ICTP-SAIFR School on NP-QCD: Gauge/string duality
Lightning review of GR1
This note will cover the basic notions about
1. Special Relativity
2. General Relativity
3. Black holes
4. Anti de Sitter space
Useful references on GR are
• James B. Hartle, Gravity: An Introduction to Einstein’s General Relativity, B. Cummings (2003) — undergraduate level textbook;
• Sean Carroll, Spacetime and Geometry: An Introduction to General Relativity, B. Cummings (2003), also available as gr-qc/9712019 — more advanced level.
1.
Special relativity
In Special Relativity (SR) space and time are unified into the notion of spacetime. The time
lapse between two events (dt) or their spatial separation (d~x) are not invariant notions by
themselves (different observers are going to measure different dt and d~x), what is invariant
is the combination
ds2 = −c2 dt2 + d~x .
(1.1)
We can set c = 1, i.e. we can measure time in units of length and 1 cm-time is the time light
needs to travel 1 centimeter (∼ 10−10 seconds). The line element above can be also written
as


−1 0 0 0


 0 1 0 0
2
µ
ν
ds = ηµν dx dx ,
ηµν = 
(1.2)
 ≡ diag(−1, 1, 1, 1) ,
 0 0 1 0
0 001
1
This is, with only minor changes, a write-up of a review lecture on GR given by Gary Horowitz at a
KITP workshop in 2009: http://online.kitp.ucsb.edu/online/adscmt m09/horowitz/
1
Figure 1: The lightcone.
where the repeated indices µ, ν = 0, 1, 2, 3 are summed over. The symbol ηµν is called
Minkowski metric. More generally, a metric is something we use to measure distances between events in spacetime.
Particles in SR follow worldlines, which are denoted by xµ (λ). This specifies the 4 coordinates (1 time coordinate + 3 space coordinates) of the particle as a function of the parameter
µ
λ. The tangent vector to the world-line is given by the “time derivative” ẋµ (λ) = dxdλ(λ) .
Massless particles (e.g. photons, the light quanta) follow null worldlines (i.e. worldlines with
zero length)
ηµν ẋµ ẋν = 0 ,
(1.3)
and the set of all null worldlines starting from and ending at a point is the light-cone of that
point, see Fig. 1. Massive particles follow instead time-like worldlines
ηµν ẋµ ẋν < 0 ,
(1.4)
i.e. curves inside of the particles’ light-cones (the red curve in Fig. 1), travelling at less than
the speed of light. The physical time experienced by a massive particle is called proper time
and is defined as
Z
p
τ = dλ −ηµν ẋµ ẋν .
(1.5)
The time we experience depends then on our state of motion.
2
2.
General Relativity
The key idea of General Relativity (GR) is that since gravity is universal (it affects all objects
in the same way) we should identify it with something equally universal:
gravity
=
geometry of spacetime.
(2.1)
The metric need not take the simple form ηµν above, i.e. need not be flat, but is allowed
to depend on the spacetime coordinates xµ
ds2 = gµν (x) dxµ dxν .
(2.2)
Once we allow the metric to be a generic function of xµ we end up with a huge redundancy,
because we can label points with different sets of coordinates
xµ → x̃µ (x) .
(2.3)
As long as x̃µ is smooth, differentiable and invertible, it is as good a coordinate as xµ . This
is called the “gauge freedom” of GR. Of course the form of the metric will change if we
change coordinates. To describe the geometry in a more invariant way we need to talk about
curvature.
2.1.
Curvature
As a simple example, let’s consider a sphere. Take a point on the equator and a vector at
that point orthogonal to the equator. Moving the vector along some closed curve keeping
it parallel to itself we see that, after doing a loop, the vector has rotated, because of the
curvature of the sphere. See Fig. 2(a).
