Global Isometric Embeddings
Mathematical Tripos, Part III
Easter Term 2012
Abstract
We give a review of mathematical concepts and important existence theorems relating to isometric embeddings. These provide an alternative viewpoint to pseudo-Riemannian ­geometry,
which is the basis of General Relativity. A number of spacetime embeddings from the l­ iterature
are discussed and analysed in detail, with particular focus on the g­ eometry near a horizon
and particle worldlines. We next give an overview of the theory behind H
­ awking and Unruh
temperatures and discuss how global isometric embeddings provide a c­ onnection between
them, forming the basis of the so-called GEMS method. A formal ­argument in ­support of the
general validity of the GEMS method is given for static spacetimes.
Saran Tunyasuvunakool, April 2012
and their application to Black Hole Thermodynamics
1
Introduction
1
2
Mathematical Background
2
2.1
Immersions and embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2.2
Isometric embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.3
Extrinsic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.4
Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3
Ricci-Flat Geometries
8
3.1
A note on notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.2
The Rindler spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.3
The Schwarzschild spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.4
The Reissner-Nordström spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.5
Extremal Reissner-Nordström . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.6
Higher dimensional generalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.7
Rotating solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
4 Geometries with a Cosmological Constant
5
19
4.1
The de Sitter and anti de Sitter spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
4.2
The BTZ black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
4.3
Reissner-Nordström geometry in adS background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Black Hole Thermodynamics
24
5.1
Hawking and Unruh temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
5.2
The GEMS approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
5.3
Recent development and possible directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
References
29
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Saran Tunyasuvunakool, April 2012
Contents
Introduction
1
Introduction
General Relativity is often noted for its comparatively high level of abstraction. As is well known,
the groundbreaking idea of the theory was to relate gravitational effects to the intrinsic structure of
spacetime itself, allowing physical laws to be expressed in a manner which is entirely independent
of a preferred observer. Roughly speaking, the notion of a curved space is formalised by defining
a manifold as a topological space which is locally diffeomorphic to some open neighbourhood of
ℝ𝑛 , while distances are introduced by means of a pseudo-Riemannian metric tensor. This intrinsic
approach has one clear advantage: it allows us to discuss curved spacetimes without the need to
invoke the notion of an ambient space. While this may make a strong case for adopting GR as a
fundamental theory1 , it is arguable that these definitions don’t lend themselves readily to a visual
or intuitive interpretation. In fact, the picture that springs into one’s mind when contemplating a
curved space is probably that of a smooth surface floating in an otherwise-empty ℝ3 . In this picture,
on the plane. The metric on the surface is then naturally inherited from the standard dot product of
ℝ3 . Given that this picture is somewhat easier to grasp, it is natural to ask whether we could study
an intrinsically-defined manifold more concretely by embedding it in some higher-dimensional
ambient space while preserving its geometrical properties.
This question applied not only to General Relativity, but also to the underlying subject of differential
geometry. At the topological level, a positive answer was provided by Whitney in 1936 [1]. For
Riemannian manifolds, a similarly positive conclusion came two decades later in the form of Nash’s
theorem [2]. On the GR side, the idea of studying spacetimes as embedded submanifolds started
at least as early as 1921 with Kasner’s local embedding of the Schwarzschild metric [3], which was
perhaps the only known exact solution at the time. This was refined to a global embedding by
Fronsdal [4] some years later. By 1965, the larger collection of exact solutions were matched by a
correspondingly large number of embeddings, which were conveniently catalogued by Rosen [5]. At
a more theoretical level, Clarke showed that pseudo-Riemannian manifolds can always be globally
realised as an embedding [6].
In a seemingly unrelated development, the subject of black hole thermodynamics emerged in the
early 1970s with Bekenstein’s suggestion that an entropy should be associated with the area of a black
hole’s horizon [7]. What began as a curious set of analogies culminated in Hawking’s celebrated
result in 1974 [8] that a spherically symmetric gravitational collapse can indeed spontaneously emit
a thermal spectrum of quantum particles. This provided a physical foundation for assigning the
Hawking temperature to a black hole, thus confirming a real connection to thermodynamics. It
was soon realised that this applies even in static geometries, as the temperature arises due to the
existence of a Killing horizon. By looking at the Rindler metric, Unruh showed that an accelerated
observer in Minkowski space should measure a nonzero Unruh temperature even in vacuum [9].
1
Avoiding the often repeated question: the universe expands into what?
1
Saran Tunyasuvunakool, April 2012
a tangent space is quite simply a plane tangent to the surface, and vectors are realised as arrows lying
Mathematical Background
Global isometric embeddings and black hole temperatures were found to have a surprising connection in 1997 [10], when Deser and Levin proposed their Global Embedding Minkowski Space
(GEMS) method. The crucial observation was that, in many cases, an embedding maps a detector’s
worldline to a curve of constant acceleration in the flat ambient space. When this happens, the
Unruh temperature corresponding to the acceleration seemed to agree with the local Hawking
temperature as measured intrinsically in the spacetime. This matching between the ‘extrinsic’ and
‘intrinsic’ temperatures has been verified for many geometries [11][12][13][14][15]. Their approach
was recently generalised to apply to a wider class of observers and embeddings [16][17], and it looks
to be a promising alternative to the conventional calculation of black hole temperatures.
The plan of this essay is as follows. We begin by reviewing the mathematics of embedded manifolds
in section 2, including an overview of major existence theorems establishing the equivalence between
the two viewpoints of differential geometry. This is followed by a detailed analysis of various global
Explicit calculations related to particle acceleration are also given. In section 5, we give a brief
account of quantum field theory in curved spaces and how temperatures arise from a horizon in
this context. Finally, we discuss Deser and Levin’s GEMS approach. Having performed relevant
calculations in previous sections, the literature review here will be relatively brief. We finish by
giving a sketch of an argument which shows that the GEMS approach should work for any static
spacetime.
2
2.1
Mathematical Background
Immersions and embeddings
Our starting point is to formally define the concept of an embedding. In this context, there is also a
closely-related notion of an immersion, which arises from the observation that a smooth surface
is locally ‘fixed’ by its tangent vectors. A map which preserves the geometry should therefore map
tangent spaces faithfully. This leads us to the following definitions:
Definition 2.1. Let 𝑀, 𝑁 be smooth manifolds and 𝑓 ∶ 𝑀 → 𝑁 be a smooth map. Its derivative
at 𝑝 is the linear map d𝑓𝑝 ∶ 𝑇𝑝 𝑀 → 𝑇𝑓(𝑝) 𝑁 defined by d𝑓𝑝 (𝑋) = 𝑓∗ 𝑋, where 𝑋 ∈ 𝑇𝑝 𝑀 and
𝑓∗ 𝑋 ∈ 𝑇𝑓(𝑝) 𝑁 is the pushforward of 𝑋 by 𝑓.
Definition 2.2. A 𝐶𝑘 -immersion is a 𝐶𝑘 -map whose derivative is injective everywhere in 𝑀. A
𝐶𝑘 -embedding is a 𝐶𝑘 -immersion whose image 𝑓(𝑀) ⊆ 𝑁 is diffeomorphic to 𝑀.
Let us pause for a moment to dissect these definitions. Suppose dim 𝑀 = 𝑚 and dim 𝑁 = 𝑛. Given
an arbitrary point 𝑝 ∈ 𝑀, 𝑓 can be regarded as a function ℝ𝑚 → ℝ𝑛 in some open neighbourhood
2
Saran Tunyasuvunakool, April 2012
isometric embeddings in sections 3 and 4, where we highlight geometric features of the results.
Immersions and embeddings
Mathematical Background
of 𝑝 via a chart. Since d𝑓 is pointwise injective, we must have 𝑚 ≤ 𝑛 as one would intuitively expect.
Furthermore, since d𝑓 is a full-rank linear map, it is a consequence of the inverse function theorem
that there is some open neighbourhood of 𝑝 which is diffeomorphic to its image in 𝑁 under 𝑓 [18].
This is a good start, but the main problem here is that we have not precluded an immersed manifold
from self-intersecting in the ambient space. A striking example of this is the familiar picture of a
Klein bottle: a compact two-dimensional smooth manifold which can be regarded as a cylinder
whose ends are “glued together with a twist”.
Figure 1: An immersion of the Klein bottle in ℝ3 , clearly showing self-intersection.
Unfortunately, even demanding that an immersion is injective is not quite enough to make it a global
diffeomorphism2 , and so we impose this as the defining property for an embedding. It should be
noted, though, that for compact manifolds one can prove that any injective immersion is also an
embedding. [19]
At first glance, it might appear as if the intrinsic definition allows for a more general class of objects
than those which are embedded in a flat space. But remarkably, in 1936 Whitney established [1] that
the two approaches are completely equivalent. A few years later, he also managed to improve the
bound on the dimensionality of the ambient space [20], giving the following result:
Theorem 2.3 (Whitney Embedding Theorem). Any 𝑛-dimensional smooth manifold can be 𝐶∞ -immersed in ℝ2𝑛−1 and 𝐶∞ -embedded in ℝ2𝑛 .
A key feature here is that one typically requires more than a single extra dimension in the ambient
space to achieve embedding. Indeed, the Klein bottle is an example of a 2-dimensional manifold
which can only be embedded in an ambient space of 4-dimensions or higher. The dimensional
bound of 2𝑛 is also tight: one can show that the 𝑛-dimensional real projective space ℝℙ𝑛 cannot be
smoothly embedded in ℝ2𝑛−1 [21].
2
A counterexample: consider a map which wraps the half-open interval [0, 2𝜋) around the unit circle. The map is an
injective (in fact bijective) immersion of [0, 2𝜋) in ℝ2 . However, since the circle is compact while the half-open interval is
not, the map cannot be a homeomorphism, and so is not an embedding.
