INTRODUCTION TOGENERAL RELATIVITY
Summary
1. Special Relativity
2. General Relativity: How it works
3. General Relativity: Geometry and matter
J. Fernando Barbero G.
XIII Encuentro de Otoño de
Geometría y Física, Murcia,
September 2004
2
WARNING AND DISCLAIMER(the fine print)
In the following I will use a point of view on space-time processes pioneered by Milne, Bondi and Geroch in which basic concepts are described in an
operational way, i.e. by using simple sets of instructions to get the relevant information about space-time. This is done with the most basic tools: clocks and
light rays. For a while my presentation will be quite informal in a mathematical sense but don’t panic! it will be shaped in a (more or less) proper mathematical
form at the appropriate time. I do not intend to derive special relativity but highlight its most important concepts. There is a different treatment in the three
sections. The presentation of special relativity is non-standard (though well known) but, hopefully, appealing and intuitive. In the second part, in which I
describe how gravity works in everyday situations, I will use the previous point of view. The goal is to understand how gravity can be described by means
of a space-time metric. Finally, to arrive at the Einstein equations I will follow a standard treatment (essentially the very nice text book by Wald). One can
possibly be much more original here too but (proper) time is a very limited resource...
We need to develop a relativistic intuition quite
far from our everyday experience.
3
Special Relativity
SPACE, TIME, SPACE-TIME: BASIC CONCEPTS
➢ Event: A place in space and an instant in time.
➢ Space-time: Ensemble of all the possible events.
How are these events organized and interrelated?
Several viewpoints on space and time:
❶ Aristotelian.
❷ Galilean.
❸ Einstenian.
4
❶ ARISTOTELIAN (pre-galilean)
• Space and time are not interrelated but independent .
• Each event happens at a certain place in some instant of time.
• Time and position are absolute and hence velocities too.
• There is a privileged reference frame at rest .
• Questions that make sense in this framework:
◦Where?◦When?◦ Are two events simultaneous?◦ Do they happen at the same place?◦What is the distance between them?◦What is the absolute velocity of a moving object? or◦ Is it at rest?...
5
❷ GALILEAN
• No privileged reference frame (at rest) but rather a class of equiva-
lent reference systems called inertial (personalized aristotelian fra-
meworks ).
• Space and time are independent and simultaneity has an absolute
meaning as in the Aristotelian viewpoint.
• Absolute positions only at each instant of time; we cannot tell if non-
simultaneous events happen at the same place (i.e. point in space)
or not.
• Questions that make sense in this framework:
◦ Time interval between events.◦ Distance between simultaneous events.◦ Relative velocity.
6
❸ EINSTENIAN
• Absolute simultaneity is lost and becomes observer dependent.
• The space-time interval appears as a measure of a certain “distan-
ce” (time would be better) between, not necessarily simultaneous
events.
• Pairs of events can be classified as space-like, null or time-like ac-
cording to the sign of the interval connecting them.
• The physical interpretation of the interval depends on the the type of
relation between the events that define it.
◦ Time-like physical observers.◦ Null light rays.◦ Space-like simultaneity and distance.
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RELATIVISTIC SPACE-TIME PHYSICS: CLOCKS AND LIGHT RAYS.
❶ A clock is a device that measures the proper time of the physical ob-
server that carries it.
❷ Light rays originating at a given event move in a way that is indepen-
dent of the state of motion of the source. (l-hypothesis)
• There is a single ray originating in this event for each spatial direc-
tion and there is a single one arriving at it from every direction.
• The light cone is formed by all the light rays arriving and originating
ar a certain event.
• Their trajectories encode intrinsic information about space-time (for
example about causality).
• They allow to explore space-time around a physical observer and
define geometrical magnitudes and properties in an operational way.
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THE MASTER LAW OF SPACE-TIME PERCEPTION
What you see is... rays of light!
INERTIAL OBSERVERS
✦ They are observers in the simplest state of motion.
✦ They are all completely equivalent (there is not a privileged one).
✦ They can be in motion with respect to each other. The magnitude of
this relative motion can be defined and measured ( relative velocity ).
✦ The proper time between the reception of two light rays by an inertial
observer Obs2 emitted by another one Obs1 is proportional to the pro-per time interval of emission as measured by the latter. (k-hypothesis)
✦ As none of these two inertial observers is privileged the proportionality
constant is the same if the rays are sent by Obs2 and received by Obs1.
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SPACE-TIME DIAGRAMS
time
spacelight ray
future light cone
pastlight cone
inertialobserver
observer
event
World lines: Histories of physical obser-
vers and objects in space-time.
Restrictions on their shape
✦ related to causality.
✦ time-like (clocks!).
Light rays:
✦ There is a light cone at every event.
✦ real observers “inside” the light cone.
I will suppose that I have an affine space
with points in R4 and associated real vec-
tor space R4.
✦ Inertial observers represented by time-
like straight lines.
✦ Light rays represented by straight lines
“at a fixed angle” w.r.t. the time direc-
tion.