We want to use this intuition in a more local way, i.e. we want to have a measure
of how the space is curved near every point. To do that we can consider an infinitesimal
square loop, which is characterized by two vectors, v1µ and v2ν . See Fig. 2(b). Moving a vector
parallel to itself along this small square will produce a rotation, which will involve both space
and time (since this is a loop in spacetime). This rotation in spacetime is called a Lorentz
transformation and consists of a spatial rotation plus a boost. Elements of the Lorentz
group are given by antisymmetric objects with 2 indices (tensors). The boost/rotation of
the vector will also depend on the details of the little square, i.e. it will depend on v1µ and
v2µ . All of this information can be packaged in the symbol
Rµνρσ v1ρ v2σ .
(2.4)
This Rµνρσ is a 4-index object known as Riemann curvature tensor. It contains all the
information about spacetime: the first two indices tell us what Lorentz transformation a
vector gets when it goes around the little square and the last two indices give us details of
3
(a)
(b)
Figure 2: (a) Parallel transport around a closed loop on the sphere. (b) An infinitesimal square
loop.
the loop.2 Schematically it is
Rµνρσ = “ ∂ 2 g + (∂g)2 ” .
(2.5)
We can take traces of the Riemann tensor to get pieces of the curvature. To do that we need
to define the inverse (as a matrix) metric g µν , which is such that
gµν g νρ = δµρ .
(2.6)
Then the Ricci curvature is the 2-index tensor defined by
Rµν = g ρσ Rµρνσ ,
(2.7)
while the scalar curvature (=no indices, just a function of the coordinates) is given by
R = g µν Rµν .
(2.8)
Notice that all these objects contain two derivatives of the metric.
In a curved space we also need to modify our notion of derivative and define a covariant
derivative ∇µ , which is something that makes sense independently of the coordinate choice
we adopt. The action of the covariant derivative on a vector with upper index V ν is given
by
∇µ V ν = ∂µ V ν + Γνµλ V λ ,
(2.9)
and on a vector with lower index Wν is given by
∇µ Wν = ∂µ Wν − Γλµν Wλ ,
2
(2.10)
For symmetry properties of the Riemann tensor you are invited to look at chapter 3 in Carroll’s lectures
notes on the arXiv.
4
where the symbol Γλµν is called a Christoffel symbol. Its explicit form can be obtained from
the requirement that the covariant derivative acting on the metric be zero3
∇λ gµν = 0 ,
(2.11)
and is given explicitly by
1
Γλµν = g λρ (∂µ gρν + ∂ν gρµ − ∂ρ gµν ) .
(2.12)
2
In flat space the Christoffel symbols vanish, and the same happens when ∇µ acts on a scalar
field φ:
∇µ φ = ∂µ φ .
The Laplacian in a curved spacetime is given by
q
1
µν
φ = p
∂µ
− det(gαβ )g ∂ν φ .
− det(gαβ )
2.2.
(2.13)
(2.14)
Integration on a curved spacetime
R
In GR we cannot just integrate a function with the standard d4 x measure, because the
result will be coordinate dependent. We need to introduce a proper 4-volume. Let’s assume
for simplicity that the metric is diagonal
ds2 = −gtt dt2 + gxx dx2 + gyy dy 2 + gzz dz 2 .
(2.15)
A curve that is extending in only the x-direction has t, y, z fixed and therefore dt = dy =
√
cy-plane is going
dz = 0, so that its proper length is gxx dx. Similarly a proper area in the x
√
√ √ √
√ √
to be gxx gyy dx dy, while the proper 4-volume is −gtt gxx gyy gzz dt dx dy dz. At this
point it is easy to guess the correct invariant measure for a generic, non-diagonal metric:
q
√
− det(gµν ) dt dx dy dz ≡ −g d4 x .
(2.16)
2.3.
GR action and equations of motion
We want to treat the metric as a dynamical field, for which we need a 2-derivative action.
Recall that the scalar curvature has precisely two derivatives of the metric, so that the
simplest action we can write down is
Z
√
S = d4 x −g R .
