Einstein’s theory of special relativity has fascinated the public for about a hundred
years.
People hear about clocks running at different speeds, objects shrinking, and a myriad of
peculiar other effects.
And people might be willing to accept those sorts of assertions, but when they start thinking
about relativity, they come up with what seems to be paradoxes. And then they begin to doubt
the theory, because, well, you know- paradoxes.
One such paradox is the oft-quoted statement that moving clocks tick more slowly than stationary
ones. This is the whole basis of the ideas of interstellar travel, where a person heading
at high speed on a spaceship to a distant star ages more slowly than a person stuck
on Earth. If that rings a bell, that’s the basis of the famous Twin Paradox of special
relativity.
I’ll actually make a video focusing entirely on the Twin Paradox, but in this video I want
to really drive home some math that shows that you have to be ultra, mega, extremely
careful when you claim that there is a paradox in relativity.
So let’s start with the most common time dilation equation you’ll find in a relativity
textbook. It is t sub moving equals gamma times t sub stationary.
T sub moving is the time experienced by someone who sees the clock moving, T sub stationary
is someone who doesn’t see the clock moving and gamma is a factor unique to relativity.
It's called the Lorentz factor. It is equal to one for something not moving and greater
than one when something is moving. Gamma is simply one over the square root of the quantity
one minus v squared over c squared. That'll be important in a little bit, but, for the
moment, just remember that gamma is greater or equal to one.
At this point, it's probably helpful for you to watch two other videos of mine, specifically
the one on the derivation of the gamma factor and the second one introducing Einstein’s
equations. I won’t assume that you’ve seen them as I go through the rest of this
video, but what I say here will a bit clearer if you've got those two under your belt.
Okay, so let’s get back to the time equation. What it says is simple. Given that gamma is
greater than or equal to one, it means that the time experienced by a person seeing the
clock move is more than the time experienced by someone who doesn’t see it move. And
this means that someone moving ages more slowly than a person who isn’t. Now this sounds
crazy, but it has been proven to be true. And I made a video about that as well. It’s
called Einstein’s Clocks.
So this idea is behind a lot of science fiction and it’s true to a degree. But it’s not
the entire story. There are two key points I want to make here.
The first is that this makes absolutely no sense. Not because the two clocks tick at
different rates- although that’s pretty weird- but because of an ambiguity.
Suppose you label the two people as number 1 and number 2. If we take number 1 as seeing
the clock to be stationary and number 2 as seeing it moving, we see that number 2 experiences
more time than number 1.
However, relativity requires that the laws are the same, no matter who is not moving,
and this poses a problem. Suppose that we say number 2 is stationary with respect to
his clock and number one says that the clock is moving. Then number one experiences a longer
time.
Okay, so this is a problem. If you pick person one to be stationary with the clock, then
person two’s clock ticks faster. If you pick person two to be stationary with the
clock, then person one’s clock ticks faster. And both people are equally right in saying
that they are stationary. They can’t both have clocks ticking faster than the other.
That just makes no sense at all.
Either we have logically proven that relativity is just bogus, or there's something more to
it. Given that the scientific community still embraces relativity, there has to be something
more to it. And the answer is both very subtle and technical. So let’s dig into that.
So, begin by putting up the most basic and general equations of relativity. These are
called the Lorentz transforms and they show how you can translate between the point of
view of two observers who are moving with respect to one another. Let’s put up the
time dilation equation I mentioned before as well.
Notice that the Lorentz transforms have both a position and time equation. We’re not
interested in the position one here, so we’ll get rid of it and we’ll change the symbols
for the time one so it looks more like you usually see in the textbooks.
So the first thing you see is that the time dilation equation is a special case of the
more general one. They're the same only if that extra term of v over c squared times
x sub stationary equals zero. And that is true only if either the velocity v is zero
or x sub stationary is zero. The first one is pretty trivial, as that means that the
two people aren’t moving with respect to one another and so you don’t need relativity
theory at all. But the second is a lot trickier. It says that the time experienced by the person
seeing the clock moving depends both on the time experienced by the person not seeing
the clock move and by the location in the frame of the person who sees the clock to
be stationary.