3
Saran Tunyasuvunakool, April 2012
2.1
Isometric embeddings
2.2
Mathematical Background
Isometric embeddings
Just as a ‘plain’ manifold does not have enough structure for GR, a general embedding is of little use
to us: the embedded object would induce a metric from its ambient space, but there is no reason to
expect that it would bear any resemblance to the original one. Indeed, Whitney’s argument is largely
topological, which is a sign that the result concerns only the global structure of the manifold, while
a metric governs its local properties. Instead, what we would like is a special kind of embedding
which preserves the metric in the following sense:
Definition 2.4. Let (𝑀, 𝑔) and (𝑁, ℎ) be pseudo-Riemannian manifolds. An embedding 𝑓 ∶ 𝑀 → 𝑁
is isometric if the metrics are related by 𝑔𝑝 = 𝑓∗ ℎ𝑓(𝑝) everywhere, i.e. 𝑔𝑝 (𝑋, 𝑌) = ℎ𝑓(𝑝) (𝑓∗ 𝑋, 𝑓∗ 𝑌)
for all 𝑋, 𝑌 ∈ 𝑇𝑝 𝑀.
For Riemannian manifolds, the question of whether such an embedding always exists was answered
in 1954, when Nash published a construction [22] which takes a given embedding and perturbs so
that the induced metric on the embedded manifold approximates the original metric more closely.
The perturbation increases all distances, so our original embedding needs to have shortened actual
distances. This process is then iterated until the two metrics eventually agree. The dimensionality
bound was later tightened by Kuiper [23].
Some terminology is in order before we state the theorem properly: we will write 𝐸𝑚,𝑛 for the
manifold ℝ𝑚+𝑛 equipped with the flat pseudo-Euclidean metric
d𝑠2 = −d𝑇21 − … − d𝑇2𝑚 + d𝑋21 + … + d𝑋2𝑛
Special cases include the Euclidean space 𝐸0,𝑛 , which we will abbreviate to 𝐸𝑛 , and the Minkowski
space 𝐸1,𝑛 .
Theorem 2.5 (Nash-Kuiper 𝐶1 -Embedding Theorem). Let (𝑀, 𝑔) be a smooth 𝑛-dimensional Riemannian manifold and 𝑓 ∶ 𝑀 → 𝐸𝑘 be a 𝐶∞ -immersion/embedding with 𝑘 ≥ 𝑛 + 1. If 𝑓 is a short3
map whose image is open in 𝐸𝑘 , then there exists an isometric 𝐶1 -immersion/embedding of 𝑀 in 𝐸𝑘 .
For compact manifolds, the embedding given by Whitney’s theorem can be made short by a
simple rescaling, while for non-compact manifolds, Nash provided a construction [22] of short
𝐶∞ -embeddings in 𝐸2𝑛+1 , leading immediately to the following result:
Corollary 2.6. Any 𝑛-dimensional Riemannian manifold has a 𝐶1 isometric embedding in 𝐸2𝑛 if it is
compact, or in 𝐸2𝑛+1 otherwise.
3
The notion of a short map comes from the theory of metric spaces: let (𝑋, 𝑑𝑋 ) and (𝑌, 𝑑𝑌 ) be two metric spaces,
then a map 𝑓 ∶ 𝑋 → 𝑌 is short if 𝑑𝑋 (𝑝, 𝑞) ≥ 𝑑𝑌 (𝑓(𝑝), 𝑓(𝑞)) for all 𝑝, 𝑞 ∈ 𝑋.
In a Riemannian manifold, the length of a smooth curve 𝛾(𝑡) with tangent vector 𝑋 is given by the standard formula
𝐿(𝛾) = ∫𝛾 d𝑡√𝑔(𝑋, 𝑋). The positive-definiteness of 𝑔 ensures that 𝐿(𝛾) is always non-negative, allowing us to define a
distance between any two points as 𝑑𝑔 (𝑝, 𝑞) = inf{𝐿(𝛾) ∶ 𝛾 joins 𝑝 and 𝑞}. This turns (𝑀, 𝑑𝑔 ) into a metric space.
4
Saran Tunyasuvunakool, April 2012
2.2
Isometric embeddings
Mathematical Background
Nash’s perturbation process controlled the first derivative, but higher derivatives were allowed to
grow arbitrarily, and the resulting isometric embedding tends to be rather pathological (in general
it could be nowhere twice-differentiable). A couple of years later, Nash published another theorem
[2] establishing the existence of a smooth embedding in a flat space of much higher dimensionality:
Theorem 2.7 (Nash 𝐶𝑘 -Embedding Theorem). Any smooth 𝑛-dimensional manifold with a 𝐶𝑘
1
Riemannian metric, where 3 ≤ 𝑘 ≤ ∞, can be 𝐶𝑘 -embedded isometrically into 𝐸 2 𝑛(3𝑛+11) if it is
1
compact, or into 𝐸 2 𝑛(𝑛+1)(3𝑛+11) otherwise.
The case of a general pseudo-Riemannian metric with indefinite signature remained unsettled for
over a decade following Nash’s papers, perhaps partly due to the geometers’ preference in working
with spaces where distances are sensible and positive, but also because it is inherently more difficult:
for start one cannot naturally turn the manifold into a metric space anymore. However, the renewed
interest and rapid development of GR during the 1960s no doubt drew attention to the topic, and
in 1970 Clarke succeeded [6] in generalising Nash’s theorem to include indefinite metrics. In the
process, he also tightened the dimensional bound for the isometric embedding of Riemannian
manifolds. His result is as follows:
Theorem 2.8 (Clarke Embedding Theorem). Any smooth 𝑛-dimensional manifold with a 𝐶𝑘 pseudoRiemannian metric 𝑔 of rank 𝑟 and signature4 𝑠 can be 𝐶𝑘 -embedded (3 ≤ 𝑘 ⪇ ∞) isometrically into
𝐸𝑝,𝑞 where
1
𝑝 = 𝑛 − (𝑟 + 𝑠) + 1
2
1
{
{ 𝑛(3𝑛 + 11)
𝑞 = { 21
{ 𝑛(2𝑛2 + 37) + 5 𝑛2 + 1
2
{6
M compact
otherwise
If 𝑔 is Riemannian, we can take 𝑝 = 0, giving an improvement to the bound in Nash’s theorem.
The proof employs a certain construction which allows the manifold to be decomposed into a
product of two submanifolds with definite signature. A particularly noteworthy point here, however,
is that while Nash’s result guarantees a smooth embedding when given a smooth metric, Clarke’s
result only gives an embedding of arbitrarily, but not necessarily infinitely high differentiability class.
Nevertheless, this is more than sufficient for our purpose.
In GR, spacetimes are taken to be Lorentzian manifolds with metric signature (−, +, … , +). For a
non-degenerate 𝑛-dimensional spacetime, we have 𝑟 = 𝑛 and 𝑠 = 𝑛 − 2, so it can be isometrically
embedded in the ‘two-time Minkowski’ space 𝐸2,𝑞 . The table below shows the required value of 𝑞
for different dimensions 𝑛.
4
In this context, the signature is the absolute difference between positive and negative eigenvalues.
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2.2
Extrinsic geometry
Mathematical Background
𝑛
𝑞 if compact
𝑞 otherwise
2
17
26
3
30
51
4
46
87
5
65
136
Fortunately, the spacetimes which we will consider can be embedded in far less dimensions than
is shown above, and a much more relevant bound is given in the following section. Furthermore,
Clarke also showed [6] that if we require our spacetime to be globally hyperbolic, i.e. if a Cauchy
surface exists, then there is an embedding into the Minkowski space 𝐸1,𝑞 and the second timelike
dimension is not required. Although the converse of this result is not proven, we shall see that all
our examples of non-globally hyperbolic spacetimes such as Reissner-Nordström and anti de Sitter
are embedded in 𝐸2,𝑞 .
2.3
Extrinsic geometry
When a pseudo-Riemannian manifold is isometrically embedded into a higher dimensional ambient
space, extra contributions from the extrinsic properties of the submanifold must be taken into
account. To illustrate this, we take the metric d𝑠2 = d𝜃2 + sin2 𝜃 d𝜙2 and embed it as the unit sphere
in 𝐸3 . Consider an affinely parametrised geodesic on this manifold: intrinsically its tangent vector
is covariantly constant along the curve by definition, however its image under the embedding is a
unit circle, so extrinsically the curve has a ‘centripetal acceleration’ orthogonal to the embedded
surface. Another good example is the cylinder, where the manifold is intrinsically flat everywhere
yet its embedded image clearly has an extrinsic curvature.
Let 𝑀 be a submanifold embedded in 𝐸𝑚,𝑛 . At each point 𝑝 ∈ 𝑀, the tangent space of 𝐸𝑚,𝑛 can be
decomposed into a direct sum 𝑇𝑝 𝐸𝑚,𝑛 = 𝑇𝑝 𝑀⊕𝑇𝑝 𝑀⟂ , where in this context 𝑇𝑝 𝑀 refers to its image
under the pushforward by the embedding. The extrinsic properties of 𝑀 are then encapsulated into
the following object5 :
Definition 2.9. The second fundamental form of a submanifold 𝑀 embedded in a flat space is the
(12)-tensor field on 𝑀 defined by 𝐾(𝑋, 𝑌) = (∂𝑋 𝑌)⟂ for all 𝑋, 𝑌 ∈ 𝑇𝑝 𝑀, where 𝑌 is locally extended
to a vector field on 𝑀. (Here 𝑊⟂ means the projection of 𝑊 onto 𝑇𝑝 𝑀⟂ )
In plain words, the extrinsic curvature is measured as the degree by which two infinitesimally close
tangent vectors differ in the directions outside the submanifold. A result in extrinsic geometry
which is of particular relevance to us is the formula for the acceleration of an embedded curve.
5
We are only considering a special case here where the ambient space is flat and 𝐾 is only defined for vectors tangent
to 𝑀. This can be generalised by replacing ∂ → ∇ and projecting general vectors onto 𝑀 appropriately.
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2.3
Further remarks
Mathematical Background
Consider a curve in 𝑀 affinely parametrised by 𝜆 with tangent vector 𝑈. Its acceleration in 𝐸𝑚,𝑛 is
related to its intrinsic acceleration by [24]
d𝑈
= ∇∗𝑈 𝑈 + 𝐾(𝑈, 𝑈)
d𝜆
(1)
where ∇∗ means covariant differentiation in 𝑀. The magnitude 𝑎𝐸 of the total acceleration is
obtained by squaring the above formula. Since ∇∗𝑈 𝑈 ∈ 𝑇𝑝 𝑀 while 𝐾(𝑈, 𝑈) ∈ 𝑇𝑝 𝑀⟂ , the cross term
vanishes, leaving just the following simple formula, where 𝑎𝑀 is the intrinsic acceleration:
𝑎2𝐸 = 𝑎2𝑀 + ‖𝐾(𝑈, 𝑈)‖2
2.4
(2)
Further remarks
There is a convenient abuse of terminology in the literature concerning this subject, namely that
the word ‘embedding’ is used even when the spacetime is not diffeomorphic to its image. Indeed,
the majority of entries in Rosen’s vast collection [5] belong to this class. In contexts where such a
mapping needs to be differentiated from a genuine embedding, the former is termed local while
the latter is termed global. In connection with thermodynamic applications, authors often consider
an embedding which only covers a region containing a Killing horizon. Where further analytic
extensions have not yet been found, we will consider such an embedding to be a global one.
