can go faster than the speed of light.” (The speed of light is denoted “c“, and is better described as the “cosmic speed limit”.) This quip is true, and it is false, because the word “nothing” is ambiguous, and so is the phrase “go faster”.
What essential truth lies behind this sound bite?
Faster Than Light? An Example.
Let’s first see how it can lead us astray.
At the Large Hadron Collider [LHC], the world’s largest particle accelerator, protons collide with other protons. Just before a collision happens, a proton that has been accelerated to 99.999999% of c comes in from the right, and another accelerated to the same speed comes in from the left.
From your perspective, observing these two protons (using fancy electronics, not your eyes) from the LHC control room, the distance between the two protons is decreasing by almost the speed of light from the right and almost the speed of light from the left. So it would seem that the distance between them is closing by almost twice the speed of light — by 1.99999998 c, if you want to be precise. And if the distance between them is decreasing by almost twice the speed of light, then, well, their relative speed (as you see it) is faster than the speed of light.
That must be wrong, somehow, mustn’t it? Otherwise it would violate relativity… right?
But no. It’s not wrong. From your perspective, the two protons are approaching each other at nearly twice the speed of light.
Even simpler: point two laser pointers at one another, and turn them on at the same moment. The light beams will each be traveling at the speed of light, and the distance between them, from your perspective, will be decreasing at twice the speed of light.
When a flash of lightning occurs, light rushes off in all directions. The light moving north moves at c. The light moving south moves at c. From our perspective, standing on the ground, the distance between the light moving north and the light moving south grows at 2c. That’s all there is to it.
But I thought — wait… — huh? — that can’t be ri… — didn’t Einst…? — you’re full of… — I just don’t believe… — … …can’t be true!?! — well I [sound of head exploding.]
This is the kind of problem that sound bites lead to. Let’s take the two-proton example apart, and see what “nothing” and “go faster” actually mean.
The Can’t-Exceed-c Rule
What Einstein’s relativity actually says is this:
- From the perspective of any observer who measures the speeds of physical objects (using a careful laying out of aligned rulers for distance and synchronized clocks for time), no objects will ever be measured to be moving faster than the cosmic speed limit c, also known as “the speed of light in empty space.”
Let’s call this the can’t-exceed-c rule. [This statement of the rule is okay as long as gravity isn’t too important; if gravity matters a lot, then it needs revision… which I’ll return to in a future post.]
The two protons individually don’t violate this rule: we, as observers in the LHC control room, view both of the colliding protons as moving slower than c. And notice: the can’t-exceed-c rule says nothing about the relative speed of two objects as observed by a bystander.
But still, there is a potential threat to the rule lurking here. Suppose we somehow accelerated an observer (let’s name him Peter) to the same speed and direction as the proton coming from the left. Peter would then move along with that proton, and would view it as stationary. From our perspective, as illustrated in Figure 1, we might think Peter would then view the proton from the right as approaching at nearly twice the speed of light. That would violate the can’t-exceed-c rule.
Similarly, if Paula were traveling with the proton coming from the right, we might think she’d view the proton from the left as moving at nearly twice c.
Here’s where relativity of space and time as Einstein intuited it, and as experiments confirm, steps in to save the can’t-exceed-c rule. The point is this: because Peter is in motion relative to us, Peter’s view of space and time is not the same as ours. This is key. Because of his differing views, Peter will lay out clocks and rulers differently from how we, sitting in the LHC control room, would do so. Therefore, the way that Peter measures speed — the distance covered by a moving object over a certain amount of time — is different from how we are doing so.
And that’s why, even though we measure the two protons approaching each other at nearly twice light speed, Peter will measure the right-moving proton approaching him below light speed. (The analogous statement would be true for Paula, traveling along with the other proton.)
In summary: from the perspective of an observer traveling with either proton,
- we and the LHC control room are approaching at a speed of 0.99999999 c
- the other proton is approaching at a speed of 0.9999999999999998 c .
Meanwhile, from our perspective
- Both protons are approaching us (from opposite directions) at a speed of 0.99999999 c
Different observers simply disagree. Yet all are simultaneously correct. This is characteristic of things that are relative (i.e. perspective-dependent.) [You and I think the Sun is bright, but an observer out at Pluto would think the Sun is dim; there’s no logical contradiction, and all of us are correct.] The difference, again, stems from different (but equally valid) ways of measuring distances and durations.
[One thing we do have to agree on: if A views B as approaching with speed v, then B must also view A as approaching with speed v. Our perspective and those of Peter and Paula are indeed all consistent with this requirement.]
Notice that no one’s perspective violates the can’t-exceed-c rule. The universe preserves this rule, as described by the math of relativity, no matter how you try to trick it into an exception. Yet that math doesn’t promise or assure that this rule applies to the “relative speed” of two objects — if “relative speed” is defined as the rate of change of the distance between two objects as measured by an bystander.
As always, though, definitions matter. If instead we defined the “relative speed” of two objects to be the speed of the second object as measured by an observer traveling with the first, or vice versa, independent of any bystander, then indeed that “relative speed” cannot exceed c. This is the definition implicitly used by physicists most of the time. But as with all definitions, it’s an arbitrary choice. It’s not the intuitive one most non-physicists would use.
From Sound Bite to Understanding
So what does “nothing can go faster than the speed of light” really mean? “Nothing” means “No physical object”, and “Go Faster…” means “can be measured by an observer to be moving…” And altogether the sound bite means,
- “no physical object, from the perspective of any particular observer, can ever be measured to be moving faster than the speed of light in empty space, c.”
Even this comes with additional fine print. (The observer should be inertial and must make the measurement using an inertial frame of reference, and gravity had better be irrelevant; and I’ve left out some details about what “observer” really means and how the measurement is really made.) I’ll return to some of that fine print another time. But I hope it’s clearer now what the initial glib statement doesn’t mean.
Fundamentally, a sound bite or TL;DR approach to science can’t ever work. Many of the remarkable and non-intuitive features of the universe are within the grasp of non-scientists, but they require more than a single sentence, or even a paragraph. What sound bites do in relativity is similar to what they do in politics: they make us think we understand something, while actually obstructing the path to knowledge.