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LIGHT RAYS ORIGINATING AT A GIVEN EVENT MOVE IN A WAY THATIS INDEPENDENT OF THE STATE OF MOTION OF THE SOURCE.Astronomical evidence (de Sitter) T(t) = t + d−R cos ωt√
c2−R2ω2 cos2 ωt−Rω sin ωt, y(t) = R sin ωt
y
T
y
T
multipleimages
y
T
Infinite light speed. Light speed independent of the state of motion
of the emitter.
Relative velocity of light w.r.t. the emitter is c.
11
PROPER TIME DIFFERENCE (THE RELATIVISTIC INTERVAL)Can a given inertial observer find out the proper time interval between
events not lying on his own world line? YES, using the k-hypothesis
A
B
T−
T+
TAB
Obs1Obs
✦ T−: difference between proper emission timesby Obs1.
✦ T+: difference between proper reception timesby Obs1.
✦ TAB = kT− and by symmetry T+ = kTAB sothat
TAB =√
T−T+
✦ Notice that T+ = k2T− ≡ (1 + z)T− wherez > −1 is called the redshift (it can be blue!)
12
COMMENTS
✦ A and B cannot be anything, they must be such that the last light rayto leave Obs1 arrives last. When this happens we say that the intervalbetween these events is time-like .
✦ If A and B lie on the worldline of an inertial observer; the redshift is ameasure of their relative motion. It can be obtained directly by any of
two observers by a simple measurement.
✦ Consistency:
Is this prescription observer independent? i.e. Would the proper time
difference measured by a second observer Obs2 coincide with theone obtained by Obs1?
Yes under some restrictions
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Is it√
T+T− =√
T++T−− with T− =√
T−−T−+ and T+ =√
T+−T++?
T−−
T−+
T+−
T++
T
T+
T−
Obs1Obs2
✦ This happens iff
T++T+−
=T−+T−−
where the l.h.s. involves the time mea-
surements performed by Obs1 to de-termine T+ and the r.h.s. those usedto get T−.
✦ For inertial observers this is a simple
consequence of the k-hypothesis .
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PROPER TIME IS OBSERVER-DEPENDENT (THE TWINS PARADOX)The fact that the geometric mean is always less or equal than the arithmetic mean
gives rise to the phenomenon of time dilation.
T−
T+
T̃+
T̃−
C
B
A
√T+T− ≤
T+ + T−2
,√
T̃−T̃+ ≤T̃− + T̃+
2
⇓√
T+T−+√
T̃+T̃− ≤12
[T− + T+ + T̃− + T̃+
]=
= T− + T̃− = T+ + T̃+
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SIMULTANEITY (MILNE’S RADAR METHOD)
✦ One inertial observer.
✦ One event A on its worldline.
✦ Another event B outside its worldline where we place a mirror.
eB: emission to B
τ
rB: receptionfrom B
τ
B
A
eB
τ
rB
τ
B
rC
eC
A
C
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✦ They are simultaneous if the observer can send a light ray to B andrecord its reflection in such a way that the time interval measured by
his clock from emission to A is the same as the one from A to recep-tion. This definition assumes a certain symmetry between departing
and returning rays.
✦ If both A and B are outside the observer’s world line one can follow theprocedure outlined in the second picture to define simultaneity relative
to the observer.
COMMENTS:
✦ This concept of simultaneity defines an equivalence relation whose
classes are called simultaneity surfaces.
✦ This definition could be extended to non-inertial observers
✦ In this case it becomes a local definition (i.e. valid not far from the
observers world line) because, for example, some parts of space-time
may not be reachable by light rays emitted by the observer.
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✦ The simultaneity notions for two observers may coincide in a neigh-
borhood of their world lines but not everywhere.
τ
τ′
τ
B
Obs2
Obs1
A
C
D E
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LENGTH (SPATIAL DISTANCE)
T−
τ
T+
τ
A
Obs
B
✦ Two events A and B where we placemirrors.
✦ One inertial observer Obs such that Aand B are simultaneous .
✦ We define the spatial length from A to Bas lAB = cτ (c can be considered as aconversion factor).
✦ If we measure time in meters we can put
c = 1 (often in the following).
✦ Can another inertial observer determine
this proper length? Yes because T− = kτand τ = kT+ τ =
√T−T+.
✦ This is the same expression that we
found for the proper time interval but now
the first emitted ray arrives last. When
this happens we say that the interval bet-
ween the events A and B is space-like .
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VELOCITY
A
B
A′
B′
Obs1
Obs2
τ0
τ0 + h
τ0+τ1+(z+2)h2
τ0+τ12
τ1
τ1 + (z + 1)h
Measure of change in spatial distance.
lAA′ =τ1− τ0
2c, lBB′ =
τ1− τ0 + zh2
c
v = lı́mh→0
lBB′ − lAA′(1 + z2)h
=z
2 + zc
1 + z =c + vc− v
✦ Obs1 is moving w.r.t. Obs2; this changes thereception interval to (1 + z)h where h is the(small) proper time delay between the emis-
sion of the two light pulses.
✦ z is measured at the time of reception.
✦ If we demand that the second emitted ray must
arrive second we have the restriction z >−1 ⇒ v ∈ (−c, c).
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RADAR SPEED
τττττ
τ′
τ′
τ′
τ′
τ′
✦ Instead of two light rays we can
consider using periodic waves
satisfying
∂2t φ− c24φ = 0
(Lorentz invariance!).
