# I Unruh temperature

#### PrashantGokaraju

A uniformly accelerating observer registers a temperature

T = 1/2π

Or an observer hovering near a black hole. According to current understanding, what are the best ideas about the origin of this temperature?

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Staff Emeritus
According to current understanding, what are the best ideas about the origin of this temperature?
Mine is that you miswrote something. The title doesn't match the message, there are no units for your temperature, and what you wrote is independent of acceleration. Maybe you miswrote more than one thing.

#### PrashantGokaraju

The dimensionless character of this temperature is due to the fact that the Rindler time is dimensionless.

A conventional temperature with units of energy or inverse length would be recorded by a thermometer, and this is

T = a/2π

where a is the acceleration.

#### Tghu Verd

A uniformly accelerating observer registers a temperature
Are you perhaps referring to the so-called Unruh effect?

If not, is there a reference to your assertion that describes what you are referring to in more detail?

#### PrashantGokaraju

Yes, unruh effect. For example if you observe a black hole by it's unruh radiation, you always get a blurry image.

Staff Emeritus
For example if you observe a black hole by it's unruh radiation, you always get a blurry image.
Huh? "observe a black hole by it's unruh radiation" is a non-sequitur.

#### Nugatory

Mentor
According to current understanding, what are the best ideas about the origin of this temperature?
The Wikipedia article on “Unruh effect” will be a pretty good starting point.

#### PeterDonis

Mentor
The dimensionless character of this temperature is due to the fact that the Rindler time is dimensionless.
First, you didn't just write a dimensionless number, you wrote a constant, $T = 1 / 2 \pi$. The Unruh temperature is not a constant; it varies with acceleration. When you write the temperature correctly, as you did in post #3, of course it has units of acceleration, i.e., inverse length.

Second, Rindler time is not dimensionless; it has units of time, since it's just proper time along the worldline of the Rindler observer at $x = 1$ in Rindler coordinates.

#### DarMM

Gold Member
This is a hard subject to explain. How is your handle on QFT?

#### PrashantGokaraju

First, you didn't just write a dimensionless number, you wrote a constant, $T = 1 / 2 \pi$. The Unruh temperature is not a constant; it varies with acceleration. When you write the temperature correctly, as you did in post #3, of course it has units of acceleration, i.e., inverse length.

Second, Rindler time is not dimensionless; it has units of time, since it's just proper time along the worldline of the Rindler observer at $x = 1$ in Rindler coordinates.
The Rindler hamiltonian is

H = ∫ ρT00

The Rindler energy of the ith state of the thermometer is ρεi where εi are the proper rest energy levels. So if the Rindler temperature is 1/2π, then the probability of the ith level is ~ e-2πρεi

This means the thermometer registers a proper temperature `1/2πρ = a/2π

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#### PrashantGokaraju

This is a hard subject to explain. How is your handle on QFT?
I do know the basic ideas of canonical quantization and path integrals, feynman diagrams etc. mostly in the special case of scalar fields. Did not study too much other field theories like QCD etc. I want to know for example, what does string theory say about the Unruh temperature and the entropy, microstates etc? Also the various other facts about black holes, like the blurry image, visibility of space behind them, etc. What is the most current understanding of these things, in string theory, or any other modern ideas

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#### DarMM

Gold Member
The origin of the Temperature is that the decomposition of a quantum state into particles is not unique.

A particle is associated with positive and negative frequency modes of solutions to the free field equation. Thus note: particles are intrinsically associated with free quantum fields and only emerge as asymptotic concepts in interacting theories.

The time coordinate of an accelerating observer produces a completely different set of functions as the positive and negative frequency decomposition of solutions to the free field equation. This gives rise to a different notion of particle, Rindler particles say. The Unruh effect is due to the vacuum containing no intertial particles, but being a Thermal bath of Rindler particles.

#### Michael Price

Huh? "observe a black hole by it's unruh radiation" is a non-sequitur.
Not a non-sequitur. Just bloody difficult. Bryce DeWitt derives the temperature of a black hole, observed at infinity, from the surface Unruh temperature, in General Relativity: an Einstein Centenary.

#### PeterDonis

Mentor

"Unruh temperature"

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