Seeking God in Science, part 11b: Time (part 2 of 2)

Don Geddis pointed out a problem with my relativity animation:

... with either Linda or Ralph stationary, you have the angled light hitting the end clock at "the same moment" as the light pulse that just goes straight up and down. That's not what would happen. If light was going at a constant speed, the light on the angled path would need to take longer to reach the end point.

That sounds plausible, but it's wrong.  The light that moves along the angled path must arrive at the same moment as the light that moves along the vertical path.  Why?  Because it is literally the same light just viewed from two different points of view, what in the trade are called "frames of reference".

Now, there is a different experiment that we could imagine where Ralph and Linda each send out two flashes of light.  One flash, as before, travels vertically in their frame of reference, and the other is launched at an angle so that its trajectory traces out the same path as the light that is vertical in the other's frame.  It seems plausible to say that that light would not arrive "at the same moment" because it is traveling along a longer path.  And it also seems plausible to say that therefore the other light must also "not arrive at the same moment" because those two beams are tracing out the same path in space.  But this is wrong, and the way in which is it wrong is the whole point of this thought experiment!  Here is a hint: think about how you would actually have to modify the experimental setup in order to demonstrate that Ralph's (or Linda's) angled beam did not arrive "at the same moment" as his (or her) vertical one.  (I'm going to leave that as an exercise for now.)

Nevertheless, there is indeed a problem with this animation: it is showing the light moving as if it were a classical object like a baseball, and so it appears to be moving at different speeds.  But that is not possible with light.  Light is only ever seen to move at one speed, namely 186,000 miles per second, a speed universally denoted by the letter C.  So how can something moving at the same speed cover different distances in the same amount of time?  Obviously it can't.  Something has got to give.

The "thing that gives" in this example is, as noted in the first part of this two-part episode, the clocks.  Ralph and Linda agree that each of their clocks reads two time units when the light returns, but they don't agree on how far the light traveled.  If we can't change the speed the only thing left is the time.  Ralph thinks Linda's clock is wrong and Linda thinks Ralph's clock is wrong.  Each sees the other's clock as reading less than it should, i.e. each sees the other's clock as running slow.  And the reason for that is that each one sees the other's clock as moving.

I tried to extend the animation to illustrate this more directly but it turns out to be really hard.  The problem is that the point I actually want to make here has to do with the fact that it is impossible to measure the one-way speed of light.  Young-earth creationists try to use this fact (and it is a fact) to explain how we can see stars and galaxies further than 6000 light years away.  My goal here is to show why their argument doesn't work.  The problem that leaves me with, however, is that I cannot assume that the one-way speed of light is c because that would be begging the question.  Because I cannot assume that the one-way speed of light is c, I cannot rely on most of the usual pedagogical tools for explaining relativity because they do assume that the one-way speed of light is c.  This assumption (and it is an assumption) is deeply ingrained in the minds of most people because it is the foundation of Einstein's relativity theory.  But it is an assumption, not a physical fact.  The physical facts are that the round-trip speed of light is c, and the one-way speed of light along any particular trajectory is constant.  It seems like you should be able to prove from those facts that the one-way speed must be c, but you can't.  It's not true.  It is possible to build so-called anisotropic models of light where the one-way speed is faster in one direction and slower in the opposite direction.

Lest you think, as many physicists do, that I am merely entertaining a transparently crackpot theory, the fact that it is fundamentally impossible to measure the one-way speed of light was noted by Einstein himself in his 1905 paper:

[The laws of physics] possess no properties corresponding to the idea of absolute rest. ... the same laws of electrodynamics and optics will be valid for all frames of reference ...  We will raise this conjecture ... to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.  [Emphasis added]

and later:

If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighborhood of B.  But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an “A time” and a “B time.” We have not defined a common “time” for A and B, for the latter cannot be defined at all unless we establish by definition that the “time” required by light to travel from A to B equals the “time” it requires to travel from B to A.  [Emphasis added.]