✦ Light rays correspond to charac-
teristic curves.
✦ In this case z is measured as a fre-quency shift (Doppler effect)
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LENGTH CONTRACTION
τ1
Tβ
Tα
τ1
τ
AB
Obs1
ruler
From τ1 = Tα(1 + z) and Tβ = τ1(1 + z)we obtain that
τ1 + Tα = τ1
(2 + z1 + z
),
τ1 + Tβ = τ1(2 + z)
and, hence,
2τ =√
(τ1 + Tα)(τ1 + Tβ)
so that
τ = τ12 + z
2√
1 + z=
τ1√1− v2
l1 = lprop√
1− v2 < lprop
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VELOCITY COMPOSITION
T0
T′0
T1
T′1
T2
T′2
T3
T′3
τα
τβ
τ′β
Obs2
Obs0
Obs1h
(1 + z1)h
1+z21+z1
h
(1 + z2)h
h√
1 + z1 = j
1+z2√1+z1
h = 1+z21+z1 j
τ′α
✦ T′0 = T0 + h, h is a delay of our choice.
✦ T′1 = T1 + (1 + z1)h “redshifted” reflec-tion by Obs1.
✦ T′3 = T3 + (1 + z2)h “redshifted” reflec-tion by Obs2.
✦ T′2 = T2 +1+z21+z1
h so that the reflection atObs1 gives rise to the rays arriving at T3and T′3.
✦ τ′α = τα + h√
1 + z1 ≡ τα + j obtainedby using the geometric mean formula
for the interval.
✦ τ′β = τβ + h1+z2√1+z1
≡ τα + j1+z21+z1 againobtained by using the geometric mean
formula for the interval.
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c + v21c− v21
=1 + z21 + z1
=c+v20c−v20c+v10c−v10
→ 1 + β201− β20
=(1 + β21)(1 + β10)(1− β21)(1− β10)
→
β20 =β21 + β10
1 + β21β10
where βij = vij/c
COMMENTS
✦ This is a rather strange addition law: it is impossible to go beyond c bycomposing smaller velocities.
✦ Let β1, β2 ∈ (−1, 1) and ∗ : (−1, 1)2 → (−1, 1) : (β1, β2) 7→ β1+β21+β1β2,
then this defines a group.
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THE INTERVAL REVISITED
A
B
τA
τB
Obs
O
tB
tA
A′
B′
✦ Pick an observer Obs and an event O in hisworld line.
✦ Let us name or label events by assigning some
“coordinates” to them in a physical way . We
could, for example, use the T− and T+ definedbefore and measured with the help of O (“nullcoordinates”).
✦ Instead we can choose to label each event by
assigning the numbers t, and x = cτ to it whe-re
➢ t is the proper time measured by the obser-ver from O to the event B′ in its world linewhich is simultaneous to B.
➢ cτ measures the spatial distance betweenB′ and B.
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✦ If we have two such events A and B is is straightforward to computethe differences between emission times and reception times to get the
square of the interval.
T− = (tB − tA)− (τB − τA) ≡ ∆t−1c
∆x
T+ = (tB − tA) + (τB − τA) ≡ ∆t +1c
∆x
T−T+ = ∆t2− 1c2∆x2
✦ If the interval from A to B is space-like we get in a completely analo-gous way that the square of their spatial distance is given by
T−T+ = 1c2∆x2− ∆t2
✦ Finally if it is lightlike we have ∆t = 1c∆x and, again
T−T+ = ∆t2− 1c2∆x2 = 0
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We find that intervals can be classified by using a 1+1 dimensional non
degenerate quadratic form in space time and proper times and spatial
distances are determined by it!
This leads to the
The Minkowski metric
ds2 = dt2− 1c2dx2
and the recognition of the fact that a 3+1 dimensional point of view is
much better that the traditional separation of space and time.
According to Penrose special relativity cannot be considered as
complete until the introduction by Minkowski of the concept of
space-time and the realization of the fact that it is completely des-
cribed by the so called Minkowskian metric.
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PROPER TIME CANNOT ORIGINATE IN A UNIVERSAL TIME
✦ Let us take the coordinate system defined before.
✦ Can we find a scalar function Φ(t, x) such that
TAB =√
(tA − tB)2− 1c2(xA − xB)2 = Φ(tA, xA)−Φ(tB, xB)?
NObecause we would have then Φ(t, x)−Φ(0, 0) =
√t2− x2c2 and
Φ(tA, xA)−Φ(tB, xB) =
√t2A −
x2Ac2−
√t2B −
x2Bc26= TAB
The existence of a universal, observer independent, time is in con-
tradiction with hypotheses l and k. If we want to keep any one of
them we have to abandon the other!
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LORENTZ TRANSFORMATIONS (1+1 dimensions)
0
s
τ
τ
τ′
τ′
s′ = ks
A
t = τ + s, t′ = τ′ + s′
1 + k2
2k=
1√1− v2c2
,1− k2
2k=
−v√1− v2c2
k(2τ′ + ks) = 2τ + s, s = t− τ
⇓τ′ =
(1 + k2)τ + (1− k2)t2k
t′ = τ′ + ks = k(t− τ) + τ′
x′ =(1 + k2)x + (1− k2)ct
2k=
x− vt√1− v2c2
t′ =(1 + k2)t + (1− k2)τ
2k=
t− xvc√1− v2c2
29
MATHEMATICAL MODEL FOR SPECIAL RELATIVITY SPACE-TIMEA slight change from the Minkowskian affine space introduced above. Let
us instead use a differentiable manifold.