(2.17)
This is the Einstein-Hilbert (EH) action. When we extremize this action with respect to the
metric we get the Einstein equations in the vacuum (i.e. without any matter)
1
Rµν − gµν R = 0 .
2
3
This defines a metric connection.
5
(2.18)
There are two modifications of this action:4
• We can add a constant
Z
S=
√
d4 x −g(R − 2Λ) ,
(2.19)
resulting in these equations of motion
1
Rµν − gµν R + Λgµν = 0 .
2
(2.20)
This Λ is called cosmological constant. One of the main unanswered questions in physics
today is to understand the tiny but non-zero (positive) value of Λ that we observe.
• We can add matter. This requires the introduction of Newton’s constant GN (which
controls the coupling of matter to the metric)
√
Z
−g R
4
S= dx
+ Lmatter (gµν , φ, Aµ , · · · ) .
(2.21)
16πGN
The equations of motion contain now the matter stress tensor Tµν (i.e. the derivative
of Lmatter with respect to gµν )
1
Rµν − gµν R = 8πGN Tµν .
2
2.4.
(2.22)
Motion of test particles
Test (or probe) particles move in the geometry without modifying it (i.e. they do not
“backreact” on the geometry). This happens when their mass is much smaller of the mass
of the object (e.g. the star or black hole) that creates the geometry in the first place. The
worldline of these particles is the “straightest” possible trajectory and it is called a geodesic.
There are two ways to think about a geodesic. First, it is a curve such that if I take a
tangent vector to the curve and I move it parallel to itself this will stay tangent to the curve.
It is easy to see this in flat space, where the only geodesics are straight lines. Second, it is
the curve that extremizes (in fact, maximizes) the proper time, so that a geodesic has the
longest proper time between two events.
3.
Black holes
Just months after Einstein wrote his equations, Karl Schwarzschild found an exact (non
linear) solution (for the vacuum case, with zero cosmological constant, in units where GN =
4
Of course one could also think of adding higher derivative terms, i.e. terms with higher powers of the
curvature invariants. String theory predicts such terms. Still, we can regard the EH action as the leading
term at low energies of these more complicated actions.
6
1):
2M
ds = − 1 −
r
2
dt2 +
dr2
+ r2 dΩ2 ,
2M
1− r
dΩ2 = dθ2 + sin2 θ dϕ2 .
(3.1)
This metric only depends on one parameter, M , the mass of the object giving rise to the
geometry.5 When M = 0 the metric reduces to flat space, and also the region r → ∞ is flat.
We say that this solution is asymptotically flat. This solution is static (nothing depends on
t) and spherically symmetric (we have a round 2-sphere at each point with constant t and
r), and was interpreted as the geometry outside a static, spherically symmetric star. Notice
that this is a solution of the vacuum equations of motion (Tµν = 0), no matter here, so that
this is the solution outside of the star.
Some classic tests of GR were performed using this solution:
• bending of light, nowadays a central tool in gravitational lensing;
• gravitational red-shift. The energy of a photon emitted at some r and propagating
to a larger r goes down, so that the photon looks redder. This is because the photon
loses energy as it climbs out of the potential well of the star, just like a massive particle
would as it climbs out of the potential of Newtonian gravity;
• motion of planets. Orbits are not exact ellipsis, they do not close (cf. precession of
the perihelion of Mercury).
GR has been tested to 1 part in 105 . GPS systems would not work without taking into
account of GR.
Notice now that something funny happens at r = 2M . One could think that we need not
worry about this point in practice: for example, in the case of the Earth it would correspond
to r ∼ 1 cm, while for the Sun it would be r ∼ 1 km. In both cases it is inside the Earth or
Sun, where we are not supposed to trust the Schwarzschild solution (which is a solution of
the vacuum equations).
But what happens if the star uses up all of its nuclear fuel and undergoes a gravitational
collapse? Then the system is no longer static and one would think that the Schwarzschild
metric does not apply anymore (since that is static). But there is Birkhoff ’s theorem which
states that
The only spherically symmetric vacuum solution is Schwarzschild.
