In this course we will seek to “understand Einstein,” especially focusing on the special theory of relativity that Albert Einstein, as a twenty-six year old patent clerk, introduced in his “miracle year” of 1905. Our goal will be to go behind the myth-making and beyond the popularized presentations of relativity in order to gain a deeper understanding of both Einstein the person and the concepts, predictions, and strange paradoxes of his theory. Some of the questions we will address include: How did Einstein come up with his ideas? What was the nature of his genius? What is the meaning of relativity? What’s “special” about the special theory of relativity? Why did the theory initially seem to be dead on arrival? What does it mean to say that time is the “fourth dimension”? Can time actually run more slowly for one person than another, and the size of things change depending on their velocity? Is time travel possible, and if so, how? Why can’t things travel faster than the speed of light? Is it possible to travel to the center of the galaxy and return in one lifetime? Is there any evidence that definitively confirms the theory, or is it mainly speculation? Why didn’t Einstein win the Nobel Prize for the theory of relativity? About the instructor: Dr. Larry Lagerstrom is the Director of Academic Programs at Stanford University’s Center for Professional Development, which offers graduate certificates in subjects such as artificial intelligence, cyber security, data mining, nanotechnology, innovation, and management science. He holds degrees in physics, mathematics, and the history of science, has published a book and a TED Ed video on "Young Einstein: From the Doxerl Affair to the Miracle Year," and has had over 30,000 students worldwide enroll in his online course on the special theory of relativity (this course!).
The twin paradox (part 4)
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Paradoxes to Ponder
Week 7: Paradoxes to Ponder
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Larry Randles LagerstromAcademic Director
Okay, the twin paradox, now part four.
want to do a brief review of where we've been with this.
And, also as you can see,
we will do something with the Lorentz transformation here.
If this is going to distract you, I suggest pause the video right now, write
that down so you can focus on other things we're going to be talking about here.
And we'll get to this is a few minutes.
But remember, here's the situation, in a smaller diagram this time.
We had Alice in the rocket going at 0.6 c, gamma factor of 1.25.
Bob in the lab frame, and
this time we're going to specifically talk about the rocket and lab frame.
And as you can see over the equations here, we've got R and L standing for
rocket and lab.
Not right and left, but rocket and lab, just to try to keep things straight here.
Because sometimes we've have Alice moving and
sometimes we have Bob moving in previous analysis and examples.
And so sometimes, if you just keep using A and B, Alice and Bob,
then we forget who's moving and who isn't, it gets even more confusing perhaps.
So, Alice is moving this time in the rocket frame,
we measure things in light years and years,
and went out to a star three light years away and then traveled back again.
So, here's what Alice's world line looks like and
Bob's frame of reference, out to three light years, and then back.
So back in the negative direction, leftward direction and
ends up, according to Bob, 10 years later, he observes her,
takes 5 years to get to the star, 5 years times .6 time
the speed of light is 3 light years and then, another five years back.
So this analysis, remember is fairly straight forward, he expects,
he'll see 10 years tick off on his clock.
And because of time dilation, the factor that 1.25, 1 over gamma factor,
he sees Alice's clocks tick off four years in each case.
So that was Bob's analysis via time dilation.
We did Alice's analysis, where she sees Bob, since she's going away and
the star coming toward her and then back again.
And for that, it's a little more complicated.
We had to use, not only time dilation, but length contraction,
because she sees the distance to the star not as 3 lightyears as Bob does, but
a contractive distance.
So we had to take that into account.
And once we did that, we got the correct number of years ticking off on her clock,
4 years.
And then, though, the real puzzle was
we also saw that Alice of course saw Bob's clocks running more slowly and
the number we got for that was 3.2 years, again the gamma factor.
4 years on Alice's clock, she see's Bob's clock ticking more slowly
compared to her lattice of clock, she compared it to by factor 1.25,
which will lead 3.2 years, so why does an Alice see Bob only age 6.4 years?
Remember, the key was that when we analyze the turn-around here, essentially,
Alice's changing frames of reference.
And so, her lines of simultaneity were parallel to this,
the green dash line that slopes down like this.
And when she changes frames of reference,
then her lines of simultaneity are paralleled this way.
And at that turn-around, there's a jump that takes place between
frames of reference and therefore, Bob's clock speed up
compared to Alice's clock at that point and tick faster as it where.
You can actually delve into the details of that turn around and
analyze it and show in more details if you want, but that actually occurred.
But for our purposes, we just said that hey,
change in frame of reference, therefore, we can see on our diagrams that we did
that we get this jump in time from 3.2 years up to 6.8 years.
