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
𝑑𝑠 2 = 𝑑𝑡 2 − 𝑑𝑧 2
The Inertial Observer
Thus, our Klein-Gordon equation becomes
𝜕𝜇 𝜕𝜇 𝜑 = 𝜕𝑡2 − 𝜕𝑧2 𝜑 = 0
Allowing solutions of the form
𝑢𝑘 𝑡, 𝑧 =
1
2𝑘
𝑒 𝑖𝑘𝑧±
Where
𝐸 = 𝜔𝑘 = 𝑚2 + 𝑘 2 = 𝑘
𝑧± = 𝑧 ± 𝑡
The Inertial Observer
So, for the Inertial Observer, the
massless scalar field becomes
∞
𝜑 𝑡, 𝑧 =
0
†
𝑑𝑘
(𝑏(𝑘)𝑢𝑘 + 𝑏 𝑘 𝑢𝑘∗ )
2𝜋
With
†
𝑏𝑘 , 𝑏𝑘′
= 𝛿(𝑘 − 𝑘 ′ ) and
𝑏 𝑘 |0 = 0
Rindler Space
Our metric is invariant under a Lorentz boost
𝑡 → 𝑡 cosh 𝛽 + 𝑧 sinh 𝛽
𝑧 → 𝑡 sinh 𝛽 + 𝑧 cosh 𝛽
We may Re-parameterize our coordinates as
𝑡(𝜏) = 𝜌 sinh 𝜏
𝑧 𝜏 = 𝜌 cosh 𝜏
Our metric Becomes (the Rindler metric)
𝑑𝑠 2 = 𝜌2 𝑑𝜏 2 − 𝑑𝜌2
Rindler Space
Now we make the transformation
𝜌 = 𝑒 𝑎𝜉 , 𝜏 = 𝑎𝜂
1
1 𝑎𝜉
𝑡 = 𝜌 sinh 𝜏 = 𝑒 sinh 𝑎𝜂
𝑎
𝑎
1
1 𝑎𝜉
𝑧 = 𝜌 cosh 𝜏 = 𝑒 cosh 𝑎𝜂
𝑎
𝑎
→ 𝑑𝑠 2 = 𝑒 𝑎𝜉 (𝑑𝜏 2 − 𝑑𝜉 2 )
The Rindler Observer
Based on the transfromed Rindler metric
𝜕𝜏 2 − 𝜕𝜉 2 𝜑 = 0
Is our new field equation, allowing
𝑢𝑘 𝜏, 𝜉 =
1
2𝑘
𝑒 ±𝑖𝑘𝜉∓ =
1
2𝑘
(𝑎𝑧∓ )
𝑖𝜔
±𝑎
Where
𝜉± = 𝜉 ± 𝜏 =
1
𝐿𝑛
𝑎
𝑎𝑧±
and
𝑧± > 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 𝑒 𝑖𝜔𝜉− into region IV
where z- > 0
• Additionally, we may extend 𝑒 −𝑖𝜔𝜉+ 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, (𝑡, 𝑧) →
(−𝑡, −𝑧)
• This exchanges RI for RII and RIII for RIV
We get two (Unruh) modes
(1)
𝑢𝑘
(2)
𝑢𝑘
1
=
=
2𝑘
1
2𝑘
𝑒
𝑖𝑘𝜉−
1
=
0
0
𝑒
′
−𝑖𝑘𝜉+
=
2𝑘
1
2𝑘
𝑒 −𝑖𝑘(𝜏−𝜉)
𝑒
𝑖𝑘(𝜏′ −𝜉 ′ )
𝑅𝑒𝑔𝑖𝑜𝑛 𝐼
,
𝑅𝑒𝑔𝑖𝑜𝑛 𝐼𝐼𝐼
𝑅𝑒𝑔𝑖𝑜𝑛 𝐼
,
𝑅𝑒𝑔𝑖𝑜𝑛 𝐼𝐼𝐼
The Rindler Observer
We now have all the parts of the
Field equation for the Rindler observer
∞
𝜑 𝑡, 𝑧 =
0
𝑑𝑘
(𝑐
2𝜋
1
(1)
(𝑘)𝑢𝑘
+𝑐
2
2
(𝑘)𝑢𝑘
+𝑐
1
†
1 ∗
(𝑘)𝑢𝑘
+𝑐
2
†
We must now relate the Unruh modes to the
modes of the Inertial observer
2 ∗
(𝑘)𝑢𝑘
)
The Bogoliubov
Transformation
We define new solutions
(1)
𝑈𝑘
(2)
𝑈𝑘
=
=
1
𝑢𝑘
2
𝑢𝑘
𝑎𝑢 2 ∗
𝑘
−𝜋𝜔
𝑎𝑢 1 ∗
𝑒
𝑘
+𝑒
+
−𝜋𝜔
Leading to the updated scalar field
∞
𝜑 𝑡, 𝑧 =
0
𝑑𝑘
(𝐵
2𝜋
𝐵
𝑟
1
(1)
(𝑘)𝑈𝑘
𝑘′ , 𝐵
𝑠
†
+𝐵
2
2
(𝑘)𝑈𝑘
𝜋𝜔
𝑘
+𝐵
1
†
1 ∗
(𝑘)𝑈𝑘
+𝐵
𝑒− 𝑎
=
𝛿 𝑟𝑠 (2𝜋)𝛿(𝑘 − 𝑘 ′ )
𝜋𝜔
2 sinh
𝑎
2
†
2 ∗
(𝑘)𝑈𝑘
)
The Bogoliubov
Transformation
Now define
𝑑
𝑟
𝑘 =
−𝜋𝜔 2𝑎
𝑒
2 sinh 𝜋𝜔 𝑎 𝐵
𝑟
𝑘
We may re-write the Rindler modes as
𝑐
𝑟
𝑘 =
1
2 sinh
𝜋𝜔
𝑎
𝜋𝜔
𝑒 2𝑎 𝑑 𝑟
𝑘 +𝑒
𝜋𝜔
− 2𝑎
𝑑
𝑠
†
𝑘
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
𝑁 𝑘 =𝑐
1
†
𝑘 𝑐
1
𝑘
We are interested in the expectation
value of the number operator
The Unruh Temperature
We get
𝜋𝜔
− 𝑎
𝑒
2 𝑘 𝑑
0|𝑑
𝜋𝜔
2 sinh
𝑎
1
= 2𝜋𝜔
2𝜋 𝛿(0)
𝑎−1
𝑒
0𝑁 𝑘 0 =
2
†
𝑘 |0
The factor looks surprisingly like Planck's Law
1
𝐵 𝑇 ~ ℎ𝜔
𝑒 𝑘𝐵 𝑇 − 1
The Unruh Temperature
We can compare the arguments of the exponentials
in the denominator of both equations to find that...
ℎ𝑎
𝑇𝑢 ~
2𝜋𝑘𝐵
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