No non-flat vacuum spacetime of (1+3) dimensions can be locally embedded in a flat space of five
dimensions. Although this result was known since the early days of GR [25], Szekeres claimed that
no correct proof had been given until his paper appeared in 1966 [26]. Another useful bound is the
following due to Friedman [27]:
Theorem 2.10. Let (𝑀, 𝑔) be an 𝑛-dimensional pseudo-Riemannian manifold where 𝑔 has 𝑝 negative
and 𝑞 positive eigenvalues and is non-degenerate, then there is an isometric immersion of 𝑀 into
1
(ℝ 2 𝑛(𝑛+1) , ℎ) where ℎ is a flat metric with at least 𝑝 positive and 𝑞 negative eigenvalues.
In particular, a (1+3) dimensional spacetime can be locally isometrically embedded in a flat space
of at most 10 dimensions. Although this is not a bound for a global embedding, we shall find that
in many cases a local embedding can be ‘untangled’ into a global one using only one or two extra
dimensions.
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Saran Tunyasuvunakool, April 2012
2.4
Ricci-Flat Geometries
3
Ricci-Flat Geometries
Having discussed the relevant mathematical concepts, we now review a number of examples of
isometric embeddings found in the literature. Often, the embedding is simply given as a magic
formula, but where possible we will try to give some geometric motivation which could lead to the
result.
In absence of a cosmological constant, a matter distribution which is describable by a traceless
energy momentum tensor gives rise to a geometry which is Ricci-flat6 , i.e. one which satisfies
𝑅 = 0 everywhere. Our first examples are spacetimes belonging to this class, which include vacuum
solutions and those with an electromagnetic field.
A note on notation
For the sake of brevity, when a coordinate is squared, we shall label it with a subscript, i.e. we write
2
𝑋21 rather than the technically correct form (𝑋1 ) . This applies to the dual coordinate basis as well.
Unless stated otherwise, 𝑇𝑖 denotes timelike coordinates in a flat spacetime 𝐸𝑝,𝑞 , while 𝑋𝑗 denotes
spacelike ones. When we wish to consider a general spacetime coordinate of the embedding space,
we shall denote it by 𝑍𝜇 ≡ (𝑇𝑖 , 𝑋𝑗 ) where the Greek index ranges over all spacetime dimensions and
the Latin indices range over temporal/spatial dimensions as appropriate.
3.2
The Rindler spacetime
Let us start with the simplest possible case by working in (1+1) dimensions. Here the symmetries of
the Riemann curvature tensor restrict it to depend only on a single parameter. A Ricci-flat geometry
is thus necessarily flat, and hence is embeddable in the Minkowski space 𝐸1,1 . An example which
will be of great importance to our later discussions is the Rindler metric, which has an apparent
singularity at 𝑥 = 0:
1
d𝑥2
2𝑎𝑥
0<𝑥<∞
d𝑠2 = −2𝑎𝑥 d𝑡2 +
−∞ < 𝑡 < ∞
(3)
In this very special case, the ‘embedding’ is simply the following coordinate transformation:
d𝑠2 = −d𝑇21 + d𝑋21
1
1
𝑇1 = √2𝑎𝑥 sinh 𝑎𝑡 , 𝑋1 = √2𝑎𝑥 cosh 𝑎𝑡
𝑎
𝑎
6
This is not true in 2 dimensions. In this case we can have vacuum solutions with a background curvature [28].
8
(4)
Saran Tunyasuvunakool, April 2012
3.1
The Schwarzschild spacetime
Ricci-Flat Geometries
𝑡
𝑇¹
1.5
1.0
1.0
0.5
0.5
1.0
1.5
𝑥
2.0
▶
0.5
0.5
1.0
1.5
2.0
𝑋¹
2.5
−0.5
−0.5
−1.0
−1.0
−1.5
Figure 2: Transformation from the (𝑡, 𝑥) Rindler chart with 𝑎 = 1 to the (𝑇1 , 𝑋1 ) Minkowski chart. Red and
blue lines correspond to 𝑥 = const and 𝑡 = const respectively. A future null cone is shown in orange and the
green curve is a null geodesic starting from rest at 𝑥0 = 1/2𝑎.
Consider a particle following the trajectory 𝑥 ≡
hyperbola 𝑋21 − 𝑇21 =
1
.
𝑎2
1
2𝑎
in the chart (3). Its worldline in 𝐸1,1 is the
One can easily check that the coordinate 𝑡 is precisely the proper time for
such a particle, and so its 4-acceleration is given simply by differentiating (4) twice:
𝑎 sinh 𝑎𝑡
𝐴𝜇 = (
)
𝑎 cosh 𝑎𝑡
By squaring this, we can see that the particle is undergoing a uniform acceleration 𝑎 starting from
rest at 𝑡 = 0. In this case an intrinsic calculation of the acceleration would also yield the same result
since the embedding is extrinsically flat.
Hyperbolic trajectories will be a common feature in our subsequent embeddings. Most importantly,
it forms the basis for Deser and Levin’s GEMS approach to black hole thermodynamics, as it allows
one to directly read off the corresponding Unruh temperature of the embedded worldline.
3.3
The Schwarzschild spacetime
The Schwarzschild spacetime is undoubtedly familiar to anyone studying General Relativity. As
a description of the geometry due to a neutral point source of mass 𝑀, it played a vital role in
the development and wide acceptance of GR. It is also the unique spherically symmetric vacuum
solution in (1+3) dimensions. The ‘canonical’ form of the metric, originally written down in 1916, is
as follows:
2𝑀
2𝑀 −1 2 2
) d𝑡2 + (1 −
) d𝑟 + 𝑟 (d𝜃2 + sin2 𝜃 d𝜙2 )
𝑟
𝑟
−∞ < 𝑡 < ∞
2𝑀 < 𝑟 < ∞
0≤𝜃≤𝜋
0 ≤ 𝜙 < 2𝜋
d𝑠2 = − (1 −
9
(5)
Saran Tunyasuvunakool, April 2012
3.3
The Schwarzschild spacetime
Ricci-Flat Geometries
An isometric embedding of (5) into 𝐸2,4 was construct by Kasner [3] in 1921. The idea is to exploit the
spherical symmetry and start by fixing 𝑟. The metric then decouples into two standard parts which
can be easily recognised: 𝑟2 (d𝜃2 +sin2 𝜃 d𝜙2 ) can be embedded as the sphere 𝑋21 +𝑋22 +𝑋23 = 𝑟2 in 𝐸0,3
and −𝑓(𝑟)2 d𝑡2 can be embedded as the circle 𝑇21 + 𝑇22 = 𝑓(𝑟)2 in 𝐸2,0 , where we let 𝑓(𝑟)2 = 1 − 2𝑀/𝑟.
With these constraints imposed, we now let 𝑟 vary and fix everything else, giving −d𝑇21 − d𝑇22 +
d𝑋21 + d𝑋22 + d𝑋23 = (−𝑓′(𝑟)2 + 1) d𝑟2 . To recover (5), we simply use the remaining dimension to
impose one extra ‘compensating’ constraint d𝑋24 = (𝑓(𝑟)−2 + 𝑓′(𝑟)2 − 1) d𝑟2 . Integrating this and
writing everything out in full parametrisation, the embedding can be given as follows:
d𝑠2 = −d𝑇21 − d𝑇22 + d𝑋21 + d𝑋22 + d𝑋23 + d𝑋24
𝑋1 = 𝑟 sin 𝜃 cos 𝜙
𝑋2 = 𝑟 sin 𝜃 sin 𝜙
,
𝑋4 = ∫
𝑟
d𝜌√
2𝑀
𝑇1 = √1 −
2𝑀
sin 𝑡
𝑟
𝑋3 = 𝑟 cos 𝜃
,
2𝑀(𝜌3 + 2𝑀/4)
𝜌3 (𝜌 − 2𝑀)
,
𝑇2 = √1 −
(6)
2𝑀
cos 𝑡
𝑟
One way to visualise this embedding is by considering its projection onto the 3-space (𝑇1 , 𝑇2 , 𝑋4 ).
Each point on the surface then corresponds to a 2-sphere whose radius shrinks as 𝑋4 decreases.
The surface has a sharp apex at 𝑟 = 2𝑀 where (5) is singular7 . Various features of physics in the
Schwarzschild spacetime can be realised geometrically in this diagram. For example, an observer
who maintains a constant spatial position very close to 𝑟 = 2𝑀 moves along a tiny circle, but since
the 4-velocity has the same magnitude everywhere, he moves at a much higher ‘angular frequency’
around the surface compared to an observer further away from the horizon. This translates to the
infinite redshift of the signal emitted by a particle approaching the horizon as observed by a detector
in the asymptotically flat region.
7
Technically speaking, 𝑟 = 2𝑀 is not part of (5) nor is the apex part of the embedding.
10
Saran Tunyasuvunakool, April 2012
3.3
The Schwarzschild spacetime
Ricci-Flat Geometries
Figure 3: The Kasner embedding with 2𝑀 = 1, projected onto the (𝑇1 , 𝑇2 , 𝑋4 ) subspace. Each point on the
surface corresponds to a 2-sphere. Red circles are constant-𝑟 curves with spacing Δ𝑟 = 0.25. Yellow curves
are radial null geodesics originating from 𝑟 = 1.75 and the green curve is a timelike null geodesic starting
from rest at the same point.
