Einstein’s theory of special relativity is just chock-a-block full of odd features.
Different observers see clocks running at different rates and see the shapes of objects
differently.
It’s all very hard to get your head around.
However, the thorniest problem for people is what's called The Twin Paradox.
I’ll describe the problem and the solution in a moment.
I should warn you- there is some math ahead, but, well, that’s what you guys asked for.
Now, I’ve made two videos about how time is viewed differently by two different observers.
One is called Einstein’s Clocks and it is the experimental proof that special relativity
isn’t crazy.
And the other one talks about how to use time dilation properly and shows that you really
have to be careful about how to use Einstein's equations.
I’ve also made a couple of videos that introduce the equations, both the Lorentz transforms
and the meaning of the Lorentz gamma factor.
If you want the strong background you need to understand this video, I recommend that
you view these videos and maybe in that order.
They’re all on the Fermilab YouTube channel.
Okay, so the twin paradox.
What’s that all about?
Let’s start by explicitly stating the two core reasons that this seems so paradoxical.
First is that all observers can completely accurately claim that they are the single
unmoving person in the universe and everyone is moving around them.
And the second thing is that moving clocks tick more slowly than stationary ones.
Let’s imagine that you have a pair of twins, called Ron and Don.
Don stays here on Earth, while Ron heads off to Alpha Centauri, call it four lightyears
away, at a speed of 99.9 percent the speed of light.
So for Don, the amount of time the trip takes is basically 8 years- four years for Ron to
get to Alpha Centauri and four years to get back.
But Ron is traveling at high speed, so his clocks run slower.
For every time T don experiences, Ron experiences a duration of T over gamma.
And gamma is just equal to one over the square root of one minus v squared over c squared.
C is, of course, the speed of light, and I showed where this comes from in the Lorentz
gamma video.
If Ron is traveling at 99.9 percent the speed of light, then v over c is 0.999.
Substituting that in, you get a gamma of about 22.4.
And, finally, what you find is that while Don experiences 8 years for the round trip,
Ron experiences just a bit over 4 months.
So that’s the usual thing with relativity and some people hope to use this as a way
to actually conduct interstellar travel.
The astronaut ages more slowly and lives to get to their destination.
Okay- so far so good.
But now we have the paradox part.
The paradox part comes from what I said early on in the video, which is that both observers
can claim that they are stationary.
So why couldn’t Ron say that he is stationary and Don moves away from him and back?
If that’s the case, then Ron is the stationary guy and Don is the moving guy.
Then Ron should experience a long time and Don should experience a short time.
So that’s the paradox.
If each twin can say that they aren’t moving, then they both can say that the other twin
is experiencing shorter time.
And that doesn’t make any sense.
They can’t both be experiencing short time.
Now if you rummage around on the internet, you’ll find several responses.
One is that this proves that relativity is bogus and therefore you shouldn’t believe
it.
But that’s completely wrong and you absolutely shouldn’t believe that answer.
But that answer is out there.
A more reasonable explanation is that there is something that breaks the symmetry between
the two- something that makes Ron and Don’s experience different.
This approach uses the fact that Ron had to decelerate at Alpha Centauri and accelerate
to come back.
Further, in order to stop back at Earth, he needed to decelerate yet again.
Don experienced no such acceleration.
And, according to this line of thinking, it is the acceleration that makes the difference.
All of the time dilation- which is to say the shortening of the time experienced by
the traveler- occurs during the acceleration periods.
However, this explanation is totally wrong.
Now if you thought this was the explanation, don’t feel bad.
Many physicists who don’t work with relativity a lot believe the same thing.
But I can prove to you why this isn’t true.
So let’s do a thought experiment in the spirit of Einstein.
I want to set up a situation in which no acceleration occurs and show that time dilation does occur.
And I’m going to do this with some equations.
I won’t go into the gory details, but I’ll show the key points.
Before I do, I want to remind you of Einstein’s equations - what are called the Lorentz Transforms.
They simply convert the amount of space and time one person observes to the amount of
space and time another person experiences.
If we just label two people as 1 and 2 and person 2 is moving with respect to person
1, then the Lorentz equations are just what we see here.
I’ll pause for a moment so that you can look at them.
In my non-accelerating scenario, there are three observers, with the unimaginative names
A, B, and C. There are also three locations, which we call 1, 2 and 3.
Location 1 is on the left, location 2 is a distance L to the right of 1, and location
3 is a distance 2L to the right of 1.
When we start out, we put ourselves in the point of view of observer A. Observer A just
sits, unmoving, at location 1.