And then, during the second leg of the trip,
again Bob's clocks, as far as Alice observes, tick off 3.2 years.
Add it all together and you get 10 years.
So both of them come up with the same answers,
all of by different reusing with them.
Another thing, just to mention here, and as because some of you
are probably still thinking, well, It just seems, aren't they symmetrical?
In other words, Bob watches Alice go away and come back in.
Alice and her spaceship, from her frame reference, watches Bob go that way,
the star came this way, and then back again.
Just think about it, it seems very symmetrical.
But we made the point that only Alice has acceleration involved,
and Bob actually does not feel any acceleration.
We mentioned of course, if you're in a car that's accelerating, you can feel being
pushed back in the seat, or decelerating you're thrown forward a little bit, or
even moving side to side around around the curve.
Another, maybe, more physics way of doing that is to do a simple experiment.
Drop a ball, and if you're moving at constant velocity motion, and there's no
wind or anything like that, if you drop a ball, it will drop right down at your
feet, because the ball participates, again we're talking about slower speeds here.
We obviously did the analysis with relativistic speeds,
but the ball participates in the motion of the object as you're moving.
If you're accelerating,
however, what happens is the ball falls behind you for large enough acceleration.
For smaller acceleration, it actually does,
but you just can't notice it as much.
And so, essentially, what happens is, as you drop it,
the ball has the velocity at that point, so it would be going forward with you, but
you sort of accelerate out from underneath it and therefor, it falls behind you.
Or if you decelerate, it falls ahead of you, because it still has that initial
velocity, you've slowed down now, and so it moves ahead of you.
So Bob could do that dropping ball experiment all he wants,
and during this whole time, it would just fall at his feet.
He would not undergo acceleration, whereas Alice would
when she was at the turn around point, decelerating and accelerating.
So if that helps a little bit to see what's going on,
that even though it seems symmetrical, it actually is not.
And of course, the acceleration or
deceleration acceleration at the turnaround point is the key factor,
because that's what changes the frames of reference for Alice.
And then you get the jump going on there,
the jump in time as Alice observes Bob's clock.
So in one sense, we've talked about how the special theory of relativity is all
about inertial frames of reference.
It only applies to that, not In the case of acceleration and deceleration, but
you really don't need, say, the general theory of relativity here which deals with
acceleration and deceleration because you just assume the acceleration and
deceleration occurs over a short enough time
that you can essentially analyze it using the special theory.
And it's really the change in reference frame,
which is covered by the special theory, that gives us the results there.
So, yes, acceleration deceleration is involved, but
we can analyze the results and get the correct results using the special theory.
So, all that being said, what about the Lorentz transformation?
What we want to show is that you can get the same numbers using Lorentz
transformation, perhaps a little faster as well.
The downside is, first of all, you've got to use the correct formulas of course,
the correct versions of the formulas, so you've got to be careful about that.
But the downside is, you don't get as much feel for
what's going on as you do when you break it down via time dilation,
length contraction, and leading clocks lag, the relativity of simultaneity.
I do want to show that you can do this.
Again, our basic equation that we suggested you memorize,
if you're in the mood for that and want to use this a lot, is this one here
where we have the rocket moving away in a positive x direction l 4 lab And
one way to memorize whether or remember whether it's a plus or minus here.
Is just think about, okay.
The rock is moving away from me If I'm in the lab.
That mean's any rocket value for X would be a bigger value for me.
because it's moved away and so, If they measure something ten meters ahead of them
that's going to be maybe thirty meters, for me it's a twenty meters In that time.
So, if you can remember that basic situation, rocket moving away,
it's going to be a plus sign here in the basic forms of the equation.
Then from that you can derive pretty easily just in your head
that the other forms of the equation here.
So let's start over here with A.
Calculating Alice's results using Bob's Values and
so Bob's values are the lab frame values and
to Alice here Bob is moving in the negative x direction to the left.
And that's why we got minus signs in this versions.
But otherwise Bob's values reason it's nice just off of that
is Bob's values are Pretty obvious what's going on here, so
on the outbound trip Bob's value right here is
x we'll just put this in so we'll say x l
equals 3 light years right, right there and the time is 5, 5 years.
T l Equals 5 years.
And so that's for the outbound trip.
So the question is, okay, those are Bob's measurements right there.
What are the measurements x and t for Alice in the rocket at that point.
And if you plug the numbers in here,
gamma's 1.25, plug these values in for xL and tL.
V is 0.6 c.
We're using c = 1 because we're doing light years per year for c.