The most striking feature of (6) is that it is periodic in 𝑡, and so countably infinitely many distinct
events are identified with the same point in the embedding space. Since the constant-𝑟 curves
have compact images, the map is not a diffeomorphism of the Schwarzschild spacetime, and thus
cannot be a global embedding. Once this is realised, the obvious fix would be to somehow unroll
𝑡-coordinate lines out in the ambient space. Intuitively, one might also expect such a construction
to remove the singular tip of the surface as well8 .
One way to achieve this was presented by Fronsdal [4] in 1959. The crucial observation here is that in
the complex plane trigonometric functions are only periodic along the real direction. If we again fix 𝑟
and perform a Wick rotation 𝑡 ↦ i𝑡/4𝑀, then d𝑇21 + d𝑇22 = 𝑓′(𝑟)2 d𝑡2 ↦ −𝑓′(𝑟)2 d𝑡2 /16𝑀2 . Hence,
if we let 𝑇𝑗 ↦ −4i𝑀𝑇𝑗 as well then d𝑇21 + d𝑇22 would be invariant. Under this transformation, we
obtain a mapping which is no longer periodic in 𝑡:
𝑇1 = 4𝑀𝑓(𝑟) sinh(𝑡/4𝑀)
𝑇2 = −4i𝑀𝑓(𝑟) cosh(𝑡/4𝑀)
Since d𝑇22 < 0 we should also define a new spacelike coordinate 𝑋5 ≡ i𝑇2 = 4𝑀𝑓(𝑟) cosh(𝑡/4𝑀).
Although we haven’t altered the 𝑡𝑡-component, the 𝑟𝑟-component is now changed to −d𝑇21 + d𝑋25 =
−16𝑀2 𝑓′(𝑟)2 d𝑟2 . Once again, we can use 𝑋4 to fix things up, this time by imposing the condition
8
A good way to see this is by rolling up a large sheet of paper into a cone.
11
Saran Tunyasuvunakool, April 2012
3.3
The Schwarzschild spacetime
Ricci-Flat Geometries
d𝑋24 = (𝑓(𝑟)−2 − 16𝑀2 𝑓′(𝑟)2 − 1) d𝑟2 . We should note that the constant 1/4𝑀 in the Wick rotation
was chosen so as to make this finite at the horizon. The result is an embedding of (5) into the
Minkowski space 𝐸1,5 which reads:
d𝑠2 = −d𝑇21 + d𝑋21 + d𝑋22 + d𝑋23 + d𝑋24 + d𝑋25
𝑋1 = 𝑟 sin 𝜃 cos 𝜙
𝑋4 = ∫
𝑟
d𝜌 √
2𝑀
,
𝑋2 = 𝑟 sin 𝜃 sin 𝜙
2𝑀(𝜌2 + 2𝑀𝜌 + 4𝑀2 )
𝜌3
𝑇1 = 4𝑀√1 −
,
,
𝑋3 = 𝑟 cos 𝜃
𝑋5 = 4𝑀√1 −
2𝑀
cosh(𝑡/4𝑀)
𝑟
(7)
2𝑀
sinh(𝑡/4𝑀)
𝑟
Most importantly, (7) is now bijective and smooth, since the integrand in 𝑋4 no longer diverges as
𝑟 → 2𝑀. It thus defines a diffeomorphism between (5) and its image in 𝐸1,5 . Furthermore, 𝑇1 and
𝑋5 satisfies the relation
𝑋25 − 𝑇21 = 16𝑀2 (1 −
2𝑀
)
𝑟
(8)
which describes a perfectly well-behaved hyperbola for all 𝑟 > 0. Fronsdal’s construction is therefore
a genuine global embedding which also provides a means by which we can analytically continue
(5) beyond 𝑟 = 2𝑀. The original spacetime (7) then corresponds to the portion of this embedding
inside the wedge 𝑋5 > |𝑇1 |. Again, we can consider its projection onto the (𝑇1 , 𝑋5 , 𝑋4 ) subspace.
This is now a hyperboloid without any singularity.
Figure 4: The Fronsdal embedding with 2𝑀 = 1, projected onto the (𝑇1 , 𝑋5 , 𝑋4 ) subspace. Again, each point
on the surface corresponds to a 2-sphere. The curves correspond to the same paths as in Figure 3.
12
Saran Tunyasuvunakool, April 2012
3.3
The Schwarzschild spacetime
Ricci-Flat Geometries
Since time runs along the 𝑇1 direction in the embedding space, one can immediately see from
the diagram that any worldline inside the wedge {𝑟 < 2𝑀, 𝑇1 > 0} must have a monotonically
decreasing 𝑟-coordinate. The wedge is thus a black hole region, and similarly the corresponding
wedge in 𝑇1 < 0 describes a white hole region. This is reminiscent of the standard Kruskal extension
[29] which shares the same structure. Indeed Kruskal’s (𝑉, 𝑈) coordinates are related to (𝑇1 , 𝑋5 )
1
𝑟
𝑉
𝑟/4𝑀 𝑇
simply by ( ) ≡ √
e
(
) where 𝑟 is determined by (𝑇1 , 𝑋5 ) through (8).
𝑈
𝑋5
32𝑀3
Let us now turn to the subject of particle trajectories. Consider a detector which is firing a rocket
to maintain its spatial position outside the event horizon. Its velocity and acceleration are given
intrinsically by:
𝑈=√
∂
𝑟
𝑟 − 2𝑀 ∂𝑡
𝐴 𝑀 = ∇𝑈 𝑈 = Γ𝑟𝑡𝑡
𝑟
∂
𝑟 − 2𝑀 ∂𝑟
𝑀 ∂
𝑟2 ∂𝑟
𝑀2
= 3
𝑟 (𝑟 − 2𝑀)
=
⇒ 𝑎2𝑀
However, we already know that its worldline in the ambient space is given precisely by the hyperbola
(8), and so the total acceleration of the embedded worldline is simply
𝑎𝐸 =
1
4𝑀√1 − 2𝑀/𝑟
(9)
Using (2), we can find the extrinsic contribution to this acceleration:
‖𝐾(𝑈, 𝑈)‖2 =
(2𝑀 + 𝑟)(4𝑀2 + 𝑟2 )
16𝑀2 𝑟3
Geodesic observers are more difficult to deal with in this context because their embedded trajectories
are no longer pure Rindler. However, for a radially infalling observer starting from rest, its initial
trajectory is locally the same as for a fixed observer [24], and so the contribution from the second
fundamental form must be the same as in the previous case. Since the intrinsic acceleration is zero
by definition of geodesic motion, the initial acceleration in the ambient space for such an observer
is given by:
𝑎𝐸 = √
=
(2𝑀 + 𝑟)(4𝑀2 + 𝑟2 )
16𝑀2 𝑟3
1 √
2𝑀 4𝑀2 8𝑀3
1+
+ 2 + 3
4𝑀
𝑟
𝑟
𝑟
13
Saran Tunyasuvunakool, April 2012
3.3
The Reissner-Nordström spacetime
3.4
Ricci-Flat Geometries
The Reissner-Nordström spacetime
If we let the point source in the Schwarzschild solution gain a charge 𝑄, then our spacetime would be
filled with a spherically symmetric electric field. In absence of any other form of energy-momentum,
such a spacetime is sometimes called an electrovacuum. Just as in the vacuum case, a spherically symmetric electrovacuum solution in (1+3) dimensions is static and belongs to the Reissner-Nordström
family:
2𝑀 𝑄2
2𝑀 𝑄2
2
d𝑠 = − (1 −
+ 2 ) d𝑡 + (1 −
+ 2)
𝑟
𝑟
𝑟
𝑟
2
≡−
−1
d𝑟2 + 𝑟2 (d𝜃2 + sin2 𝜃 d𝜙2 )
(𝑟 − 𝑟− )(𝑟 − 𝑟+ ) 2
𝑟2
d𝑡
+
d𝑟2 + 𝑟2 dΩ2
𝑟2
(𝑟 − 𝑟− )(𝑟 − 𝑟+ )
−∞ < 𝑡 < ∞
𝑟+ < 𝑟 < ∞
[𝑟± ≡ 𝑀 ± √𝑀2 − 𝑄2 ]
0≤𝜃≤𝜋
(10)
0 ≤ 𝜙 < 2𝜋
Owing to its similarity with the Schwarzschild solution, we can go through Kasner’s process as before
to obtain an immersion of (10) into 𝐸2,4 , which is also listed in Rosen’s catalogue [5]:
d𝑠2 = −d𝑇21 − d𝑇22 + d𝑋21 + d𝑋22 + d𝑋23 + d𝑋24
𝑋1 = 𝑟 sin 𝜃 cos 𝜙
𝑟
,
𝑋4 = ∫ d𝜌 √
𝑟+
𝑇1 = √1 −
𝑋2 = 𝑟 sin 𝜃 sin 𝜙
,
𝑋3 = 𝑟 cos 𝜃
𝑀2 𝜌2 + (𝑄2 − 𝜌4 )(𝑄2 − 2𝑀𝜌)
𝜌4 (𝜌2 − 2𝑀𝜌 + 𝑄2 )
2𝑀 𝑄2
+ 2 sin 𝑡
𝑟
𝑟
,
𝑇2 = √1 −
(11)
2𝑀 𝑄2
+ 2 cos 𝑡
𝑟
𝑟
Indeed, setting 𝑄 = 0 in the above mapping recovers (6) as a special case. In this region, the
embedding has features which are virtually identical to its Schwarzschild counterpart. Assuming
that we are in a non-extremal configuration with 𝑀2 > 𝑄2 , we can now attempt to perform the
Wick rotation trick again with 𝑡 ↦ i𝜅𝑡 for some constant 𝜅. Proceeding exactly as before, we find
that the 𝑋4 constraint has a simple pole at 𝑟 = 𝑟+ unless 𝜅 =
constraint reads:
d𝑋24 =
𝑟+ −𝑟−
.
