0:10

In 1907, two years after the miracle year, Einstein had been asked to write

a review article on the special theory of relativity for a science yearbook,

is really designed, not for the general public, necessarily but for

other scientists, just summarizing some of the key results of previous years.

And, he wrote later, reminisced later that he

had what he called the happiest thought, or most fortunate thought of his life.

In fact, here's how he described the situation.

0:46

He had been bothered as he thought more about his special theory of relative,

he's bothered by a couple of things.

And one thing was there's only constant velocity motion and therefore, it wasn't

general enough and also didn't take in the count any gravitational effects.

So, he had been mulling those things over and

later he reminisced that I was sitting in a chair in the padded office in Behrns,

Switzerland when all of a sudden a thought occurred to me,

if a person falls freely, he will not feel his own weight.

And he says this revelation startled him.

And as he thought more about it over the next eight years,

the end result was the development of his general theory of relativity.

So even though in one sense it's beyond the course,

we want to get at least a little insight into what that is all about,

and see actually, in a couple instances, well one instance at least,

the one we're going to talk about in this video,

it gives us a result that is similar to the special theory of relativity but

different in a couple key respects, that is a time dilation effect due to gravity.

He talked about that if a person as he says,

if a person falls freely he will not feel his own weight.

So the classic example of doing that in Einstein himself

use this would be in an elevator.

So let's imagine here's just an, it's a thought experiment,

sort of an elevator out there in the middle of space.

2:15

If it's falling freely, or even just a real elevator,

if you cut the cable, and there are no other safety mechanisms.

As you fall freely, you will not feel your weight.

If you drop a ball at that point, it'll just stay there.

Because your all falling down towards the center of the earth down

to it's surface of the earth.

So, it'll be like gravity doesn't exist anymore and

2:42

Einstein started analyzing that situation a little bit more and

noted also that if you say for your elevator again, you attach a cable to it

and maybe there's a crane or something, [INAUDIBLE] space, so much [INAUDIBLE]

here but the idea is you have something where now you can lift the elevator up.

In fact you can accelerate it, in upward direction at some acceleration a.

Again, just like we would do in a real elevator, as the elevator goes up,

you feel that momentary push downward.

You feel heavier.

Okay, it's like gravity has actually increased.

And so, an acceleration upwards,

if you're in the elevator is therefore equivalent to a gravitational field.

And this was essentially Einstein's happiest thought and

it came to be known as the Equivalence Principle.

The Equivalence Principle.

3:38

That essentially a gravitational field is equivalent to

an accelerated frame of reference, okay.

That this person inside here, if they're walled off from everything, they could be

in the [INAUDIBLE] space for all they know, if the crane or the elevator or

whatever, the cable, pulls ups on them at an acceleration that's equivalent to

the acceleration due to gravity, then it feels like gravity to them.

They couldn't tell whether they were in the elevator just on the surface

of the earth feeling gravity pulling them down or they're actually out in middle

of space and crane is actually pulling them at an acceleration such that they

press down into the floor as it were and you feel that normal gravitational pull.

So that's the equivalence principle,

gravitational field is equivalent to an accelerated frame of reference.

And so you see that opened up some things for Einstein because he was interested in

going beyond the Special Theory of Relativity,

which was only constant velocity inertial frames of reference.

And here was something that enabled him to perhaps, if he pursued this, to

bring in accelerated frames of reference and also tie it into gravity, as well.

And so again, over the next eight years, and a number of stops and starts and dead

ends and so on, and so forth, plus he's working on other things along the way.

He was able to come up with his general theory of relativity.

And the key thing here, for analysis, is that what the Equivalence Principle

enables you to do is, if you do an analysis involving acceleration,

5:12

the conclusions of that analysis also apply to a gravitational field.

A situation where you just have a gravitational field occurring.

We want to do two examples of that.

In this video we are going to do a time dilation example.

In the next video will do the bending of light example.

And so, let's imagine a situation like this.

So we're going to have an elevator here, and

we'll have our two observers, or really two clocks.

5:39

So put a clock at the top of the elevator, a clock u for upper, and

down here we'll have clock L for lower.

And these clocks, emit little pulses of light.

So, here's one,

and we'll get some nice red pulses of light coming up here, in that direction.

And then, up here, another little laser, and pulses of light going down.

And we'll have a detector here and there, okay.

And so, the clock, each clock and these are synchronized clocks,

identical clocks and they omit these little pulses of light

that maybe they emit ten pulses light per one tick of each clock.

Okay, now another factor here is that we're going to assume

any velocities involved are much less than the speed of light.

So this example will have nothing at all to do with the special

theory of relativity.

We'll assume the velocities are low enough that we can ignore

all relativistic effects.

That they're very, very small compared to anything else going on here.

So, that the clock's.

So, that means the two clock's are insinc with each other.

Right, we don't have to worry about leading clock's like in all left because

new velocity is involved or much less and see and were going to now put this

into acceleration in an upward direction.

So we are going to accelerate it in upward direction but again at very,

very small velocity.

Just sort of ordinary every day velocity so

we can ignore any special relativistic effects there.

7:26

They will be observing these pulses of light coming down to their detector here.

And meanwhile, their pulses go up for clock U.

As these pulses are emitted here, of course, they're traveling along now.

But while they're traveling the elevator moves up slightly.

It's accelerated up a little bit here such that these,

let's say there are, again, ten pulses per

7:57

tick of each clock, okay?

So, clock L detector, though, because it's accelerated upward a little bit while

these pulses are in motion, will collect more pulses per tick of its clock, or

collect more of these pulses coming down, because it's sort of moving up a bit and

sweeping through them a little bit faster because of the acceleration involved here.

And so, maybe instead of receiving ten pulses,

which it normally would do per tick of its clock,

it receives 12 of these pulses of light, coming down from the upper clock.