(BTW, that quote is a huge hint for the exercise above.)

So the constant speed of light is a postulate, an assumption, exactly like taking the sun to be the center of our solar system.  It is not a physical fact, it is just a convention that makes the math easier.  And, as we shall see, the fact that it is impossible to measure the one-way speed of light is actually an enormous clue to what is really going on under the hood.

Because we want to specifically avoid the assumption that the one-way speed of light is c, we need to be more precise about what it actually means to assume that "the one-way speed of light along any particular trajectory is constant".  As before, we turn to Einstein:

... light is always propagated in empty space with a definite velocity ... which is independent of the state of motion of the emitting body.

There is a lot of subtlety packed into those 23 words, but there are two main things to notice.  First, the velocity is "independent of the state of motion of the emitting body".  That is really the crux of what it means for the speed of light to be the same for all observers.  We'll get back to that in more detail later.

The second thing to notice is that we're talking about light propagating in empty space i.e. in a vacuum.  Note that we can never actually see light moving in a vacuum.  The only thing we can actually see is light that arrives at some kind of sensor -- a camera, or our eyes.  When light is moving in a vacuum, by definition there can't be anything there to sense it.  So the only things we can actually measure when light moves through a vacuum is the time and place the light was emitted, and the time and place it arrived.  From this we can infer a trajectory and a speed, but -- again by definition -- we cannot measure them.  The only way to "see" light actually move is to put some physical objects along its path, like dust particles.  Here is an extraordinary video clip that lets us "see" light move using this technique.

Actually there are two different clips with two different viewing angles.  In the first clip (at the 1:10 mark) the camera is viewing the light perpendicular to its motion.  In the second (at 1:33) the camera angle is oblique, and the light appears to move faster while moving towards the camera and slower while moving away, which is a visceral illustration of anisotropic light.  You can, of course, explain (or, as a creationist would say, "explain away") this observation using the standard assumption.  But you can also explain the fact that light is appearing to move at different speeds in different directions by hypothesizing that it actually is moving at different speeds in different directions.  No experiment can disprove this.  (But note that you then have to explain -- or "explain away" why the light appears to move at the same speed in both directions when "viewed" at 90 degrees!)

It is worth watching the entire video to see the details of how these clips were made.  Around the 23 minute mark you will find the standard explanation of why the light appears to move at different speeds when viewed from an oblique angle: you are not actually seeing the light move.  What you are seeing is light that has traveled from the source, reflected off dust particles in the air, and then traveled to the camera.  What the camera is measuring is the time it takes the light to traverse that entire path, not just the first segment.

In philosophy-speak, the one-way speed of light being c works epistemologically but not ontologically.  It makes the math work, but if you want to understand what is actually going on it really ought to bother you that there is no experiment you can do that will allow you to demonstrate that light moves at the same speed in all directions.  And actually, the fact that it is impossible to rule out anisotropic models experimentally, the fact that it is fundamentally impossible to measure the one-way speed of light, is actually a Big Clue that there is something else going on under the hood.  What is it?

Here is a hint: notice that it is also possible to make anisotropic models of time.  We can show experimentally that two clocks sitting next to each other remain synchronized, but how do we know that the length of one tick is the same as the next?  Maybe clocks actually tick at different rates when it is Tuesday in New York, but because all clocks are equally affected, including all of the physical and biological processes that cause us to have a subjective experience of time passing, there is no way to tell.  That seems silly, but there is no experiment you can do that will rule it out.  And this, too, is a clue.

(In point of actual fact, clocks do tick at different rates depending on the circumstances!  It is only clocks that are not moving or accelerating relative to each other that will remain in sync!)

So let's go back to basics: what does it actually mean for the speed of light to be the same in all reference frames in terms of things we can actually observe?  Here again, Einstein provides the answer, or at least the foundation:

[A]ll our judgments in which time plays a part are always judgments of simultaneous events. If, for instance, I say, “That train arrives here at 7 o’clock,” I mean something like this: “The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events.”