2𝑟2+
With this condition, the 𝑋4
𝑟2 (𝑟+ + 𝑟− ) + 𝑟2+ (𝑟 + 𝑟+ )
4𝑟5+ 𝑟−
−
𝑟2 (𝑟 − 𝑟− )
𝑟4 (𝑟+ − 𝑟− )2
(12)
Crucially, the RHS changes sign at some value 𝑟 between 𝑟− and 𝑟+ , and so we cannot use this to
embed (10) all the way down to 𝑟 = 𝑟− . The fix is immediate once we realise that each term in (12)
is individually positive, so following [12] we add another timelike dimension to the embedding
space. The constraint is then split up into spacelike and timelike parts, giving the following global
14
Saran Tunyasuvunakool, April 2012
3.4
The Reissner-Nordström spacetime
Ricci-Flat Geometries
embedding of the 𝑟− < 𝑟 < ∞ region of the Reissner-Nordström geometry into 𝐸2,5 :
d𝑠2 = −d𝑇21 − d𝑇22 + d𝑋21 + d𝑋22 + d𝑋23 + d𝑋24 + d𝑋25
𝑋1 = 𝑟 sin 𝜃 cos 𝜙
𝑟
𝑋4 = ∫ d𝜌 √
𝑟+
,
𝑋2 = 𝑟 sin 𝜃 sin 𝜙
,
𝑋3 = 𝑟 cos 𝜃
𝜌2 (𝑟+ + 𝑟− ) + 𝑟2+ (𝜌 + 𝑟+ )
𝜌2 (𝜌 − 𝑟− )
,
𝑋5 = 𝜅−1 √1 −
2𝑀 𝑄2
+ 2 sinh 𝜅𝑡
𝑟
𝑟
,
𝑇2 = ∫ d𝜌 √
𝑇1 = 𝜅−1 √1 −
𝑟
𝑟+
2𝑀 𝑄2
+ 2 cosh 𝜅𝑡
𝑟
𝑟
(13)
4𝑟5+ 𝑟−
𝜌4 (𝑟+ − 𝑟− )2
We now look at the projection of (13) into the (𝑇1 , 𝑋5 , 𝑇4 ) subspace. In the region near the outer
horizon 𝑟 = 𝑟+ , the surface looks very much like Fronsdal’s embedding of Schwarzschild. However,
we see that rather than extending down indefinitely, the surface curves back around as 𝑟 decreases
inside the inner region and finally join up into crossing lines again at the inner horizon 𝑟 = 𝑟− . As
duly noted by Deser and Levin in [12], an observer in the asymptotically flat region is unaware of the
geometry inside the outer horizon, and so for the purpose of calculating the black hole’s temperature
this embedding is indeed sufficiently global.
Figure 5: An isometric embedding of RN outside the inner horizon, with 𝑀 = 5 and 𝑄 = 4 (𝑟+ = 8, 𝑟− = 3),
projected onto the (𝑇1 , 𝑋5 , 𝑋4 ) subspace. Red curves correspond to constant 𝑟, with spacing Δ𝑟 = 1.
15
Saran Tunyasuvunakool, April 2012
3.4
Extremal Reissner-Nordström
Ricci-Flat Geometries
Just like the Schwarzschild case, a fixed observer follows a hyperbolic trajectory. Here the acceleration
is given by
𝑎𝐸 =
𝜅
(14)
√1 − 2𝑀/𝑟 + 𝑄2 /𝑟2
2
In this case, the intrinsic acceleration is 𝑎𝑀
=
2
2
(𝑀−𝑄 /𝑟)
.
𝑟 (𝑟2 −2𝑀𝑟+𝑄2 )
2
Using (2), we can deduce the initial
acceleration of a radially infalling observer starting from rest at distance 𝑟:
𝑎2𝐸 =
2𝑀 𝑄2
1
+ 2)
(1 −
4
𝑟
𝑟
−1
(
𝑀2 − 𝑄2 4(𝑀𝑟 − 𝑄2 )2
−
)
𝑟6
𝑟4+
(15)
Before we move on, it is interesting to consider why this embedding fails at 𝑟 = 𝑟− , as this gives
us a good idea of how we could continue it. Recall that the Reissner-Nordström solution can be
maximally extended to contain infinitely many asymptotically flat regions. This global geometry
is represented by a 2D Carter-Penrose diagram, where two copies of asymptotically flat ‘Region I’
are connected to two distinct copies of ‘Region II’ which, unlike in our embedding, remain separate
away from the outer horizon. To obtain a maximally extended embedding of RN, we want to follow
this picture and construct an embedding for the 0 < 𝑟 < 𝑟+ portion, i.e. one which covers two copies
of ‘Region II’ and ‘Region III’ connected by a Cauchy horizon. We would then glue each half of its
Cauchy horizon to the corresponding half of the bottom edge of different copies of Figure 5. However,
doing this in-place would result in self-intersection, so we will need to use a higher-dimensional
ambient space to actually complete this construction. The process can then be repeated ad infinitum.
Finally, we note that the ‘inner’ embedding is easily obtained by following the same steps as before,
but setting 𝜅 =
𝑟+ −𝑟−
2𝑟2−
instead. Its projection near 𝑟 = 𝑟− is similar to Figure 4 with 𝑇1 and 𝑋5
swapped.
3.5
Extremal Reissner-Nordström
The extremal case of RN geometry presents a difficulty as the nature of the horizon is markedly
different. The topology of the Carter-Penrose diagram changes: there is no region in which 𝑟 is
timelike and the Killing horizon becomes degenerate. Furthermore, since the Killing horizon is no
longer bifurcate, we cannot expect the embedding to retain the Rindler-like characteristic as in the
previous cases.
Let us consider the (𝑇1 , 𝑇2 , 𝑋4 )-projection of the embedding (11) as we approach the extremal limit.
The conical singularity gets stretched into a long thin filament as the outer horizon stretches to an
infinite spacelike distance from any given point in the asymptotically flat region.
16
Saran Tunyasuvunakool, April 2012
3.5
Higher dimensional generalisations
Ricci-Flat Geometries
Figure 6: Kasner-type embedding of the outer RN geometry approaching the extremal limit.
If we look at the embedding (13) in the same limit, we see that the bottom portion is squeezed
up towards the Killing horizon as expected, but more pertinently the top portion of the surface
flattens down. The problem, then, lies in the fact that this embedding involves a coordinate which
is a function only of 𝑟 when 𝑟 = 𝑀 needs to go off to infinity. If we wish to retain most of the
characteristics of our previous embeddings, an obvious fix would then be to try and add some
𝑡-dependence to 𝑋4 . However, this introduces a cross term d𝑡d𝑟 which is not easy to compensate
for.
For completeness, it should be noted that Carter’s extremal RN extension in terms of two null
coordinates (𝑢, 𝑣) in [30] does lend itself immediately to an embedding using the ‘dictionary’ given
in the last page of [5]. However, this embedding would require eight dimensions with three timelike
coordinates, and hence not minimal. Fixed observers would not follow a hyperbolic trajectory in
this scheme.
3.6
Higher dimensional generalisations
The spherically symmetric electrovacuum solution has a generalisation in (1 + 𝐷) dimensions
commonly known as the Tangherlini spacetime. Unsurprisingly, the metric [31] takes the same form
as the Reissner-Nordström solution:
−1
̃2
̃2
̃
̃
2𝑀
𝑄
2𝑀
𝑄
2
d𝑠 = − (1 − 𝐷−2 + 2(𝐷−2) ) d𝑡 + (1 − 𝐷−2 + 2(𝐷−2) ) d𝑟2 + 𝑟2 dΩ2𝐷−1
𝑟
𝑟
𝑟
𝑟
2
−∞ < 𝑡 < ∞
𝑟+ < 𝑟 < ∞
dΩ2𝐷−1
17
is the round metric on 𝑆
𝐷−1
(16)
Saran Tunyasuvunakool, April 2012
3.6
Rotating solutions
Ricci-Flat Geometries
̃ are related to the ADM mass and charge by [32]:
̃ and 𝑄
The parameters 𝑀
̃=
𝑀
4 Γ(𝐷/2)
𝑀
(𝐷 − 1)√𝜋𝐷−2
̃=√
𝑄
2
𝑄
(𝐷 − 1)(𝐷 − 2)
The previous embedding (13) is applicable here with minimal tweaking. We embed a (𝐷 − 1)-sphere
of radius 𝑟 into (𝑋1 , … , 𝑋𝐷 ) using standard parametrisation. The hyperbolic embedding into
(𝑇1 , 𝑋𝐷+2 ) carries through with surface gravity
𝜅=
(𝐷 − 2)(𝑟𝐷−2
− 𝑟𝐷−2
+
− )
𝐷−1
2𝑟+
The integrals for 𝑇2 and 𝑋𝐷+1 are given in [15] as
𝑟+
𝑟𝐷−2
4𝑟3𝐷−4
+
−
2 𝜌2(𝐷−1)
(𝑟𝐷−2
− 𝑟𝐷−2
)
+
−
𝑟
𝐷−2 + 𝑟𝐷−2 ) + 𝑟2(𝐷−2) (𝜌 + 𝑟 )
[𝒲(𝐷) (𝜌) − 𝑟𝐷−2
+ (𝜌 + 𝑟+ )] (𝑟+
+
−
+
𝑟+
𝜌2 (𝜌𝐷−2 − 𝑟𝐷−2
− )𝒲(𝐷−2) (𝜌)
𝑟
𝑇2 = ∫ d𝜌√
𝑋
𝐷+1
= ∫ d𝜌√
where 𝒲(𝐷) (𝜌) ≡
(17)
𝐷
𝑟𝐷
+ −𝜌
𝑟+ − 𝜌
This gives an embedding of the 𝑟 > 𝑟− region of the charged Tangherlini spacetime into 𝐸2,𝐷+2 . Its
features are essentially the same as in the RN case, in particular a fixed observer has an embedded
acceleration
𝑎2𝐸 =
3.7
𝜅2
̃2 /𝑟2(𝐷−2)
̃ 𝐷−2 + 𝑄
1 − 2𝑀/𝑟
(18)
Rotating solutions
The problem of isometric embedding becomes significantly more difficult when we lose spherical
symmetry, and there is virtually no known result in this situation. From our point of view, the
standard approach fails because the metric now depends on multiple coordinates, preventing a clean
separation. One important case which falls under this category is the Kerr-Newman spacetime
which describes the geometry outside a charged rotating point source. In absence of a complete
embedding, successful attempts have been made at embedding the event horizon, which is a 2D
Riemannian manifold, into 𝐸4 [33] or the hyperbolic space 𝐻3 [34]. Since this kind of embedding
cannot be used in our later thermodynamic application, we shall not present these results here.