(R4, g)
✦ R4 with the usual differential manifold structure.
✦ g: Minkowski metric on R4.
✦ This metric can be taken as a twice covariant symmetric tensor with
the following form at every point in R4 in the coordinate basis.
−1 0 0 0
0 1 0 00 0 1 00 0 0 1
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✦ All the issues discussed before are easily described in this framework:
➢ At each point in R4 tangent vectors v are classified as:
✧ time-like if g(v, v) < 0.✧ null if g(v, v) = 0.✧ space-like if g(v, v) > 0.
➢ Physical observers are described by curves parametrized by proper
time (i. e. with time-like tangent vectors T satisfying g(T, T) = −1).➢ Light rays are null geodesics (geodesics with null tangent vectors).
➢ Inertial observers are defined by time-like geodesics.
➢ The Minkowski metric has certain symmetries described by Killing
fields . It is always possible to choose four of these that commute
and allow the construction of coordinate systems in which the me-
tric takes the form written above. These are the inertial reference
systems .
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➢ Transformations that preserve the form of the metric correspond to
the choice of a different set of commuting Killing fields to build the
inertial frame; they are the Lorentz transformations .
➢ In each of these it is possible to have inertial observers “at rest” (i.e.
with constant spatial coordinates).
➢ Given a space-like parametrized curve γ(s) defined by events thatare simultaneous w.r.t. a given inertial observer its length is∫
ds√
g(γ̇, γ̇)
✦ Given a time-like parametrized curve γ(s) describing an observerthat carries a clock the proper time that it measures is given by∫
ds√|g(γ̇, γ̇)|
Let us generalize and introduce a general (−+ ++) signature metric.
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What kind of physical phenomenon can be described by such a model?
RELATIVISTIC GRAVITYMATHEMATICAL MODEL FOR AN EINSTENIAN SPACE-TIME
(M, g)
✦M: four dimensional differentiable manifold.✦ g: metric on M with (−+ ++) signature.✦ We say that (M, g) and (M′, g′) are isometric if there exists θ : M→M′ diffeomorphism from one to the other such that g′ = θ ∗ g.
✦ This defines an equivalence relation. We take equivalence classes as
space-time models.
✦ In this mathematical setting the different frameworks (Aristotelian, Ga-
lilean, Einstenian) can be described by introducing different mathema-
tical structures.
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NOTATION AND CONVENTIONS
✦ Tensors on a vector space V on a field K (usually R or C) of type (k, l)are multilinear maps
T : V∗× · · · ×V∗︸ ︷︷ ︸k
×l︷ ︸︸ ︷
V × · · · ×V → K
where V∗ is the dual vector space of V. I will only consider finite di-mensional vector spaces.
✦ Tensor fields are defined at each point P of a differentiable manifoldM by using the tangent and cotangent vector spaces ( TP and T∗P).
✦ By taking a basis of V and its dual basis on V∗ one can define compo-nents of a tensor. In physics one often works with these components.
✦ One can define the usual operations for tensors: contraction exterior
products, defining other tensors by “filling in the slots with vectors”,
and so on...
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In the following I will use the abstract index notation of Penrose:
✦ A tensor of type (k, l) is denoted as Ta1...akb1...bl.
✧ A contraction is denoted by repeating indices. For example Tabcbe (nosumming of repeated indices meant [!]).
✧ Outer product, for example TabcdeSfg.
✧ Symmetries: for example if Tabvawb = Tabwavb, ∀va, wa ∈ V then wesay that Tab is symmetric and denote it as Tab = Tba.
✦ The metric tensor is gab [non-degenerate symmetric (0, 2) tensor].
✦ The inverse metric gab is defined to satisfy gabgbc = δac.
✦ I will “raise and lower indices” with the metric (that is, I will use the
vector space isomorphism g : V → V∗ : va 7→ gabvb) whenever I feellike to.
35
General RelativityRELATIVISTIC FREE FALL Let us consider the Schwarzschild metric(G = 1, i.e. mass measured in meters).
dτ2 = −(
1− 2Mr
)dt2 +
(1− 2M
r
)−1dr2 + r2(dθ2 + sin2 θdφ2)
Two hypotheses:
❶ Test particles move in geodesics.
❷ Light rays move in null geodesics (geodesics with null tangent vec-
tors).
Test particles are in the simplest state of motion; they are the best candi-
dates to become what we can call inertial observers in general relativity.
36
RADIAL GEODESICS
3 4 5 6 7 8 9
2.5
5
7.5
10
12.5
15
r
t
r0
radialnullgeodesic
radialtimelikegeodesics
M = 1c = 1
We see radial geodesics that go “up” and then “down” (in r and t) andothers that go “up forever". We see also how null geodesics look like.
37
Why do we see things falling when we stand on the ground?
Does the ground really correspond to r = constant?
