### The Unruh Temperature

```The Unruh
Temperature
For a Uniformly Accelerated Observer
Cory Thornsberry
December 10, 2012
The Unruh Effect
• Two inertial observers in the Minkowski vacuum will
agree on the vacuum state
• We add a non-inertial observer accelerating with
constant acceleration, a.
• The accelerating observer will “feel” a thermal bath
of particles.
The Unruh Effect contd.
• “…an accelerated detector, even in flat
spacetime, will detect particles in the vacuum”
Unruh, 1976
• There is a physical temperature associated with the
particle bath, Tu.
• For simplicity, we assume…
o Uniformly accelerated observer
o Acceleration is only in the z-direction
The Inertial Observer
• The accelerating observer is moving through socalled Rindler Space, but first…
We begin in Minkowski Space
2 =  2 −  2
The Inertial Observer
Thus, our Klein-Gordon equation becomes
= 2 − 2  = 0
Allowing solutions of the form
,  =
1
2
±
Where
=  = 2 +  2 =
± =  ±
The Inertial Observer
So, for the Inertial Observer, the
massless scalar field becomes
∞
,  =
0
†

(() +   ∗ )
2
With
†
, ′
= ( −  ′ ) and
|0 = 0
Rindler Space
Our metric is invariant under a Lorentz boost
→  cosh  +  sinh
→  sinh  +  cosh
We may Re-parameterize our coordinates as
() =  sinh
=  cosh
Our metric Becomes (the Rindler metric)
2 = 2  2 − 2
Rindler Space
Now we make the transformation
=   ,  =
1
1
=  sinh  =  sinh

1
1
=  cosh  =  cosh

→  2 =   ( 2 −  2 )
The Rindler Observer
Based on the transfromed Rindler metric
2 −  2  = 0
Is our new field equation, allowing
,  =
1
2
±∓ =
1
2
(∓ )

±
Where
± =  ±  =
1

±
and
± > 0
The Rindler Observer
• Our trajectory (world)
curves are restricted to
Region I
• We need to cover all of
Rindler space for valid
solutions
• We may “extend” our
solutions into the other
regions
• (t,z) may vary in all
space. (τ,ξ) is restricted
to RI
Region
I
II
III
IV
z+ = z+t
>0
>0
<0
<0
z- = z-t
>0
<0
<0
>0
Table 1: Values of z± vs. Region
Fig 1: Rindler Space
The Rindler Observer
• We required that z± > 0
• We may analytically extend  − into region IV
where z- > 0
• Additionally, we may extend  −+ into region II
where z+ > 0
• z± is never positive in Region III
• We may not extend the solutions into RIII. We do
not have a complete set of solutions
The Rindler Observer
• We perform a time reversal and a parity flip, (, ) →
(−, −)
• This exchanges RI for RII and RIII for RIV
We get two (Unruh) modes
(1)

(2)

1
=
=
2
1
2

−
1
=
0
0

′
−+
=
2
1
2
−(−)

(′ − ′ )

,

,

The Rindler Observer
We now have all the parts of the
Field equation for the Rindler observer
∞
,  =
0

(
2
1
(1)
()
+
2
2
()
+
1
†
1 ∗
()
+
2
†
We must now relate the Unruh modes to the
modes of the Inertial observer
2 ∗
()
)
The Bogoliubov
Transformation
We define new solutions
(1)

(2)

=
=
1

2

2 ∗

−
1 ∗

+
+
−
Leading to the updated scalar field
∞
,  =
0

(
2

1
(1)
()
′ ,

†
+
2
2
()

+
1
†
1 ∗
()
+
−
=
(2)( −  ′ )

2 sinh

2
†
2 ∗
()
)
The Bogoliubov
Transformation
Now define

=
− 2

2 sinh

We may re-write the Rindler modes as

=
1
2 sinh

2
+

− 2

†

The Bogoliubov
Transformation
• Those two modes are known as a Bogoliubov
Transformation. They relate the modes of the
inertial and Rindler observers.
The Unruh Temperature
• Assume the system is in the Minkowski vacuum, |0
The number operator is given by
=
1
†

1

We are interested in the expectation
value of the number operator
The Unruh Temperature
We get

−

2
0|

2 sinh

1
= 2
2 (0)
−1

0  0 =
2
†
|0
The factor looks surprisingly like Planck's Law
1
~ ℎ
− 1
The Unruh Temperature
We can compare the arguments of the exponentials
in the denominator of both equations to find that...
ℎ
~
2
Conclusion
• So, an observer moving at a constant acceleration
through the vacuum, will experience thermal
particles with temperature proportional to its
acceleration!
• This does not violate conservation of energy. Some
of the energy from the accelerating force goes to
creating the thermal bath.
• The observer will even be able to "detect" those
thermal particles in the vacuum!
References
• Bièvre, S., Merkli, M. “The Unruh effect revisited”. Class. Quant.
Grav. 23, 2006 pp. 6525 – 6542
• Crispino, L., Higuchi, A., Matsas, G. “The Unruh effect and its
applications”, Rev. Mod. Phys. 80, 1 July 2008 pp. 787 – 838
• Pringle, L. N. “Rindler observers, correlated states, boundary
conditions, and the meaning of the thermal spectrum”. Phys.
Rev. D. Volume 39, Number 8, 15 April 1989 pp. 2178 – 2186
• Siopsis, G. “Quantum Field Theory I: Unit 5.3, The Unruh effect”.
University of Tennessee Knoxville. 2012 pp. 134 – 140
• Rindler, W. “Kruskal Space and the Uniformly Accelerating
Frame”. American Journal of Physics. Volume 34, Issue 12,
December 1966, pp. 1174
• Unruh, W. G. “Notes on black-hole Evaporation”. Phys. Rev.
D. Volume 14, Number 4, 15 August 1976 pp. 870 – 892
```