18
Saran Tunyasuvunakool, April 2012
3.7
Geometries with a Cosmological Constant
4
Geometries with a Cosmological Constant
Let us now relax the Ricci-flat condition on our spacetime but still restrict our attention to those
with a constant scalar curvature. This amounts to introducing a nonzero cosmological constant
Λ. In particular, we consider the de Sitter and anti de Sitter spacetimes along with black holes in
these backgrounds. The embedding of these geometries have found application in thermodynamics,
but we should stress that there are many other spacetimes whose embeddings9 are known [5]. A
number of these are archetypal pathological examples in GR with interesting causal structures.
4.1
The de Sitter and anti de Sitter spacetimes
The de Sitter spacetime dSn is the natural Lorentzian generalisation of the hypersphere 𝑆𝑛 , and like
d𝑠2 = −d𝑇21 + d𝑋21 + … + d𝑋2𝑛−1
(19)
−𝑇21 + 𝑋21 + … + 𝑋2𝑛−1 = ℓ 2
The hyperboloid above solves the Einstein Field Equation with a cosmological constant Λ =
(𝑛−1)(𝑛−2)
,
2ℓ 2
and its scalar curvature is everywhere constant and positive with value 𝑅 =
2𝑛Λ
.
𝑛−2
The
geometry clearly possesses an SO(1, 𝑛) symmetry group. An immediately visualisable case is dS2
which appears as a hyperboloid in 𝐸1,2
Figure 7: A global isometric embedding of dS2 in 𝐸1,2 . Solid black lines indicate the static coordinate chart
(23). Higher dimensional dSn can be realised in the same picture by associating an (𝑛 − 1)-sphere to each
point.
9
At least the for the portion which is covered by the original chart
19
Saran Tunyasuvunakool, April 2012
its Riemannian counterpart, the most convenient definition of dSn is via its embedding:
The de Sitter and anti de Sitter spacetimes
Geometries with a Cosmological Constant
A number of coordinatisations for dSn are in common use, however we shall be interested in the
one which bears a strong resemblance to our earlier cases:
𝑇1 = √ℓ 2 − 𝑟2 sinh(𝑡/ℓ)
𝑋1 = √ℓ 2 − 𝑟2 cosh(𝑡/ℓ)
,
𝑋2 , … , 𝑋𝑛 = 𝑟 ⋅ (standard parametrisation of 𝑆𝑛−2 )
−1
𝑟2
𝑟2
⇒ d𝑠 = − (1 − 2 ) d𝑡2 + (1 − 2 )
ℓ
ℓ
2
−∞ < 𝑡 < ∞
,
d𝑟2 + 𝑟2 dΩ2𝑛−2
(20)
−ℓ < 𝑟 < ℓ
In this case, the metric is static and 𝑡 is proportional to the proper time of a particle at constant
spatial position. We can therefore treat this metric as the one experienced by an observer comoving
with the dSn background. Again, the important feature is the existence of a Killing horizon at
𝑟2 = ℓ 2 with a Rindler-like structure. The particle follows a hyperbolic trajectory extrinsically, with
a corresponding acceleration given by
𝑎𝐸 =
1
√ℓ 2
(21)
− 𝑟2
Moreover, for a ‘radial’ trajectory we can exploit the SO(𝑛−2) symmetry and choose to set 𝑋2 = 𝑟 and
𝑋3 = … = 𝑋𝑛 = 0. Hence, in considering the extrinsic properties of such a curve we can treat the
embedding as a surface in 𝐸1,2 . The second fundamental form can be calculated straightforwardly
in this case, and is given in the (𝑡, 𝑟) basis by
𝐾𝛼𝛽 = (
− 1ℓ (1 −
𝑟2
)
ℓ2
0
1
ℓ
0
(1 −
𝑟2 −1
)
ℓ2
)
Therefore, we have 𝐾𝛼𝛽 𝑈𝛼 𝑈𝛽 = 1ℓ 𝑈𝛼 𝑈𝛽 = − 1ℓ for any radially moving particle, and by using (2) we
can also write 𝑎𝐸 in a chart-independent manner in terms of its intrinsic acceleration:
𝑎2𝐸 = 𝑎2𝑀 +
1
ℓ2
(22)
The anti de Sitter space adSn is obtained from dSn by replacing one spacelike coordinate in the
ambient space with a timelike one. The hyperboloid (19) has circular cross-sections when projected
onto the (𝑇1 , 𝑇2 ) plane, allowing for the existence of closed timelike curves. Since these violate
causality, one usually takes the universal cover when discussing adSn . The easiest way to discuss this
is by looking at a particular parametrisation: take (23), replace 𝑋1 by 𝑇2 and replace the hyperbolic
functions by trigonometric ones. The embedding now identifies points 𝑡 ∼ 𝑡 + 2𝜋𝑛/ℓ, 𝑛 ∈ ℤ, so by
20
Saran Tunyasuvunakool, April 2012
4.1
The BTZ black hole
Geometries with a Cosmological Constant
taking the universal cover we simply drop this identification. The corresponding metric is:
d𝑠2 = − (1 +
𝑟2
𝑟2
2
d𝑡
+
+
)
(1
)
ℓ2
ℓ2
−∞ < 𝑡 < ∞
,
−1
d𝑟2 + 𝑟2 dΩ2𝑛−2
0<𝑟<∞
The curvature and cosmological constant are the negative of the corresponding values for dSn . There
is no longer an apparent horizon, however we note that a radially moving observer still follows a
hyperbolic path in the ambient space, with a corresponding acceleration
𝑎2𝐸 = −𝑎2𝑀 +
1
ℓ2
(23)
Notably, 𝑎2𝐸 is negative if 𝑎2𝑀 < 1/ℓ 2 . This is a clearly a contradiction if 𝑎2𝑀 ≥ 0 since it would imply
that we have isometrically mapped a timelike curve into a spacelike one. We can thus conclude that
timelike curves in adS𝑛 are subject to a minimum acceleration of 1/ℓ.
4.2
The BTZ black hole
In three dimensions, the Riemann tensor has six independent components, and is therefore completely determined by the Ricci tensor. Without a cosmological constant, the vacuum Einstein
equation demands that the spacetime must be completely flat. Remarkably, there does exist a spherically symmetric vacuum solution with an event horizon if we include a negative cosmological
constant. Moreover, since the ‘spherical’ symmetry group in this case is just SO(2), such a solution
is also allowed to rotate. The solution is known as BTZ, after Bañados, Teitelboim and Zanelli who
first communicated the result in 1992 [35]. The metric is as follows:
d𝑠2 = −𝑁2 d𝑡2 + 𝑁−2 d𝑟2 + 𝑟2 (𝑁𝜙 d𝑡 + d𝜙)
𝑁2 = −𝑀 +
𝑟2
𝐽2
+
ℓ 2 4𝑟2
,
𝑁𝜙 = −
𝐽
2𝑟2
2
(24)
Here 𝑀 and 𝐽 are the ADM mass and angular momentum, and the parameter ℓ is related to the
cosmological constant by Λ = −1/ℓ 2 . Like the Kerr solution, a non-extremal BTZ metric with
|𝐽| > 𝑀ℓ has two horizons at 𝑟 = 𝑟± with
𝑟2± =
𝑀ℓ 2 [
𝐽 2]
)]
[1 ± √1 − (
2
𝑀ℓ
[
]
In a follow-up paper [36], it was shown that the BTZ solution can be realised as a quotient space
of adS3 . Firstly, recall that adS3 possesses an SO(2, 2) symmetry group. The authors of [36] start
21
Saran Tunyasuvunakool, April 2012
4.2
The BTZ black hole
Geometries with a Cosmological Constant
by taking a certain linear combination10 𝜉 of the SO(2, 2) generators, which by definition forms
a Killing vector field of adS3 . They then constructed a parametrisation of adS3 in 𝐸2,2 with one
coordinate 𝜙 satisfying 𝜉 =
∂
∂𝜙
globally. The pulled back flat metric is then precisely (24). The
appropriate parametrisation in the region 𝑟 > 𝑟+ is as follows:
d𝑠2 = −d𝑇21 − d𝑇22 + d𝑋21 + d𝑋22
𝑟
𝑟
𝑟2 − 𝑟2+
sinh ( +2 𝑡 − − 𝜙)
2
2
ℓ
ℓ
𝑟+ − 𝑟−
𝑇1 = ℓ √
𝑟
𝑟
𝑟2 − 𝑟2−
cosh (− −2 𝑡 + + 𝜙)
2
2
ℓ
ℓ
𝑟+ − 𝑟 −
,
𝑇2 = ℓ √
𝑋1 = ℓ √
𝑟
𝑟2 − 𝑟2−
𝑟
sinh (− −2 𝑡 − + 𝜙)
2
2
ℓ
ℓ
𝑟+ − 𝑟 −
,
𝑋2 = ℓ √
(25)
𝑟
𝑟
𝑟2 − 𝑟2+
cosh ( +2 𝑡 − − 𝜙)
2
2
ℓ
ℓ
𝑟+ − 𝑟 −
The BTZ solution is obtained by quotienting out adS3 by the equivalence relation 𝜙 ∼ 𝜙+2𝜋𝑛, 𝑛 ∈ ℤ.
This means that (25) does not quite constitute an embedding of (24) into 𝐸2,2 , and thus we cannot
use these equations to plot a surface which shows global structures.
Although the above adS3 embedding suffices for our purposes, we should also note that recent work
[37] had succeeded in globally embedding the non-rotating (𝐽 = 0) BTZ solution into 𝐸2,3 . Indeed,
here we can apply the same technique we used for Schwarzschild to obtain the following embedding,
whose projection looks unsurprisingly like Figure 4:
d𝑠2 = −d𝑇21 − d𝑇22 + d𝑋21 + d𝑋22 + d𝑋23
𝑇1 = √1 +
𝑋1 =
1
𝑟
𝑀
,
𝑇2 =
𝑟2
ℓ √
−𝑀 + 2 sinh(√𝑀 𝑡/ℓ)
√𝑀
ℓ
ℓ √
𝑟2
−𝑀 + 2 cosh(√𝑀 𝑡/ℓ)
√𝑀
ℓ
,
𝑋2 = 𝑟 cos 𝜙
,
(26)
𝑋3 = 𝑟 sin 𝜙
Interestingly, this embedding is also algebraic: it is the intersection of quadric hypersurfaces −𝑇21 −
𝑇22 + 𝑋21 + 𝑋22 + 𝑋23 = −1 and 𝑋23 + 𝑋24 =
𝑀
𝑇2 .