ISOMETRIES OF THE METRIC
✦ Give a metric gab on a manifold M an isometry is a diffeomorphismφ : M→M such that (φ ∗ g)ab = gab.
✦ These isometries are the symmetries of the metric, and by extension
of space-time.
✦ If we have a one parameter group of diffeomorphisms φt we can ge-
nerate it by means of a vector field and we can use φt to carry along
any smooth tensor field Ta···b··· . By comparing Ta···b··· with φ−tT
a···b··· we can
define the Lie derivative of this tensor field.
✦ Finding the symmetries of the metric gab boils down to finding the socalled Killing vector fields satisfying the Killing equation ∇(aξb) = 0(where ∇a is the derivative operator associated with the metric gab).
38
✦ In the case of the Schwarzschild metric introduced above it is straight-
forward to show that the time-like vector field ∂t is a Killing vector field
or equivalently that (t, r, θ, φ) 7→ (t + T, r, θ, φ) is an isometry for everyvalue of T.
✦ Metrics for which such time-like (and hypersurface orthogonal) Killing
vector fields exist are known as static ➢ Schwarzschild is static.
✦ Any observer whose world line is an integral curve of the vector field
∂t will perceive a static space-time in the sense that any experiment
performed by him to explore its properties will yield the same result if
repeated at different instants of time (if the same setup is used).
✦ These are not inertial as inertial observers should be in free fall . In this
respect everyday gravity is a fictitious force very much as the centri-
petal or Coriolis forces are.
39
Let us consider observers with constant r = r0 (and forget about θ and φin the radial case). In a free fall experiment such an observer would stay
on the ground.
2.25 2.5 2.75 3 3.25 3.5 3.75 4
2.5
5
7.5
10
12.5
15
r
t
ξa = ∂t
M = 1
c = 1
40
WARNING What is the physical meaning of r and t? Aren’t they just coor-dinates? Are we entitled to assign a physical meaning to them such as
vertical distance to the floor or time in flight?
2.25 2.5 2.75 3 3.25 3.5 3.75 4
2.5
5
7.5
10
12.5
15
r
t
∝T(r)
∝ y(r)
PQ
(r, t)
ξa = ∂t
M = 1c = 1
r0
✦ We can do better: let us follow the
space-time philosophy of the first
talk on special relativity
✦ Let us pick an event P on the world-line of the falling object.
✦ Let us trace back to the floor the
two light rays (i.e. null geodesics)
that arrive and start at P. In this ca-se there is a discrete time symmetry
that implies that for an observer with
constant r all the events with the sa-me t are simultaneous.
41
EQUATIONS FOR THE TIME-LIKE RADIAL GEODESICS
In the following the dot denotes derivative w.r.t. the affine parameter.
ṫ =E
1− 2Mr, ṙ =
[E2−
(1− 2M
r
)]1/2
where E is a certain real parameter to be interpreted below.
EQUATIONS FOR THE NULL RADIAL GEODESICS
ṫ =1
1− 2Mr, ṙ = ±1
where the + is for outgoing null geodesics and − for the ingoing ones.
42
✦ From them we get
dtdr
=E(
1− 2Mr) [
E2−(
1− 2Mr)]1/2 time− like
dtdr
=±1(
1− 2Mr) null
with solutions given by
ttime(r) = t0 + E∫ r
r0
dρ(1− 2Mρ
) [E2−
(1− 2Mρ
)]1/2 time− liketnull(r) = t0 + r− r0 + 2M log
r− 2Mr0− 2M
null, outgoing
43
✦ The proper time elapsed at the ground from launch at r0 to the event Q
is given by T(r) =(
1− 2Mr0)1/2
ttime(r) (as can be easily read from themetric).
✦ The distance to the floor is proportional to the proper time elapsed at
the floor from emission to reception of the light ray reaching the free
falling object at P.
y(r) =(
1− 2Mr0
)1/2 [r− r0 + 2M log
r− 2Mr0− 2M
]
✦ It is straightforward to get
v(r) ≡ dydT
=1E(E2− 1 + 2M
r)1/2
a(r) ≡ d2y
dT2= −
M(
1− 2Mr)
E2r2(
1− 2Mr0)1/2
44
✦ We see that E = 1√1−v20
(1− 2Mr0
)1/2[v0 ≡ v(r0)].
✦ We have E ' 1− Mr0 +12v
20 for r0 >> 2M and v0
45
GRAVITATIONAL REDSHIFT
2.75 3 3.25 3.5 3.75 4 4.25 4.5
1
2
3
4
5
6
r
t
∆τ = 1, ν1
∆τ = 1, ν2
R2
ν2 < ν1
R1
Le us consider two observers at two
different heights R1 and R2 > R1 (radialcase) and suppose that R1 emits radia-tion at a certain frequency ν1.
What is the frequency observedby R2?
At R1 the proper time between theemission of two pulses is proportional
to (1− 2M/R1)1/2 whereas the propertime at R2 (reception) is proportional to(1− 2M/R2)1/2. We hence find
ν2ν1
=(1− 2M/R1)1/2(1− 2M/R2)1/2
46
This has been measured:
➢ Pound and Rebka (1960) Using Mössbauer effect.
➢ Vessot and Levine (1979, GPA) using hydrogen masers on a sounding
rocket (0.01 %).