Plug that all in and you'll find you get 0 for here, and you get 4 for that.
Think a minute, does that make sense?
Well yes, because this is Alice's world line right?
She in her frame of reference hasn't moved.
She is at 0 in her frame of reference.
And her clocks tick off four years.
And that's what we got before.
So I guess that does make sense there.
Now we get to the inbound trip in a minute.
That's a little bit trickier.
Let's do the outbound trip for Bob's results using Alice's value, okay?
Well we could either do Alice's analysis as we did with the length contraction and
time dilation or even just use these results here.
We know that Alice Here is going to be xR at
the turnaround point there at the parameter star, xR equals 0 tR equals 4.
And if you plug those values in,
and you get as you should 3 And 5 in other words.
Bob's measurement of that space time point is x l equals 3
3 light years and time 5 years there.
So that checks out as it should.
Now what about the inbound trip here.
We have to be a little careful here because these Lorentz transformation
equations apply when the origin is the same for both of them.
In other words, when we're starting things at the origin and
at that point Bob's clock and measurements are at zero and same thing for Alice's.
Then these equation apply.
In this case we're out here starting point is out here at the 0.35,
and we're going back to the left.
So, what we have to do to make it easy to use our
general equations here is to redefining this is the origin point.
And then what happens here is, if you think about it, if this now is our new
(0,0) point just for the second leg, the inbound leg analysis right there.
That means if we think about it, XL is going to be negative three.
Okay, so this is now zero, count off negative three back to here.
And so, we can write XL equals negative 3 in there and TL's the same.
TL still goes 5, up 5.
So, it's really negative 3, tL equals 5 [Us?].
And if you plug in those values,
negative 3 and TL equals 5 just from the diagram here.
Again, you're going to get you're going to get x r equals 0 and
t r equals 4 another 4 years as you'd expect.
So 8 years there and
same thing inbound trip over here although in this case we just use x r.
Again x r equals 0 and t r equals 4 we've redefined the origin to be a 0 so,
and again, on the rocket ship for Alice she's just at 0,
she hasn't moved as far as she's concerned.
And TR = 4 and you plug those in.
And notice what happens here, for this one actually you get XL=-3 okay,
and cL equals five again.
And so you say, well wait a minute, xL equals negative three.
Well remember again, we've defined this as the origin,
so really as far as Bob is concerned,
Alice has moved negative three in the negative direction, in the left direction.
Okay, and once again, L and R here are not left and right,
they stand for lab and Rocket.
So it just shows us yes we can get the same answers using Lorentz's
transformation as long as we're careful with our plus and minus signs.
And think about the origin for the inbound leg there.
Let's also, one other way we could do this just to Review a number of things we've
done in the past and used for calculations and that is the invariant interval.
So let's just do a quickly put that up there and see how that works and
get the same results, or we should.
Okay so remember the basic form of our invariant interval equation.
Between two spacetime points
essentially is c squared delta t.
And here again we'll use lab and rocket frame again.
So l and r so i'll just do L squared minus
excuse me delta X just to remind ourselves
it's a difference in coordinates
delta XL squared equals C squared delta
TR squared minus Delta X R squared.
And again what's nice if we're using everything in light years per year,
light seconds per second then C just becomes 1 so we don't have to worry about
the, the C we're just looking at the difference in the time coordinates and
so for the outbound trip, outbound For
the outbound trip, we have delta xL is three.
Okay, so let's write that down, delta xL.
Now as it goes to three, minus zero, so
difference is three light years in this case.
Delta tL Again, we're assuming we've done Bob's analysis.
Delta tL is five years for the outbound case.
The other nice thing with the invariant interval if we're doing delta xR for
Alice As we've already mention a couple times, that's just zero, right?
Coz she stays, from her frame of reference she's not moving,
she's just in her spaceship.
And so we can see we have everything we need here except, so we got delta xL,
and delta t sub L, we have delta xR over here which is zero.
So that disappears.
So we're just left with delta R.
And when you do that plug in the numbers here.
We get, delta t sub R equals four.
That's easy to verify that if you want.
So, that's the outbound trip for the inbound trip very similar.
Again, so again you can see here, it's an even
easier calculation than the Lorentz transformation if you know the values.
And depending on what the situation is,
sometimes it's very easy to see what those are.
But you lose the insight behind it and that's why we started off using.
First we did our diagrams, our space time diagrams.
And then did our analysis with time dilation, and so on and so forth.
Now, for the inbound, notice what happens here,
is as far as Bob is concerned, Alice ends up at zero and starts at three.