1+𝑀 1
The same paper [37] gave a global embedding of
the non-extremal rotating case as well, but this was done in 𝐸5,5 and, in the authors own words, “[is]
not altogether satisfactory and a more economical embedding would be desirable”.
We again consider an observer at a fixed distance 𝑟 from the source. In general, such an observer
does not follow a hyperbolic motion in 𝐸2,2 unless its trajectory obeys [12]
𝜙 = 𝑟− 𝑡/𝑟+ ℓ
(27)
This constraint fixes the trajectory to be constant in the (𝑇1 , 𝑋1 ) plane11 . The corresponding accel-
To be precise, we take 𝜉 = 𝑟ℓ+ 𝐽12 − 𝑟ℓ− 𝐽03 − 𝐽13 + 𝐽23 , where 𝐽𝜇𝜈 ≡ 𝑍𝜈 ∂𝑍∂ 𝜇 − 𝑍𝜇 ∂𝑍∂ 𝜈 are the generators of SO(2, 2) acting
on 𝐸 with coordinatisation (𝑍0 , … , 𝑍3 ).
11
Note that if the geometry is non-rotating then 𝑟− = 0, and so the condition defines a fixed observer as expected.
10
2,2
22
Saran Tunyasuvunakool, April 2012
4.2
Reissner-Nordström geometry in adS background
Geometries with a Cosmological Constant
eration in 𝐸2,2 is then given by:
1 𝑟2 − 𝑟2
𝑎𝐸 = √ +2 2−
ℓ 𝑟 − 𝑟+
(28)
2
The BTZ solution can be ‘generalised’ to include a charge by adding − 𝑄2 (𝑟/𝑟0 ) to the lapse function
𝑁2 [35]. However, this does not work if the black hole is rotating as it would induce a magnetic field
which gives an additional contribution to the energy-momentum tensor [38]. The general charged
and rotating solution in (1+2) dimensions was eventually found in [39], but we will not consider
this. For the non-rotating case, we can embed it into 𝐸2,3 following the same procedure as for RN.
The full result is given in [13] and we only highlight the important points here: the surface gravity at
the outermost horizon 𝑟 = 𝑟𝐻 is 𝜅 = [(𝑟𝐻 /ℓ)2 − 𝑄2 /4] /𝑟𝐻 and a fixed observer follows a hyperbolic
path with acceleration
𝑎𝐸 =
4.3
𝑟2𝐻 /ℓ 2 − 𝑄2 /4
(29)
𝑟𝐻 √−𝑀 + 𝑟2 /ℓ 2 − (𝑄2 /2) log(𝑟/𝑟𝐻 )
Reissner-Nordström geometry in adS background
Our final example is that of a spherically symmetric electrovacuum with a negative cosmological
constant. Conveniently, this is given by adding a contribution from the adS4 background (23) to
the RN metric (10). The resulting metric takes the same form as RN, but with its ‘lapse function’
replaced by (1 −
2𝑀
𝑟
+
𝑄2
𝑟2
+
𝑟2
).
ℓ2
There are now potentially four horizons, but we shall restrict our
attention to the outermost one. The embedding which covers this region is the same as (13) for RN
with the obvious modifications. Most importantly, the surface gravity is [14] 𝜅 =
2
4
𝑟2𝐻 −𝑄2 +3𝑟4𝐻 /ℓ 2
2𝑟3𝐻
where
𝑟𝐻 is the horizon radius. The appropriate integrals for 𝑇 and 𝑋 in the embedding are also given
in [14]. In this case, a fixed observer’s acceleration in the ambient space is given by
𝑎𝐸 =
𝑟2𝐻 − 𝑄2 + 3𝑟4𝐻 /ℓ 2
2𝑟3𝐻 √1 − 2𝑀/𝑟 + 𝑄2 /𝑟2 + 𝑟2 /ℓ 2
(30)
The (1+𝐷)-dimensional charged Tangherlini solution (16) can also be made asymptotically adS(1+D)
[31] and embedded into 𝐸2,𝐷+1 in the same fashion [15].
23
Saran Tunyasuvunakool, April 2012
4.3
Black Hole Thermodynamics
5
Black Hole Thermodynamics
The formulation of the four Laws of Black Hole Mechanics strongly suggested that black holes
behave like thermal objects, but since they were obtained purely from geometrical considerations,
any connections with thermodynamics were initially regarded only as formal analogies. However,
the study of quantum field theory in curved spaces later revealed that a detector can indeed observe
a thermal spectrum of particles in the presence of a Killing horizon.
We will give a brief overview of quantum field theory in curved spaces and how a temperature could
arise in this context, before proceeding to review the application of global isometric embeddings to
temperature calculations.
Hawking and Unruh temperatures
In the canonical approach to quantum field theory, one starts with the space of solutions of the
classical field equation and promotes them to quantum operators. The theory’s Lagrangian endows
the solution space with a symplectic structure, which upon quantisation is lifted to give the canonical
commutation relations between the field and its conjugate momentum. To build up the state space
of the theory, we pick an orthonormal basis {𝑢𝑖 (𝑥)} of the classical solution space and write the
field operator as 𝜙(𝑥) = ∑𝑖 [𝑎𝑖 𝑢𝑖 (𝑥) + 𝑎†𝑖 𝑢∗𝑖 (𝑥)], where ∑𝑖 is understood to mean an appropriate
integration measure. We define the vacuum state |0⟩ to satisfy the property 𝑎𝑖 |0⟩ = 0 for all 𝑎𝑖 . From
this, the space of 𝑛-particle states is defined inductively by ℋ𝑛 = span ⋃𝑖 {𝑎†𝑖 |Ψ⟩ ∶ |Ψ⟩ ∈ ℋ𝑛−1 }
∞
where ℋ0 = span |0⟩ ≅ ℂ. The total state space of the theory is the Fock space ℋ = ⨁ ℋ𝑛 .
𝑛=0
For an ‘𝑛-particle state’ to behave as its name suggests, each 𝑎†𝑖 should act to create a state of definite
energy in some sense, and so 𝑢𝑖 (𝑥) should be eigenfunctions of some timelike Killing vector field.
Since no single choice of frame is particularly special from the point of view of GR, any timelike
Killing vector field could equally well be used to define a vacuum and a Fock space. Given any
two orthonormal bases {𝑢𝑖 (𝑥)} and {𝑢𝐼 (𝑥)}, we can always write one in terms of the other via
𝑢𝐼 = ∑𝑖 (𝐴𝐼𝑖 𝑢𝑖 + 𝐵𝐼𝑖 𝑢∗𝑖 ). To preserve the symplectic structure defined by the theory, the linear maps
𝐴𝐼𝑖 and 𝐵𝐼𝑖 need only satisfy two conditions
𝐴𝐴† − 𝐵𝐵† = 𝟙
,
𝐴⊤ 𝐵 − 𝐴𝐵⊤ = 0
Crucially, these Bogoliubov conditions allow for the basis transformation to be nontrivial. An
immediate but profound consequence is that vacua defined with respect to different bases do not
agree in general.
In 1974 [8], Hawking found that there is a spontaneous emission of massless Klein-Gordon particles
from a spherically collapsing star. The main observation is that null rays reaching ℐ+ suffer a very
24
Saran Tunyasuvunakool, April 2012
5.1
Hawking and Unruh temperatures
Black Hole Thermodynamics
large blueshift close to the horizon ℋ+ . We can therefore invoke the geometric optics approximation
in this region, and trace a family of null rays from ℐ+ back to ℐ− . The asymptotic form of the
positive frequency modes at ℐ± then correspond to the
𝑢
𝑣
+
Eddington-Finkelstein null variables.
Starting from |0⟩𝑣 at ℐ , one finds that an observer at the ℐ end of the ray sees a Planck-distributed
−
spectrum of particles corresponding to the Hawking temperature
𝑇BH =
𝜅𝐻
2𝜋
Following up on this result, Unruh considered the quantisation of a Klein-Gordon field in Rindler
coordinates [9]. He showed that, under a Bogoliubov transformation, the vacuum state defined with
respect to the usual Minkowski basis corresponds to a Planck spectrum of particles in the Rindler
basis. The corresponding temperature is now known as the Unruh temperature
𝑇U =
𝑎
2𝜋
The same analysis was also performed in the maximally extended Schwarzschild geometry. He
showed that a vacuum defined via Kruskal time (the Unruh vacuum) corresponds to a thermal state
in the basis defined via Schwarzschild time (whose vacuum is known as the Boulware vacuum).
Moreover, the temperature here agrees with Hawking’s analysis of the spherically symmetric collapse.
To see whether this temperature has any physical implication beyond merely being an artifact of
a basis change, Unruh modelled a detector as a quantum particle in a box, evolving in its own
proper time according to Schrödinger’s equation. The particle was then allowed to interact with
the background scalar field in some vacuum state. Using standard QM perturbation theory, he
calculated the probability that the detector is excited from its ground state to higher energy levels
due to its interaction with the field. Here he found that if the detector has a uniform acceleration
𝑎, then its transition rate is exactly the same as if it were immersed in a thermal bath at the Unruh
temperature. This therefore indicates that the detector could indeed feel these particles ‘created’
through the Bogoliubov transformation.
A generalised calculation based on this model is presented in [40]. The crucial result is that the
detector’s rate of transition to an energy level 𝐸 is governed by its response function
ℱ(𝐸) = ∫
∞
d(Δ𝜏) e−i(𝐸−𝐸0 )Δ𝜏 𝐺+ (Δ𝜏)
−∞
where 𝐺+ (Δ𝜏) = ⟨0| 𝜙(𝑥(𝜏+Δ𝜏))𝜙(𝑥(𝜏)) |0⟩ is known as the Wightman function of the field, and 𝑥(𝜏)
is the particle’s worldline. Through this formula, one could then calculate the Hawking temperature
from the Wightman function.