➢ Hafele and Keating (1972) carrying atomic clocks in airplanes.
➢ Every day at GPS.
NON RADIAL GEODESICSNon radial geodesics display a rich variety of behaviors:
✦ They describe Keplerian orbits, at least for values of r much larger thana certain characteristic length (the Schwarzschild radius)
✦ They also describe corrections to these orbits classical tests of ge-neral relativity (light deflection by compact masses, rotation of perihe-
lia, Shapiro time delay,...).
47
✦ The Schwarzschild metric describes the space-time metric in vacuo for
a spherically symmetric situation (i.e. outside the earth if one forgets
about its rotation).
✦ The matching metric inside (say for r < r0 where matter is present)is different; its detailed form depends on the properties of the matter
distribution.
✦ Most of our observational evidence supporting General Relativity rela-
tes to the Schwarzschild solution.
✦ Something weird happens at R = 2M (Schwarzschild radius ) with theexterior Schwarzschild metric.
Can we extend it for r < 2M?What happens at r = 0?
48
THE KRUSKAL EXTENSION
AN EXAMPLE
X
T
X > 0X
2> T
2
Consider the metric [ t ∈ R, x ∈ (0, ∞)]
dτ2 = −x2dt2 + dx2
This metric seems to be singular at x = 0because det gab = 0 there. Let us however“change coordinates” according to
t =12
log(
T + XT − X
), x =
√X2− T2
where now X > 0 and X2 > T2. In thesecoordinates it becomes
dτ2 = −dT2 + dX2
which is just Minkowski defined on a sub-
manifold of R4! (Rindler space-time) .
49
✦ We see then that the initial metric describes just a piece of Minkows-
kian space-time and suggests that:
✧ The singularity of the initial metric can be considered as an artifact
introduced by a “bad choice of coordinates”.
✧ By choosing appropriate coordinates the metric can be extended to a
larger manifold (in such a way that this extension can be considered
complete in a certain sense).
✦ There is a way to do something similar with the Schwarzschild solution
to construct the Kruskal extension.
50
Consider the coordinate change defined by
( r2M − 1
)er/(2M) = X2− T2
t = 2M log(
T+XT−X
)where now X and T are constrained to satisfy X2− T2 > −1.
In these coordinates (Kruskal-Szekeres) the Schwarzschild metric takes
the form
dτ2 =32M3e−r/(2M)
r(−dT2 + dX2) + r2(dθ2 + sin2 θdφ2)
51
A space-time diagram representing the Kruskal extension of the Schwars-
child metric (each point represents a full spherical surface).
II
IIII
IIIIII
IVIV
XX
TT
singularity (r = 0)singularity (r = 0)
singularity (r = 0)singularity (r = 0)
t const.t const.
r const.r const.
r = 2Mr = 2M
r = 2Mr = 2M
ξa = ∂t
52
COMMENTS
✦ The original Schwarzschild metric describes only the region labeled
as I in the diagram (and represents the exterior gravitational field of a
spherical body).
✦ Nothing happens at the boundary of the regions I and II as far as the
regularity of the metric is concerned.
✦ If one looks at the light cones in regions I and II one sees that time-like
curves in I can be extended to arbitrary values of r with arbitrary largeproper time whereas those in II hit the boundary of the diagram in a
finite proper time . The boundary between these two regions is called
a horizon and separates the exterior region from the black hole repre-
sented by region II.
✦ The boundary of this space-time represents a genuine singularity . It
is a very dangerous place because tidal forces in its vicinity become
arbitrarily large.
53
✦ The extension obtained is maximal in a precise mathematical sense.
There is no way to go beyond this singularity.
✦ One can study the symmetries of the Kruskal extension. The Killing
field that coincides with ∂t in region I has the curious feature of beco-
ming space-like in region II and zero at X = 0, T = 0.
✦ There is no way of having a “ground” inside region II. Hence, there is a
minimum size for a static object “supporting” a spherically symmetric
external geometry.
✦ What about regions III, IV and the other singularity?
➢ Region III is in a sense the opposite as the black hole II so it is called
a white hole .
➢ Region IV is similar in its properties to region I but it is physically
isolated from it in the sense that there is no way to send or receive
signals from there (at speeds smaller than the speed of light).
➢ There is no realistic astrophysical situation that could give rise to the
54
space-time represented by the full Kruskal extension. In the collapse
of a physical object infalling matter completely “covers” regions III
and IV.
➢ Notice, however, that horizon representing an astrophysical black
hole appears even in this case.
SPACE-TIME SINGULARITIES
✦ Physically they show their presence as the impossibility to arbitrarily
extend time-like and null geodesics in their affine parameters (that is if
there exist so called incomplete time-like and space-like geodesics).
✦ This is the property that is proved in the important singularity theorems
(of Hawking and Penrose) that show that singularities such as the big
bang or black holes are generic physical features and not artifacts of
the usual metrics that display them.
55
THE EINSTEIN FIELD EQUATIONS
✦ We know from Newtonian gravity that the gravitational field is created
by the distribution of masses in the universe.
✦ In a relativistic setting we would expect that it is the distribution of
matter and energy that creates gravity, i.e. determines the space-time
metric.