So we don’t have to assume staticity and the metric is going to stay in Schwarzschild
form as the star collapses. The radius gets smaller and smaller and at some point it will fall
5
To convince yourself that M indeed has a meaning of mass you can take the weak field limit of the
geometry and recover the Newtonian case. Reading off the gravitational potential Φ from the Newtonian
limit one finds
Φ=−
M
.
r
7
(3.2)
inside the r = 2M surface, so that we need to understand what this surface really means.
Also, what happens at r = 0?
We need to distinguish between two kinds of singularities that appear in GR:
• curvature singularities which are actual pathologies of the geometry and
• coordinate singularities which are places where the coordinates become singular
but the geometry is fine.
To do this we need to compute curvature invariants. For Schwarzschild one finds that6
48M 2
,
(3.3)
r6
which is perfectly fine at any finite r, in particular at r = 2M . This means that r = 2M
is a coordinate singularity (we can get rid of it by a coordinate change), while r = 0 is a
curvature singularity, an actual singularity in the geometry.
Rµνρσ Rµνρσ =
How do we find a good coordinate system? We use something physical, like for example
the motion of ingoing light rays. We can trade the time coordinate t with a coordinate v
which is constant along ingoing light rays (see Hartle’s book, page 258, for details). Then the
metric is described by (v, x, y, z), the so-called Eddington coordinates. In these coordinate
the solution looks like in Fig. 3 (left). The collapsing star is depicted in grey, it shrinks until
r = 0 where the density becomes infinite. The wiggly line denotes the singularity at r = 0.
At any point inside of the star there is a light-cone that is originally expanding. After a
while there is so much matter inside of the light-cone that its expansion stops (because all
the matter bends light more and more) and the light-cone remains at some finite r. This
happens at r = 2M , the event horizon. This is a null surface. Outgoing light rays are
being tipped more and more vertical until at r = 2M they are completely vertical, see Fig. 3
(right). At r < 2M they actually stay inside the event horizon and reach the singularity. The
4-dimensional space inside of the event horizon is called a black hole (BH). The conjecture
that spacetime singularities should always be surrounded by a horizon in physically allowed
solutions is known as cosmic censorship.
A quick way to find the position of the event horizon rH (at least for the metrics that we
are considering here and the metrics that we will consider in the problem sets) is to find the
solution(s) to the equation gtt = 0. This is because for r > rH one has gtt < 0 and grr > 0
while for r < rH it is the opposite and the roles of t and r get exchanged, with t being
the spacelike coordinate and r being the timelike coordinate. Decreasing r is the timelike
direction, so once you cross the horizon there is no way to avoid the singularity at r = 0 (in
the same way as you can only go forward in time and not backward) and you are doomed!
Another way to visualize the black hole and understand better causal questions is to
require that light rays are at 45◦ (unlike what we had in Eddington coordinates, where the
light rays were not at a fixed angle, in fact they were tipped more and more vertical). In the
Kruskal diagram of Fig. 4 the collapsing region is still in gray. Once we cross the r = 2M
6
This is a scalar invariant called Kretschmann invariant.
8
Figure 3: Schwarzschild in Eddington coordinates. The infalling light ray is a v = const. surface.
Outgoing light rays are pulled more and more vertical.
surface we have to reach the singularity at r = 0.
Now suppose we remove all matter and extend the solution everywhere. The result looks
like in Fig. 5. The wiggly lines are singularities and light rays are at 45◦ . Region I is the
usual exterior region of a BH. Region II is the inside of the BH. These two regions are
physical. There are two more regions that are unphysical: Region III is the time reversal of
a BH and it is called a white hole, while Region IV is an identical copy of Region I. Now the
topology of the space is no longer R4 , but we have a wormhole. Of course there is no way to
communicate between Regions I and IV, since this would require to travel at superluminal
speeds. This extension of the Schwarzschild solution is not physical, because it requires a
singularity in the past (which is the time reversal of the singularity in the future), a very
peculiar initial condition.