Notice that for two events to be considered simultaneous they have to be adjacent not only in time but also in space.  (Another hint!)  Two things that happen in the same place and at the same time are "simultaneous".  You can't assume anything about events that happen in different locations, and indeed it will turn out that different observers will have legitimate disagreements about whether or not two distant events are simultaneous or not.

In the experiment with Linda and Ralph there are three different sets of simultaneous events:

E1.  Ralph and Linda's light sources flash and their clocks start.  This is the reason I drew Ralph's train upside-down, to emphasize the fact that the two light sources are next to each other when they flash.

L1.  Linda's light arrives back at Linda's clock and her clock stops and registers a result (2 time units).

R1.  Ralph's light arrives back at Ralph's clock and his clock stops and registers a result (also 2 time units).

Note that L1 and R1 are not simultaneous because they do not happen in the same place.  In the animation they appear to happen at the same time, but this is just happenstance because I wanted to highlight the symmetry between Linda and Ralph's perspectives.  Linda and Ralph disagree on the light path lengths -- they each see the other's light path as longer than their own.  But if we assume that the speed of light is the same everywhere, then there is only one possible conclusion that each of them can come to: the actual time that it took the other's light to return to the (moving) clock was actually longer than the 2 time units that clock reported.  The moving clocks are running slow!

We can quantify this.  To keep things simple let's assume that the relative speed between the two train cars is such that the longer light paths appear to be at 45 degree angles.  In other words, Ralph sees Linda's light move at a 45-degree angle and Linda sees Ralph's light move at a 45-degree angle.  Now, the phrases "Ralph sees Linda's light move" and "Linda sees Ralph's light move" should be setting off alarm bells in your head because I spent the first half of this post explaining that it is not possible to see light move (notwithstanding the existence of videos purporting to show light moving).  So let me be more precise: let us change the experimental setup in the way alluded to in the exercise above: both Linda and Ralph set up a second mirror so that they each have a second beam of light that moves at a 45-degree angle in their train car.  Those beams will not return to the light source, they will land two distance units away.  To be precise (which we have to be because we already know that Linda and Ralph disagree about lengths) Ralph's beam will land two distance units away as measured by Ralph in his train car, and Linda's 45-degree beam will land two distance units away as measured by Linda in her train car.

We now have two more events:

R2: Ralph's 45-degree light beam lands on the floor of Ralph's train.

L2: Linda's 45-degree light beam lands on the floor of Linda's train.

Now adjust the relative speed of the trains so that L1 and R2 are simultaneous events, i.e. both 45-degree-angle beams land at the exact instant that the places they land are adjacent as the trains pass each other.  By symmetry, R1 and L2 must also be simultaneous events.  With some elementary geometry you can figure out that the speed required for this is √2/2 or about 0.7c, but the exact value doesn't really matter.  What matters is that this speed is less than c.  The reason is simple geometry: the trains are traveling along the legs of two right triangles while the diagonal light is traveling along the hypotenuses of those same triangles, so the trains will always have to cover less distance than the light in order to arrive at the same place at the same time.

As I mentioned earlier, I tried to extend the animation to illustrate all this, but that turns out to be really hard.  I spent a couple of hours with ChatGPT and was not able to get it to do the right thing, so I'm probably going to have to code it up myself.  (If there are any animators out there who want to give it a shot I'd be willing to pay you for your time!)  The reason it's hard is that the animation as it stands sweeps some details under the rug in the interest of highlighting the important parts while trying to keep things simple and intuitive.  Two details that I've swept under the rug are Lorentz contraction and the relativity of simultaneity.  When you switch perspectives in the current animation, the train cars appear to remain the same length and the clocks advance at the same rate.  (These distortions are analogous to the ones you get with a Mercator projection.)