How does this come about?
Let us go back to special relativity and consider something that we com-
pletely left aside in the first part:
56
RELATIVISTIC DYNAMICS
Dynamics can be introduced by using action principles.
EXAMPLE: The relativistic particle.
Consider a particle described by a certain worldline that we have to de-
termine dynamically.
✦ Let us fix two space-time events A and B and a sufficiently smoothcurve γ connecting them.
✦ The corresponding action is the proper time measured along γ from
A to B multiplied by −mc2 (where m is an attribute of every physicalparticle known as its rest mass).
✦ If we choose inertial coordinates (t, x) this is
S = −mc2∫ tB
tA
√1− ẋ
2
c2
57
✦ We can use the Hamiltonian formalism to study the resulting dynamics.
We obtain the following
➢ Canonical momenta are given by
p =mẋ√1− ẋ2c2
.
➢ The Hamilton equations imply that they are constant in t.
➢ The conserved energy is
E =√
m2c4 + p2c2.
➢ (Ec , p) can be considered as the components of a (four-)vector pa =
mcua proportional to the so called four-velocity
ua ≡ dxa
dτ
that satisfies uaua = −1.
58
➢ Another observer at an event (with four- velocity va) where the parti-cle is present would measure an energy given by −vapa.
If we have a swarm of non-interacting particles (dust) the action of the
system is given by the sum of the actions. We find that we can define a
total energy-momentum vector for the system as the sum of the individual
contributions of each particle.
STRESS-ENERGY-MOMENTUM TENSOR
For continuous matter distributions we define the so called stress-energy-
momentum tensor Tab. Suppose that we have an observer with four-velocitygiven by va and choose three mutually orthogonal “spatial” vectors xa1,xa2, x
a3 satisfying vax
a(i) = 0. The components of Tab have the following
meaning:
➢ Tabvavb is the energy density per unit proper volume.
➢−Tabvaxb is the momentum density of matter in the spatial direction
59
defined by xb.
➢ Tabxai xbj (i 6= j) is the ij component of the stress tensor.
For normal (ordinary, physically observed) matter there are some restric-
tions on Tab
ENERGY CONDITIONS
✦ Weak energy condition: For normal matter the energy density measu-
red by an observer with four-velocity va must satisfy Tabvavb ≥ 0.
✦ Dominant Energy Condition: If va is the four velocity of an observerwe have that Tabvavb ≥ 0 and Tabva is not space-like. Physically thismeans that the pressure does not exceed the energy density (and hen-
ce the velocity of sound is less than the speed of light).
✦ Strong Energy Condition: Tabvavb ≥ −12T. It is satisfied if no large ne-gative pressures exist. It holds for the EM field and massless scalars.
60
The most important and common types of matter distributions used in
general relativity are
❶ Perfect fluids.
❷ Electromagnetic Fields.
❸ Scalar fields.
❶ PERFECT FLUIDS
Consider a Minkowskian space-time (with metric given by gab). A per-fect fluid is a matter distribution with
Tab = ρuaub + P(gab + uaub)
where ρ, P, and ua are the mass-energy density, the pressure and thefour velocity in the rest frame of (each sufficiently small volume of) the
fluid.
61
The dynamics of a perfect fluid subject to no external forces is given
by
∇aTab = 0COMMENTS
✦ In this case this condition leads to the familiar continuity and Euler
equations for fluids (in a certain inertial coordinate system and in the
non relativistic limit in which the fluid velocity is much smaller than
the speed of light).
∂ρ
∂t+∇ · (ρv) = 0
ρ
[∂v∂t
+ (v · ∇)v]
+∇P = 0
62
✦ It implies energy-momentum conservation.
➢ Let us take a family of observers such that their four velocities
satisfy ∇avb = 0 (they are “parallel”).
➢ Let us define the vector field Ja = −Tabva then
∇a Ja = −∇a(Tabvb) = (∇aTab)vb + Tab(∇avb) = 0
and energy momentum conservation follows immediately.
➢ Conversely energy momentum conservation for all inertial obser-
vers requires that ∇aTab = 0.✦ There are prescriptions to obtain the conserved stress-energy-mo-
mentum tensor for the usual field theories such as the scalar, elec-
tromagnetic or Yang-Mills fields:
63
❷ ELECTROMAGNETIC
Tab =1
4π
[FacF cb −
14
gabFdeFde]
where Fab = ∂aAb − ∂bAa.
❸ SCALAR
Tab = ∂aφ∂bφ−12
gab(∂cφ∂cφ + m2φ2)
✦ If the metric is not Minkowski then many definitions can be easily
adapted but some of the previous statements must be modified.
➢ Particle motions are described by time-like curves.
➢ Matter is described by similar energy momentum tensors where, in
most cases, one simply substitutes partial derivatives for covariant
64
derivatives. They must satisfy ∇aTab = 0
➢ There are prescriptions to build suitable Tab for virtually any typeof matter that we want (more on this at the end).
➢ Perfect fluids continue to be represented in terms of ua, ρ, and P.
➢ Electromagnetic fields are represented by a 2-form field Fab.
LOCAL CONSERVATION OF Tab
✦ It may be impossible to find a family of observers for which ∇(avb) = 0with (vava = −1) in which case we cannot generalize the situation inMinkowski.
