Hello, welcome to the last lecture. My voice is a little bit weak today. We had discussed,

we had started to discuss the Unku effect. That is, if you have an inertial frame and

you have your usual vacuum, what happens if you have an accelerated observer?

Let me just remind you of a few things.

So this is the situation where space time, some zero point factor relations.

And we know that they are low and invariant in the sense that the statistics are the same

for all frames. But what happens if I have an accelerated observer? So this is just constant

acceleration. The asymptotex would be just the speed of light. And the goal will simply

be calculate the correlator of the yield of two points along the trajectory of the observer

and express the result in terms of the difference of properties and compare against known results

in inertial frames. Now we had already discussed this hyperbolic motion I had written down

in the formulas. Now we are now going to discuss the correlator as seen by the observer.

So I would evaluate my field phi at the second space time point, which is x of tau 2, t of tau 2,

and tau 2 is the corresponding proper time. And I would take the correlator with the corresponding

expression at point one. And this correlator we know, this is the ground state correlator

in the inertial frame. In that frame it's not difficult to have written down the expression.

So we know this is h by c over i 1 over delta x squared plus c squared delta t squared.

And delta x is obviously the difference of these two coordinates. Now we can plug in our

expressions for the trajectory here. I won't write them down again, but I'll tell you the result.

So the result is finally delta x squared, delta c squared, delta t squared, contains a sine hyperbolic

motion which is no surprise given the shape of the trajectory. So that is minus c to the fourth over i squared.

It is remarkable that this only depends on the difference of proper times. This is not guaranteed

if the trajectory had any other shape, this would not be the case. These terms just combine in just

a black manner. And now we want to interpret this, this correlator, as a correlator in the frame

of reference of the accelerating observer, that is the correlator between two field values taken at

two different proper times, two and two and one, but of course at the same coordinate that is the

co-coordinate zero state of the observer. And so if we interpret it like that and compare against

our own results, especially for the finite temperature case, we find that the correlator for the observer

looks like the correlator of the thermo.

So the vacuum correlator with respect to these two space time points, when expressed in terms of the

proper times themselves, looks like a thermo correlator. And the interesting question is what is the

temperature? And that is, well, we only have one parameter left which is a and one fundamental constant

which is c, and so the combination is h by a over 2 pi c. So this is the unwu temperature for a given acceleration.

And the meaning of this is really, for example if you think of an atom, it will see these thermal fluctuations.

Sometimes it will be excited by these thermal fluctuations and then say fall down to the ground state again

and read that. Or even if you are accelerating something more complicated, then the whole rocket

will detect you and see this thermal variation.

The interesting question is of course what are the numbers?

Well, that depends on which acceleration you want to plug in. We could plug in one g, that is 10 meter per second

squared, gravitational acceleration, and then we would find that the temperature is 10 to the minus 20 Kelvin.

And even though people are very good at cryogenics nowadays, no one has ever reached these low temperatures.

So you wouldn't be able to measure this. I can also plug in lastly greater accelerations, then of course

the temperature increases linearly. For example, I had taken a single ion accelerated by one kilo eVolts per centimeter

that I get something like 10 to the 12 meters per second squared. That's already vastly larger. But of course it helps, but still

the temperature is 10 to the minus 8 Kelvin. Now this is already in the range of what is reached in terms of cryogenics

according to atoms and so on. But it would be very hard to see if you really accelerate so strongly.

Okay, so there are some attempts of somehow simulating this in other steps. For example, maybe you get something else

than the speed of light. There's an interesting connection of this to what happens at a black hole.

So if you are at the event horizon of a black hole, you can also calculate the corresponding gravitational acceleration

at that point, the gravitational acceleration of the Schwarzschild radius. And then again you get a temperature,