Trying to add the 45-degree beams reveals some of those distortions, so it's actually impossible to simply add those to the current animation.  You have to get rid of the distortions in order to make it all work, which means that you have to put in Lorentz contraction and time dilation, and then it all gets really hairy and I have not yet been able to make it work.  Sorry.  But all that technical detail detracts from the point I really want to make here which is simply that moving clocks appear to run slow, and this is a straightforward consequence of disagreement over the length of the path that light took to travel between two events.

There is one detail that does matter, though, and that is the fact that to get an apparent path of light moving at a 45-degree angle you do not have to travel at the speed of light, but only about 7/10 of the speed of light.  This is counterintuitive at first, but becomes pretty straightforward when you start to think of it in terms of geometry: the light always appears to travel on the hypotenuses of triangles and the trains always move along the legs, and so the distance covered by the train is always less than the distance covered by the light.  Let that really sink because what I am about to tell you next is going to blow your mind even though it will be obviously true: the distance covered by the train is always less than the distance covered by the light is a fact of geometry, and so it must remain true NO MATTER HOW FAST THE TRAINS ARE MOVING!

This makes most people's heads explode when they first hear it because another truism that has been instilled into them is that it is impossible for massive objects (like trains) to travel faster than light, and this is true.  If you race light, the light will always win.  But this is not some kind of arbitrary "cosmic speed limit", it is a logical consequence, a geometric consequence, of the fact that the speed of light is the same for all observers.  If you try to race light in a straight line you will always lose.  But if the light is reflected and so takes a path that is not a straight line, you can always get to the finish line at the same time (or sooner) no matter how far away it is, and you always can do that by moving slower than light.

And here is the final mind-blowing fact: no matter how far away the finish line is, the time it will take you to get there in the setup we've described above will always be the same according to the clock on your train car: two time units.  Again, the exact number doesn't really matter.  What matters is that it's more than zero.  So: in an arbitrarily short amount of time as measured by a clock on your train car (as long as it's more than zero) you can traverse an arbitrarily long distance.  Want to go to the Andromeda Galaxy in one second?  No problem.  Just hit the gas and accelerate to 0.99999999999999999999999999999c.  (That is actually the right answer, more or less, at least if ChatGPT did the math right.  There are twenty-nine 9's there.)

But wait, you say, that's obviously impossible because I would be traveling more than 186,000 miles in a second!  Well, that's true, but remember that not only do moving observers disagree on when "now" is, they also disagree on where "here" is, and so they also disagree on where "there" is.  When you move close to the speed of light, everything around you in the direction of your motion appears to get shorter (that's the Lorentz contraction).  So from the point of view of someone back on earth, it takes you 2.5 million years to get to cover the 2.5 million light years to Andromeda.  But from your point of view, the distance to Andromeda shrinks so that you can cover it in one second according to your clock (which is running very, very slow according to observers on earth).

If we extend this to a two-way round trip we get the so-called "twin paradox": if you can get to Andromeda in one second (according to your clock) then you can also get back in one second (again, according to your clock).  So if you actually do this you will be two seconds older but earth will be five million years older.  Of course, actually carrying out this experiment with a human involves a few practical difficulties, like the fact that accelerating yourself to such speeds would require vastly more energy than in all the atomic bombs on earth and decompose the atoms in your body into their subatomic constituents many times over.

But there is something that can make such a round trip without any trouble at all: light.  What would it look like to go to Alpha Centauri and back not just close to the speed of light but at the speed of light?  This requires some pretty significant suspension of disbelief because this is not actually possible even in theory.  It is worth reiterating why this is the case: no matter how much distance you cover in a finite amount of time, your arrival will still be simultaneous with that of a light beam that reflected off a mirror, and so you will have moved slower than that light.  To move at the speed of light you have to travel an infinite distance in a finite amount of time as measured by your clock.