✦ If such a family of observers exists (the so called stationary case) then
Ja = −Tabvb is a conserved current and we have energy momentumconservation.
65
✦ More generally the symmetries of a metric (described by Killing fields)
allow us to define conserved quantities.
✦ In general we have approximate conservation in space-time regions
small compared to the curvature radius.
THE GEODESIC DEVIATION EQUATION
✦ γs(t) smooth 1-parameter family of geodesics such that ∀s ∈ R γs isaffinely parametrized by t.
✦ Let us suppose that the map f : (t, s) 7→ γs(t) is sufficiently smooth,one to one, and with smooth inverse.
✦ In these conditions
✦ We can define a two dimensional submanifold in space-time Σ span-ned by the points in the geodesics γs(t) with coordinates given by(t, s).
66
✦ The vector field defined on Σ by Ta =(
∂∂t
)ais tangent to each geo-
desic of the family and satisfies Ta∂aTb = 0.
✦ A vector Xa =(
∂∂s
)ameasuring the deviation between nearby geo-
desics can be defined.
✦ The freedom to change affine parameters in each geodesic accor-
ding to t 7→ b(s) + c(s)t can be used to get XaTa = 0 everywhere onΣ.
✦ We can define the rate of change of the displacement to a nearby
geodesic as va = Ta∇bXb and the relative acceleration betweennearby geodesics as aa = Ta∇bvb.
✦ The geodesic deviation equation
aa = −R acbd XbTcTd
67
THE EINSTEIN EQUATIONS
✦ Inspired by the Mach principle that suggests that the structure of space-
time is influenced by the distribution of matter in the universe Einstein
looked for a set of equations in which the space-time geometry is de-
termined by the distribution of matter and energy .
✦ If va is the 4- velocity and xa is the orthogonal deviation vector the tidalacceleration of two nearby particles is −R acbd x
bvcvd.
✦ In Newtonian gravity the tidal acceleration between particles separated
by a vector ~x is −(~x · ~∇)~∇φ.
✦ This suggests the correspondence R acbd vcvd ↔ ∂b∂aφ.
✦ The Newtonian Poisson equation and the fact that the energy density
is given by Tabvavb leads us to take ∂a∂aφ ↔ 4πTabvavb.
✦ So everything together for any observer Rcd = 4πTcd.
68
✦ This equation has a serious drawback originating in the Bianchi iden-
tity
∇cRcd = 12∇dR
that would imply that the trace of Tab is constant throughout space-time(unacceptable!).
✦ However as the combination known as the Einstein curvature
Gab = Rab − 12gabR
satisfies ∇aGab = 0 we find the Einstein equations
Gab = 8πTab
69
COMMENTS
✦ In a fixed coordinate system these are a system of coupled, nonlinear,
second order partial differential equations that are hyperbolic if the
metric has Lorentzian signature.
✦ For usual choices of matter fields Tab itself depends on the metric.
✦ Once the equations are solved for a certain type of matter fields the
dynamics of the matter is completely fixed by the local conservation
condition ∇aTab = 0, in particular:
➢ This is true for perfect fluids; also in the case of zero pressure (dust)
this condition implies that every particle moves along a geodesic.
➢ For sufficiently small bodies with “weak enough” self gravity the
condition ∇aTab = 0 implies that they move along geodesics. Thisis no longer a hypothesis but is a consequence of the Einstein equa-
tions . A very non trivial consistency condition is satisfied .
➢ For large enough bodies there are deviations from the geodesic mo-
70
tion (described by more complicated equations such as the Papape-
trou equation).
✦ The Einstein field equations can be derived from an action principle
by using the so-called Einstein-Hilbert action with a Lagrangian metric
proportional to the scalar curvature. If a matter Lagrangian is included
in a suitable way one automatically obtains a locally conserved energy-
momentum tensor.
✦ One can “solve” the Einstein equations by computing Gab for any me-tric and defining Tab as the resulting expression. This would lead ge-nerically to very unphysical matter (violating energy conditions or su-
ch that no known physical interaction can produce such an energy-
momentum distribution). BEWARE of exotic solutions.
✦ They are very difficult to solve, even numerically ...
71
WHAT IS NEXT?
➢ Newtonian limit.
➢ Gravitational waves and radiation.
➢ Cosmological models.
➢ Homogeneous and isotropic models.
➢ How to solve the Einstein equations?
➢ Algebraically special solutions.
➢ Perturbation theory.
➢ Causal structure.
➢ Well-posedness of the Einstein equations.
➢ Singularities and singularity theorems.
➢ Initial value formulation.
➢ Asymptotics and asymptotic flatness.
72
➢ Gravitational energy.
➢ Black holes and thermodynamics.
➢ Hamiltonian formulation.
➢ Numerical solution of the Einstein equations.
...(SOME) BIBLIOGRAPHY
✦ Geroch, R. P. General Relativity: from A to B , University of Chicago
Press (1978).
✦ Misner, C. W., Thorne, K. S., and Wheeler, J. A. Gravitation , San Fran-
cisco, Freeman (1973).
✦ Wald, R. M. General Relativity , University of Chicago Press (1984).