Finally, another useful way to visualize a Schwarzschild BH is through a Penrose diagram,
where the whole spacetime is conformally mapped to a finite region, see Fig. 6. The most
important use of a Penrose diagram is to compare spacetimes to Minkowski spacetime at
infinity, i.e. to understand when a spacetime is asymptotically flat. You are invited to read
more about this kind of diagrams in chapter 7 of Carroll’s book.
What about other BH solutions? There are none!7 This is the content of the BH unique7
Again, we are still assuming vacuum solutions, so no charges.
9
Figure 4: Schwarzschild in Kruskal coordinates.
Figure 5: Regions of a Schwarzschild black hole with all the matter removed.
10
Figure 6: Penrose diagram of the extended Schwarzschild black hole. i+ is future timelike infinity,
i0 is spatial infinity, i− is past timelike infinity, I + is future null infinity, and I − is past null infinity
(note that i+ and i− are distinct from r = 0). The external triangles are Minkowski spacetime,
consistently with the fact that Schwarzschild is asymptotically flat.
ness theorem:
The only static, asymptotically flat BH is Schwarzschild.
You can form a BH in many different ways (from collapses of stars with different multipole
moments, for example) but the end state is always Schwarzschild. One says that “BHs have
no hair”, i.e. there are no distinguishing features for static BHs. This is different from the
case of stars, which can be distinguished by observers at infinity by looking at their multipole
moments. Also if we let a particle or an e/m wave propagate in the BH background, at late
times, when they have disappeared behind the horizon, the BH will still have no hair.
In 4 dimensions there are two generalizations of Schwarzschild:
• we can add angular momentum and the resulting BHs are called Kerr and are also
unique (the only two parameters are mass and angular momentum);
• we can add charge (for example by coupling gravity to a Maxwell field) and have a
Reissner-Nordström (RN) BH.
Astrophysical BHs are probably electrically neutral (because they would otherwise attract
opposite charges and neutralize) but they do carry angular momentum.
3.1.
Black hole thermodynamics
Now just a few comments about quantum effects. So far everything was classical and nothing
could come out of the BH. But Hawking showed in ’74 that, quantum mechanically, a BH
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will radiate with an essentially thermal spectrum at a Hawking temperature given by
TH =
1
,
8πM
(3.4)
in units of ~ = 1. The way to compute this temperature is to analytically continue the
Schwarzschild metric, t → iτ , and perform the coordinate transformation r = 2M + ρ2 /8M ,
so that near r = 2M the Euclideanized metric reads
ds2 ' dρ2 +
ρ2
dτ 2 + (2M )2 dΩ2 .
(4M )2
(3.5)
This is the product of a 2-sphere with radius 2M and the Euclidean plane, provided that
τ is periodically identified with the right period (otherwise one has a conical singularity at
the origin). The period of τ must be 8πM , so that 8πM/4M = 2π, as it should be. The
temperature is identified with the inverse period of the Euclidean time, yielding the formula
written above.8 Let’s see how to do this in a generic case, where the Euclideanized metric
contains
ds2 ⊃ V dτ 2 +
1 2
dr ,
V
(3.8)
where V is some function (in the Schwarzschild case it was V = 1 − 2M/r). Calling again
the position of the horizon rH , the requirement of no conical singularity implies that it must
be
√
√
d( V )
1
1 0 2π = lim V ∆τ
→
= TH =
V .
(3.9)
r→rH
dr
β
4π r=rH
BHs also have an entropy, the so-called Bekenstein-Hawking entropy, which is given by
the area of their horizon
SBH =
A
.
4GN
(3.10)
On way to motivate this is from the first law of thermodynamics
dM = TH dSBH .
(3.11)
8
A quick justification of this statement is as follows. Consider the amplitude to go from a configuration
(g1 , φ1 ) at time t1 (where g is the metric and φ denotes all other fields) to a configuration (g2 , φ2 ) at time t2
Z
h(g1 , φ1 ), t1 |(g2 , t2 ), t2 i = Dg Dφ eiS[g,φ] .