But that can't be right because light doesn't cover an infinite distance in a finite amount of time!  It moves at a finite speed: 186,000 miles per second.  Yes, that's right.  But remember the qualification: you have to travel an infinite distance in a finite amount of time as measured by your clock.  To transfer this to a chunk of light you would have to reckon the time as measured by a clock moving at the speed of light.  It is not possible for a clock to move at the speed of light, but if it could, what would happen?  Well, as it got closer and closer to the speed of light, it would run slower and slower until, at the speed of light, it would stop altogether.  At the same time (no pun intended) the Lorentz contraction would shrink the distance between your start and end points down to zero.  At that point your speed from your perspective would be zero divided by zero.

And this is the reason that it is impossible to measure the one-way speed of light.  It is not merely physically impossible, it is mathematically nonsensical!  It is literally zero divided by zero.

So why can we measure the two-way speed of light?  Why does time pass when light reflects?  If both the outgoing and reflected beam of light think that their origin and destination are at the same place and time, how can time elapse at all?

To answer this it is best to go back to a physically possible situation with a physical clock moving at close to but not quite the speed of light, making the round-trip to alpha centauri in a short but still finite period of time from its perspective.  Actually, it is best to imagine two such clocks, one going to Alpha Centauri and the other coming back so we don't have to deal with acceleration and general relativity.  The motion of the clocks is arranged so that they pass each other at Alpha Centauri, at which point the inbound clock is set to the same time as the outbound clock.  Let's say that each leg takes one second as measured by these clocks, so the total round-trip time is measured as two seconds, during which time five million years will have passed on earth.

When did those five million years "happen" from the perspective of the clocks?

The (mind-blowing) answer is: they didn't!  If you actually work this out (and I'm not going to do that because this post is already too long) what you will find is that the apparent passage of time on earth can be ascribed entirely to the disagreement over when "now" is from the two perspectives of the two clocks.  For the outbound clock, "now" on earth is in the past.  For the inbound clock, "now" on earth is in the future.  And the difference is exactly ... five million years (minus two seconds)!

From the "perspective" of light, time never "passes".  What happens instead is that "now" changes.  But those changes "happen" instantaneously every time the light is reflected by something and changes direction and hence "perspective".

And it turns out that this is what happens with clocks too!  Clocks are collections of atoms.  Atoms interact with each other mainly through the electrons in their outer shells, and those interactions are electromagnetic ones.  In other words, there are photons -- light -- going back and forth between atoms.  Every time that "happens" (I have to put "happens" in scare quotes here because the whole idea of "things happening" is being jettisoned here) there is a change in perspective that "ratchets that atom up to a new now" (the English language is up against its conceptual limits here).  And, of course, it's not just clocks.  Anything made of atoms (which is to say, everything) works this way.

We will see all this again when we get to quantum mechanics, which we will have to do before we can answer the obvious question: if time doesn't actually pass, if things don't actually happen, why does it appear that they do?

I'll leave you with just one parting thought: the fact that it is impossible to measure the one-way speed of light can be explained in two different ways.  One, as I've done above, is to point out that the whole concept of the "one-way speed of light" is mathematical nonsense, quite literally zero divided by zero.  But there is also a practical way to understand it: think about what it would take to make this measurement.  You would need two clocks in two different locations, and you would need to measure that time that light left one location and the time it arrived at the other.  To do that you would have to make sure that the two clocks were synchronized, and there is no way to do that without assuming that the one-way speed of light is a constant.  But notice what happens if you don't make this assumption: not only is it impossible to measure the one-way speed of light, it is impossible to measure the one-way speed of anything for the exact same reason: you can't synchronize your clocks unless you assume the one-way speed of light!  For slow-moving objects you can get very close, but there will always be an uncertainty corresponding exactly to your uncertainty about the one-way speed of light.  We will see this kind of fundamental "uncertainty principle" again as well when we get to quantum mechanics.