So in this case, delta XL is negative three.
You do the second point minus the first point,
zero minus three is negative three there.
Delta tL though, equals five again.
Delta xR is zero again.
And once again you got the same thing.
You got delta tR equals four when you work that out.
So, that's the Invariant Interval approach to this.
Again, it can be useful in certain cases just to get a quick answer if you need it.
One final point here to make or remark.
And that is we've mentioned a couple of times about
experimental results that confirm all this.
Because no matter how many thought experiments we do like this, we,
there's also the lingering doubt that, is this really real, does it happen this way.
And it seems logical once you've starting from Einstein's principles and
sort of building on those.
And yes, we can see how with the light clock we get time dilation,
length contraction, relativity simultaneity, but because it's so
far out of our every day experience it's sometimes just hard to know.
So I mentioned for example the mewon experiment.
They've also done experiments with mewons in particle accelerators
where they essentially have some mewons just at rest and
then other mewons circling around them.
And from those they can show that the particles at rest
will decay faster than those in motion.
So again it shows the clocks slow down the ones in motion.
So that's well understood and
demonstrated but I've also mention these airline experiments they've done.
The first one, just if your interested a little bit.
And this was done in 1971 I believe.
Where they took they had two accurate atomic clocks on the ground.
And they took some portable one's up in commercial airliners actually and
flew around and some flew east bound, some flew west bound.
And they had to take in count a number of things going on including the rotation of
the earth.
And the fact that clocks actually run more faster at higher altitudes and lower
altitudes in a gravitational field like the earth effect next week we'll talk just
a little bit about that, to see how time dilation occurs in a gravitational field.
So they had to take those things into account as well as the velocity of
the airplanes.
And the eastbound and westbound was a little different because earth was moving
and therefore the clocks on the ground where not completely stationary, but
they were moving on the Earth, and so on and so forth.
So what did they find?
I won't give you all the numbers here, but I'll give you an idea.
For the westbound clocks that went on the planes, they had a prediction
that the difference between the clocks and the stationary clocks, stationary in
the sense they're still on the ground was 275 plus or minus 21 nanoseconds.
That was the prediction the result
was 273 plus or minus I think it was about 7.
And a second, something like that.
So pretty close, this bound result wasn't quite as good as out.
But, they predicted the clocks that flew around
would run more slowly than the clocks on the ground by this amount.
And the experimental result was very close to it.
They actually repeated this in 1996.
A form of this experiment where by that time they had better atomic clocks.
About 25 years later.
And they took one on a jet liner from Washington DC to London and
back again and the prediction there for
the result was about 30, almost 40 nanoseconds.
I think 39.8 nanoseconds.
And the result was about, I think,
39 plus or minus 2 nanoseconds.
You don't have to remember these numbers of course.
But the point is they're getting results that are very close to experimental or
the theoretical predictions.
And also just the fact that these are very very small time amounts here.
Billionths of a second are what nanoseconds are.
They've also done, in the intervening years since 1996,
not that long ago actually, but there's an experiment done in 2010
at the National Institute of Standards and Technology.
Yeah, NIST which is in Boulder, Colorado in the United States.
Other countries have these things, have these institutes as well that
are in charge of standards and making very precise measurements and things like that.
So in the United States it's the National Institute now the standards and
technology, used to be the National Bureau of Standards but
they changed the name a while back.
Anyway, 2010, they did an experiment with
a very very accurate form of clock that they had developed.
And instead of taking it on a jet airline or anything, they could just do it within
the lab at various speeds even as low as four meters per second.
Okay, so it's about 12 feet per second, if you want to think in those terms.
Up to maybe about 90 miles an hour.
So they did a variety of speeds, but slow speeds.
We're not talking about jetliner speeds.
And the experimental results
matched up nearly perfectly within the very small error bars they had.
To the predictions of the special theory of relativity.
So, the theory of relativity is well understood, theoretically.
It's a well supported, experimentally.
It doesn't mean that it's possible we could discover something that
men we had a change a little bit maybe the speed of light isn't constant,
over the time of the universe or
things like that and every once in a while in the news you'll hear something about
some experiment that says maybe something travels faster than the speed of light.
These experiments are so subtle though that often there are confounding factors
that you have to take care of and take into account and it's a very difficult at
the precision involved to really see if something is going on or not and
so far every time, Einstein and since, the theory has been proven correct.
So that's the twin paradox.
The most famous paradox of all and
just some of the analysis we've done in various ways with it and
then just the experimental results that show yes, no matter how weird it seems,
the twin paradox actually is an experimentally proven fact.
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