So let's write that down.

For clock L here, let's say it receives 12

pulses During one tick.

One tick of

its clock.

All right?

So it receives 12 pulses coming down,

where as normally it would receive only ten.

So what does that tell us then?

It tells us that this clock is therefore running slower than this clock, okay?

because it's receiving 12 pulses during one tick.

Well, they're both synchronized to do ten pulses per tick.

But now let's look at the upper clock and it's getting 12 pulses per tick.

So now we have to look at the upper clock it's running faster then it is.

It's only doing ten pulses.

Well, it looks like the upper clock is doing 12 pulses in one tick.

Therefore, the upper clock is running faster.

You can also look at the other way around if you want to do

the perspective of the upper clock.

It's accelerating but

it's accelerating away from the pulses coming up from the lower clock.

And so as it gathers in those pulses at the detector,

instead of maybe seeing ten pulses coming in during one tick of its clock.

Ten pulses from the lower clock.

Maybe it only receives eight pulses during that one tick.

So this clock, it's sending out ten pulses per second, everything is fine.

But I'm only receiving eight pulses per one of my ticks from clock L, and

therefore looks like the lower clock is running slower.

So either way you analyze it, you come to the conclusion that if you have this

acceleration involved, again velocity is much less than the speed of light,

no special relativity involved here.

But just the fact that you have even a small acceleration, you'll find that

the lower clock runs slower compared to the upper clock when they compare clocks.

The lower clock sees the upper clock running faster,

the upper clock sees the lower clock running slower in this case.

And then you say hm, got the Equivalence Principle.

Therefore this situation also applies to the gravitational field.

And you'll have got the acceleration going upwards so

if you put someone in here standing in the elevator.

10:50

If you make this equal to acceleration due to gravity on earth then that person would

feel a gravitational field it would be like there was a gravitational field and

Einstein would say according to the equivalence principle

It's the same analysis either way.

So the conclusion here is we did the analysis with an accelerated frame of

reference and concluded that the upper clock runs a little faster than the lower

clock or the lower clock runs a little slower.

In other words, lower down in a gravitational field,

11:16

clocks are going to run slower than up high.

Lower altitude, at a lower altitude, a gravitational field,

like at the surface of the Earth one clock is at the surface of the earth and

one clock is up on an airliner or up in space on a satellite.

This clock on the surface it going to run slower

than the clock on the upper part, the higher altitude.

And so this is gravitational.

11:51

Is what we're talking about here.

So, not only do we have a time dilation effect at high velocities, Special Theory

of Relativity, when we're just in a gravitational field there is time dilation

effect between a lower clock and a clock higher up in the gravitational field.

And they've actually done experiments with this.

The physics building, they just sort of from the basement to the top floor.

They have two very accurate clocks.

12:18

And you can see it's not quite a set up like this with the pulses but

something similar.

And you can actually see that the lower clock runs slower than the upper clock.

Even over just the height of a building.

Again if you have very accurate clocks involved.

Another thing, we've actually mentioned this a couple of times before.

But the global positioning system, the GPS system,

has to be designed to take this into account.

And actually, has to be designed to take into account the special theory of

relativity, and the general theory of relativity involving gravitation here.

It turns out, if you do they analysis on the satellites that are involved in

the global positioning system, they're moving at pretty high velocity compared to

the earth, compared to just a reference point on the earth.

Therefore, it turns out if you do an analysis for

the special theory of relativity, so special theory,

13:21

Says that the clocks, okay, since they're going faster time dilation, right?

If you observe the clock of a moving object they're going fast,

you the observer say, here on surface of the Earth and we

watch the satellite going by overhead, or really the satellite might be stationary,

but we're rotating on the Earth, however you want to look at it.

We're at rest, satellite is moving,

therefore we observe the satellite clock running more slowly.

And it's about seven microseconds slow per day, is the effect.

This is the abbreviation for microsecond.

Millionths of a second, okay?

Seven millionths of a second slow per day.

14:11

the general theory from gravitational time dilation, if you do the analysis there.

Remember, the lower clock is going to run slower.

The higher clock,

in this case the satellite up there in space, is going to run faster.

And so the satellite clock, from our perspective, runs about 43

microseconds fast per day.

So there are sort of competing effects there.

And you put them together and you can see that it's

roughly, what is it here?

36 microseconds, something like that.

It's in that range if we got the numbers quite right there.

But 36, 37 microseconds when you do the analysis fast per day.

14:55

And you might say, well, I mean, that's not a very big difference if you're

talking about millionths of a second, 38, 40 millionths a second, whatever it is.

But at the speed of light, okay,

as the signals go from ground to the satellites at the speed of light.

The speed of light in about 30 to 40 microseconds travels 10 or 11 kilometers.

15:16

What that means is, in designing the GPS system, if you don't take into account

the special theory contribution and then the general theory contribution

in the other direction, fast for the general theory time dilation,

gravitational time dilation, and slowing down for the special theory of relativity.

If you don't take both of those into account, and get the numbers right there,

your GPS system is going to drift by 10 or 11 kilometers per day.

In other words it's not going to be a very accurate system.

So they actually had to design the system

such that the clocks ran slow on the satellites.

In other words, it has to run slow by 36 or 37 micro seconds

16:00

per day, compared to an identical clock at the ground.

By taking that into account, they are able to get a very accurate system.

Again, that's Einstein's happiest thought, or most fortunate thought of his life,

depending on how you do the translation of the German.

Kurt in 1907 developed it over the next several

years into the general theory of relativity or

one key part of the general theory of relativity and

one of the consequences of that again is gravitational time dilation,

that a clock In a gravitational field lower down in the gravitational field

runs slower than a clock higher up in a gravitational field.