Observer B is at location 1, moving to the right at velocity v. Observer C is way over
at location 3, moving to the left at velocity minus v.
So, what’s going to happen?
Observer B will head to the right and Observer C will head to the left.
They will cross paths at location 2.
Observer B will keep heading off to the right and Observer C will pass by Observer A. All
pretty straightforward, right?
So here’s what we’re going to do.
When we start the experiment, all three observers are going to start a stopwatch.
When Observer B and C cross paths at the center, Observer B holds up a big digital clock that
Observer C can read.
That clock records how long it took for Observer B to go from location 1 to location 2.
Observer C also writes down the time they see on their own stop watch as they cross
location 2.
Observer C then travels back to position 1.
As they pass location 1, they hold up a big digital billboard that shows the reading on
their clock as they pass location 1, the reading on their clock as they passed location 2,
and the amount of time it took Observer B to get to location 2.
You can then work out the amount of time it took Observer C to get from location 2 to
1 and add it to the amount of time that Observer B took to get from 1 to 2, and then you can
figure out how long it took to travel from location 1 to 2 and back to 1 again, without
any acceleration.
So let’s define three events, which we’ll number with Roman numerals.
Event I is when they start.
Event II is when Observers B and C pass at the center.
And Event III is when Observer C passes location 1.
To work those out, we need to figure out the location and duration of each event according
to each observer.
We can start with Observer A. There are three places and times that are important.
They are when the experiment starts, when the two ships pass one another, and when Observer
C zooms by.
Remember that location 2 is a distance L away from Observer A and that, at least as far
as Observer A is concerned, Observer B is moving to the right at velocity v. Thus Observer
A would say that it took a time equal to the distance divided by the velocity (or L over
v) for Observer B to get to the meeting point.
Okay- I’m going to do some math here.
If math isn’t your thing, just let it roll over you and I’ll get to the punchline at
the end of it.
We can write those three important locations according to observer A as we see here.
The space and time coordinates for A are zero, zero for Event I, L, L over v for Event II,
and finally zero, two L over v for Event III.
If you need to take a moment to work this out, go ahead and pause the video until you’re
comfortable with what I said.
Okay, now we just use the Lorentz Transforms to figure out what position and time Observers
B and C see for those three points.
It’s important to remember to get the sign right on the velocity.
As far as Observer B is concerned, Observer A is moving with a negative velocity, but
Observer C sees Observer A moving with a positive velocity.
So we can do that, and we find that Observer B sees things differently.
She sees those locations like we see here.
And Observer C sees things in yet a different way, which we can see here.
Now these aren’t obvious.
If you want to do the calculations yourself, just put in the x comma t from observer A
into the Lorentz Equations with the appropriate velocity and it’s actually pretty easy.
Okay, so that’s the hard part.
Now we just need to figure out the total duration.
We do that by subtracting what Observer B’s clock was at event II and I.
We then need to do the same thing for Observer C, except for event III and event II.
And we get this here.
If we add the two durations for the outgoing trip and return trip we can get the total
duration for the moving observers and we find that the moving duration was equal to 2, divided
by gamma, times L over v.
Now compare that to what the stationary observer, Observer A, experienced and that’s just
a time of 2 times L, divided by v, which is the same as the moving observer, but without
the gamma.
So, when you get down to the final, nitty-gritty, you find that the duration of the moving observer
is just the duration of the stationary observer, divided by gamma.
Okay- so we’re out of the math.
What does this tell us?
Since gamma is a number that is always greater than or equal to one, that says that the moving
people’s duration is smaller than the stationary person’s duration.
That’s just what it means.
So, that gets us back to the paradox thing.
If you can’t know who is moving and who isn’t, what I’ve done here doesn’t seem
to have solved the mystery.
But it has.
Forget the math and focus on one, crucial, difference.
Observer A existed in one and only one reference frame.
The moving observers existed in two.
That’s the only difference.
So that’s the answer.
The Twin Paradox isn’t a paradox.
Further, the solution doesn’t depend on the accelerations- after all no accelerations
occurred in the example.
And the bottom line is that if a person leaves the Earth and flies to another star at high
velocity and returns, the traveler will age much slower than a person who stays on Earth.
So, if you’re the kind of person who wants to outlive your enemies, I think we have finally
have a workable plan.
Okay- so this video is a bit long and it had more math than usual, but sometimes debunking
commonly-believed wrong scientific explanations needs just that.
I hope you liked learning the truth about this very popular seeming paradox.
If you did, please like, subscribe and share.
And, of course, we love comments.
So, go and tell the world about your new knowledge and- remember that physics is everything.