(3.6)
In Schrödinger picture this becomes (g1 , φ1 )|e−i(t2 −t1 )H (g2 , t2 ) . Assuming that (g1 , φ1 ) = (g2 , φ2 ), writing
t2 − t1 = −iβ, and summing over a complete set of eigenstates (ψn , En ) of the Hamiltonian we arrive at
X
Z=
e−βEn .
(3.7)
n
We can interpret this as the partition function of the canonical ensemble at temperature T = 1/β. This
expression is the same as the Euclidean path integral obtained by Wick rotating the RHS of (3.6) if the
fields are periodic in (Euclidean) time with period equal to β.
12
Figure 7: AdS space in global coordinates.
We similarly have a second law of BH thermodynamics
d(SBH + Sradiation )
≥ 0.
dt
(3.12)
Just to have an idea of the orders of magnitude: for a solar mass, TH ∼ 10−7 K (tiny!)
and SBH ∼ 1077 in units kB = 1 (huge!).
4.
Anti de Sitter space
Finally a few words about one of the main characters of these lectures: Anti de Sitter (AdS)
space. This is the simplest, maximally symmetric solution to Einstein equations with a
negative cosmological constant, Λ < 0:
2
r
dr2
2
2
ds = −
+
1
dt
+
+ r2 dΩ2 ,
(4.1)
r2
L2
+
1
L2
where L2 = −3/Λ is the (constant) radius of curvature of the space.
One useful way to think about AdS is to do a conformal transformation of the metric
above, meaning that we are not changing any causal relation and light rays stay null. In
other words, we rescale the lengths without changing the angles. The transformed metric is
2
4
f2 = L ds2 ' −dt2 + L2 dΩ2 + L dr2 ,
ds
r2
r4
(4.2)
where the ' symbol is because we are taking large r (“dropping the 1” in the metric factors
above). This metric is called AdS in global coordinates. Notice that now the distance of a
point to r = ∞ is finite. Moreover, the metric at r = ∞ is just a cylinder, i.e. a sphere
of radius L times time, see Fig. 7. AdS can be seen as the interior of this cylinder and the
boundary of AdS is at finite distance. Light rays can reach the boundary and bounce back
13
Figure 8: The Poincaré patch.
in finite time. This is very important: it means that to evolve any system we need to impose
boundary conditions!
We will see in the lectures that according to the AdS/CFT correspondence there is
a duality between certain theories of gravity defined in the interior of AdS and certain
quantum field theories (with no gravity) living on the boundary of AdS. Most of the time
we don’t write AdS in the form above because we are not usually interested in QFTs defined
on spheres. In QFT we usually work in flat space. We can then change coordinates and
write the metric of AdS in a way that the metric at infinity is just Minkowski. This different
coordinate choice is called Poincaré coordinates. They do not cover the whole space, just a
wedge called Poincaré patch (the gray region in Fig. 8), which is given by
ds2 =
L2 2
r2
2
2
(−dt
+
d~
x
)
+
dr .
L2
r2
(4.3)
The r = 0 surface is called the Poincaré horizon. The boundary of AdS is at r = ∞. The
fact that the boundary of AdS is a timelike surface is the most important feature of this
spacetime. Sometimes it is useful to use the coordinate u = L2 /r as radial coordinate. The
metric above becomes
ds2 =
L2
(−dt2 + d~x2 + du2 ) .
u2
(4.4)
To introduce a black hole with horizon at u = uH we modify the metric above by including
a ‘blackening factor’ f (u):
L2
du2
u4
2
2
2
ds = 2 −f (u)dt + d~x +
,
f (u) = 1 − 4 .
(4.5)
u
f (u)
uH
As we will see, the Hawking temperature of this black hole can be identified with the temperature of a thermal QFT state in the boundary of AdS.
14