25
Saran Tunyasuvunakool, April 2012
5.1
The GEMS approach
5.2
Black Hole Thermodynamics
The GEMS approach
A new approach to studying black hole temperatures was suggested by Deser and Levin in 1997 [10],
which they later called the Global Embedding Minkowski Space (GEMS) approach. This stemmed
from the observation that the standard embedding of dS4 maps the worldline of a comoving observer on to a hyperbolic trajectory in the ambient space, and so we can associate to it an Unruh
temperature corresponding to the 5-acceleration given by (22). Remarkably, this temperature is in
exact agreement with an earlier result obtained by coupling a scalar field to an accelerating detector.
The same procedure was also applied to a comoving observer in adS4 with 5-acceleration (23),
obtaining for the first time a temperature of this spacetime. They then proceeded to verify its validity
by presenting an explicit calculation of the Wightman function and the detector response function.
In the following year [11], the same authors employed embedding (7) of the Schwarzschild spacetime
and considered a detector at a fixed spatial position. In the embedding space, such a detector
registers an Unruh temperature corresponding to its 6-acceleration (9) which clearly reduces to the
Hawking temperature as 𝑟 → ∞. Alternatively, we can also obtain this result by defining the black
hole’s temperature to be the invariant temperature in Tolman’s law [41]:
Proposition 5.1 (Tolman and Ehrenfest, 1930). Let 𝑇(𝑥) be the temperature measured at a fixed
spatial position 𝑥 in a static metric, then 𝑇0 ≡ √−𝑔00 (𝑥) 𝑇(𝑥) is constant everywhere.
Deser and Levin followed this up by considering yet more geometries [12], showing that the embedded Unruh temperatures for static observers in BTZ, Schwarzschild-adS and non-extremal
Reissner-Nordström spacetimes, with accelerations (28), (30) and (14) respectively, are all in agreement with ‘proper’ calculations of Hawking temperatures. The comparison in the BTZ case is slightly
more complicated, as the conventional calculation of the Hawking temperature considers a class
of observers which is ‘static at ∞’. These observers obey 𝜙 = −𝑁𝜙 𝑡 rather than conditions (27)
imposed on ‘pure Unruh’ observers. However, since there is one observer belonging to both classes,
the invariant temperature given by Tolman’s law can still be compared, and indeed they do agree.
The validity of the GEMS approach has since been verified by various other authors for the RN-adS
[14], charged BTZ [13] and Tangherlini(-adS) [15] spacetimes, using accelerations (30), (29) and (18)
respectively in the Unruh formula, and also for a number of other static geometries which we have
not considered in this essay.
Throughout their discussion, Deser and Levin stressed that their method is only meaningful if
embedding maps the detector’s trajectory to a Rindler hyperbola. We now show that, for static
spacetimes, any such embedding necessarily gives rise to an Unruh temperature which agrees with
the Hawking temperature. We first consider metrics of the form
d𝑠2 = −𝑓(𝑟)2 d𝑡2 +
1
d𝑟2 + 𝑟2 dΩ2𝐷−1
𝑓(𝑟)2
26
(31)
Saran Tunyasuvunakool, April 2012
5.2
The GEMS approach
Black Hole Thermodynamics
The local Hawking temperature is measured by a detector whose velocity is proportional to the
Killing vector 𝜉 =
∂
.
∂𝑡
The norm of this Killing vector is given by 𝜉𝜇 𝜉𝜇 = −𝑓(𝑟)2 and so there
is a Killing horizon at 𝑟 = 𝑟𝐻 iff 𝑓(𝑟𝐻 ) = 0. Assuming that 𝑓 is meromorphic, we can write
̃ 2 where 𝑓(𝑟
̃ ) ≠ 0. The surface gravity on 𝑟 = 𝑟 can be calculated via the
𝑓(𝑟)2 = (𝑟 − 𝑟𝐻 )𝑛 𝑓(𝑟)
𝐻
𝐻
formula 𝜅2𝐻 = − 12 (∇𝜇 𝜉𝜈 )(∇𝜇 𝜉𝜈 ). The only relevant Christoffel symbol here is Γ𝑡𝑡𝑟 =
𝑓′(𝑟)
,
𝑓(𝑟)
thus giving:
𝜅2𝐻 = 𝑓′(𝑟)2 𝑓(𝑟)2 |𝑟=𝑟
𝐻
= (𝑟 − 𝑟𝐻 )
2𝑛−2
2
̃ 2 ( 𝑛 𝑓(𝑟)
̃ + (𝑟 − 𝑟 )𝑓′(𝑟))
̃
𝑓(𝑟)
|
𝐻
2
𝑟=𝑟𝐻
For a non-degenerate Killing horizon, we must require 𝑛 = 1, and hence
𝜅2𝐻
̃ )4
𝑓(𝑟
𝐻
=
4
Let us now turn to the embedding side of the story. We can choose to embed the angular dΩ2 part
however we like without affecting the worldline geometry of a fixed detector. For its embedded
worldline to be Rindler, however, we must look for an embedding whose 𝑡-dependence is of the
form
𝑇 = 𝑘−1 𝑔(𝑟) sinh(𝑘𝑡)
𝑋 = 𝑘−1 𝑔(𝑟) cosh(𝑘𝑡)
Since no other coordinate is allowed to depend on 𝑡, we must set 𝑔(𝑟) = 𝑓(𝑟) in order to recover
(31) under pullback. As with all previous cases, we now need to introduce another coordinate 𝑍
satisfying d𝑍2 = 𝑓(𝑟)−2 − 𝑘−2 𝑓′(𝑟)2 − 1 ≡ 𝐹(𝑟). For the embedding to well-behave at the horizon,
̃ we get
𝐹(𝑟) must be regular at 𝑟 = 𝑟 . Expanding 𝐹(𝑟) in terms of 𝑓(𝑟),
𝐻
𝐹(𝑟) =
̃ 4
4𝑘2 − 𝑓(𝑟)
+ (regular part)
̃ 2
4(𝑟 − 𝑟 ) 𝑘2 𝑓(𝑟)
(32)
𝐻
The first term is singular at the horizon, unless the numerator vanishes at 𝑟 = 𝑟𝐻 . This can only
happen if we choose 𝑘2 = 𝜅2𝐻 . The Unruh temperature recorded by the detector is therefore
𝑇𝑈 = 𝜅𝐻 /2𝜋𝑓(𝑟). Tolman’s law then gives 𝑇BH = 𝜅𝐻 /2𝜋, the Hawking temperature.
For a general (1 + 𝐷)-dimensional static metric d𝑠2 = −𝑓(𝑥)2 d𝑡2 + 𝑔𝑖𝑗 𝑥𝑖 𝑥𝑗 , we give a somewhat
hand-waving but hopefully intuitive argument. Consider the spacelike hypersurface Σ corresponding
to some fixed 𝑡. Let ℋ be a connected component of the zero set of 𝑓 in Σ; this is a Killing horizon
for 𝜉 =
∂
.
∂𝑡
We can foliate Σ by the level sets of 𝑓, which are submanifolds of Σ if 𝑓 is analytic. Let
𝐹 be some continuous labelling of the connected components of these level sets. We assume that
there is a coordinate chart (𝑦1 , … , 𝑦𝐷−1 ) which can cover each level set in the region of interest,
and let (𝐹) ℎ𝐼𝐽 d𝑦𝐼 d𝑦𝐽 be the induced metric on a level set. By definition, d𝐹 is normal to the level
27
Saran Tunyasuvunakool, April 2012
5.2
Recent development and possible directions
Black Hole Thermodynamics
sets, and so with an appropriate choice of 𝐹 our metric becomes
d𝑠2 = −𝑓(𝐹)2 d𝑡2 +
1
d𝐹2 + (𝐹) ℎ𝐼𝐽 d𝑦𝐼 d𝑦𝐽
𝑓(𝐹)2
where 𝐹(x) ≡ 𝐹𝐻 when 𝑥 ∈ ℋ. Once again, the worldline geometry of a fixed observer is unaffected
by the choice of embedding in the 𝑦𝐼 part. We can now apply exactly the same argument as before,
noting that any differences in algebra only manifest themselves in the regular part of (32).
Finally, we note that this argument essentially precludes an extremal static black hole from possessing
an embedding which is suitable for the GEMS approach.
5.3
Recent development and possible directions
Owing to the relative ease with which a temperature could be obtained from the GEMS approach,
there have been attempts at generalising it for a detector whose embedded worldline is not pure
Rindler. In general, the corresponding particle spectrum would no longer be Planckian in this
situation [42], leading to a difficulty in defining a temperature. The primary reason is that the change
in acceleration means that the detector cannot reach an equilibrium with the particles to measure a
temperature, but if the acceleration changes sufficiently slowly then an effective temperature could
still be obtained. This is precisely the viewpoint taken in [24], where they noted that the embedded
worldline of a freely falling observer starting from rest is initially locally Rindler, and so should
measure an effective Unruh temperature there corresponding to the accelerations (10) and (15)
for the Schwarzschild and RN geometries respectively. It is interesting to note that a temperature
measured in this way does not diverge near the event horizon. Another tractable case is that of
a circularly orbiting detector, where the GEMS approach has been verified to work if we use an
appropriate ‘rotating’ generalisation of the Unruh formula [16].
The BTZ black hole remains the only specimen of a rotating solution whose embedding is known,
however recently it had been noted that near a Killing horizon the physics is effectively governed by a
two-dimensional metric, and the application of the GEMS method to this reduced metric yields the
correct temperature. The particular example given in [17] was for the Kerr-Newman geometry, whose
effective metric takes the usual form d𝑠2 = −𝑓(𝑟)2 d𝑡2 + 𝑓(𝑟)−2 d𝑟2 , where 𝑓(𝑟)2 =
𝑟2 −2𝑀𝑟+𝑎2 +𝑄2
.
𝑟2 +𝑎2
The
embedding was obtained in our standard form, but with rather more complicated integrals. Since
the reduction method is based on a general consideration of the Einstein-Hilbert action, it would
not be surprising if this is applicable to a vast array of geometries. Indeed, this could prove to be a
convenient method of calculating Hawking temperatures in general.
Nevertheless, it would still be interesting to obtain genuine embeddings for rotating spacetimes.
Aside from thermodynamic applications, these results could also be a helpful aid in understanding
the geometry. However, it is probably safe to say that such a result will be very complicated, especially
in higher dimensions where simultaneous rotations about multiple axes [32] and drastically different
horizon topologies [43] become possible.
28
Saran Tunyasuvunakool, April 2012
5.3
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