So that’s a big point. What it means is that we have to be very careful about defining
which clock we’re talking about. There are many clocks stationary with respect to each
observer. Since they are at different locations, we need to take that into account.
Since this is a core point of this video, I want to explicitly show what this means.
Let’s start out by just talking about one observer, which we’ll call number 1, who
is stationary with respect to the clock. We draw him here with his clock.
But that’s not the only clock in his world. He has clocks at every location and, since
those clocks aren’t moving with respect to him, all of those clocks are ticking identically.
Now let’s add in observer 2. He sees observer 1 moving to the right at velocity v and he
wants to know what he thinks the moving clocks will read.
To do that, we will use the Lorentz transforms but, to do that, we need to figure out the
locations. Let’s pick two locations- say the location of observer 1 and observer 2-
as seen by observer 1. Observer 1 always thinks his location is x sub 1 equals 0.
Since observer 1 thinks observer 2 is moving to the left at velocity v, he sees observer
2’s position as changing. According to observer 1, observer 2’s position is just x sub 1
equals minus v times t sub 1.
Let’s stop the motion and concentrate just on the two locations and times as seen by
observer 1. His own location is simply x sub 1, t sub 1 equals 0, t sub 1.
Now let’s write down observer 2’s location as seen by observer 1. It is just x sub 1,
t sub 1 equals minus v times t sub 1, t sub 1.
We can now use the Lorentz transform for time. Let’s first find out what Observer 2 thinks
the time is at observer 1’s location. We just put in the values seen by observer 1.
In doing so, we find that t sub 2 equals gamma t sub 1. That’s just the normal time dilation
equation.
Now let’s do the same thing for how observer 2 sees what observer 1 sees at observer 2’s
location. Remember that observer 1 sees this location as x sub 1, t sub 1 equals minus
v t sub 1, t sub 1.
We start with the equation t sub 2 equals gamma times the quantity t sub 1 plus v over
c squared times x sub 1. We put in the x and t seen by observer 1.
Thus we get that t sub 2 equals gamma times the quantity t sub 1 minus v squared over
c squared times t sub 1. So that’s pretty easy.
We can factor out the t sub 1 and get this equation here. And at this point, we can use
the definition of gamma, which I remind you is one over the square root of the quantity
one minus v squared over c squared. That means that this equation can be written as t sub
2 equals gamma, divided by gamma squared, times t sub 1.
And finally, we get t sub 2 equals t sub 1 divided by gamma. That’s divided, not multiplied
by. That’s exactly the opposite thing we saw when we looked at what was happening at
observer 1’s location.
Now we need to step back and review a bit. This means that the transformation for what
observer 2 sees compared to what observer 1 sees depends on the location. Observer 2
sees more time elapsed than Observer 1 sees at Observer 1’s location, but sees less
time elapsed than Observer 1 sees at Observer 2’s location.
This is a super subtle point, but it clearly says that comparing clocks between two moving
observers is a tricky business. You need to take into account both the time each experiences
and the location at which the comparison is going on. Further, they both agree that the
remote person’s clock is slower. You probably need to think long and hard about this and
maybe even rerun this video to get your head around this idea.
A crucial point here is that when people start grabbing equations from relativity textbooks
without understanding exactly what the equations mean, then they’ll almost certainly make
a mistake. And perhaps the most important point is that if you ever- and I mean ever-
find something about relativity that sounds weird to you, always go back to the Lorentz
transform equations. They’ll never let you down.
Okay, so that might have been a mind-bender, but I hope it taught you not to be cavalier
about using relativity. And, of course, I welcome your comments and hope you’ll like
the video and subscribe to the channel. Plus share the video with your friends so we can
help people understand that you need to be careful with relativity. In the meantime,
remember- physics is everything.