General Relativity Phys4501/6
Second term summary, 2013 Gabor Kunstatter, University of Winnipeg
Contents
I. Einstein’s Equations 3 A. Heuristic Derivation 3 B. Lagrangian Formulation of General Relativity 5 C. Variational Derivation of Einstein’s Equations 6 D. Properties of Einstein’s Equations 7
II. Energy Conditions 7
III. Spherical Symmetry 9 A. Derivation of solution and general properties 9 B. Geodesics 9 C. Experimental Tests 9 D. Black Holes and Conformal Diagrams 10 E. Charged Black Holes 10
IV. Horizons and Trapped Surfaces 12 A. Stationary vs. Static 12 B. Event Horizons in General 13 C. Trapped Surfaces 14 D. Killing Horizons 15 E. Surface Gravity 16
V. Theorems and Conjectures 17 A. Singularity Theorem 17 B. Cosmic Censorship Hypothesis 18 C. Hawking’s Area Theorem 18 D. No Hair Theorem 18
1 VI. Black Hole Thermodynamics 18
VII. Cosmology: Theory 20 A. Symmetry Considerations 20 B. FRW Metric 21 C. Hubble Parameter 22 D. Matter Content 22 E. The Friedmann Equation 23 F. Time evolution of the scale factor and density 25
VIII. Cosmology: Observations 26 A. Redshifts 26 B. Proper Distance 28 C. Luminosity Distance 29 D. Derivation of Luminosity vs Redshift Relation 29
2 I. EINSTEIN’S EQUATIONS
A. Heuristic Derivation
• So far we have learned how to derive equations of motion for matter fields from a variational principle keeping the geometry fixed (i.e. didn’t vary metric).
• We now need to understand better the equations of motion for the metric of spacetime, i.e. the geometry.
• We know that the energy momentum tensor must be conserved, and that in the New-
tonian limit h00 plays the role of the gravitational potential φ which in Newtonian physics is determined by the matter distribution by Poisson equations:
∇~ 2φ = 4πGρ (1)
The equations of motion must therefore be second order in the metric components, tensorial, and guarantee energy-momentum conservation.
• Eq. (138) of the February summary gives us a hint, since we can get the vacuum ~ 2 k Poisson equation ∇ φ = 0 by taking the trace R 00k = 0. This is the Newtonian, non-
relativistic limit of R00 = 0, so one would be tempted to postulate, as did Einstein, the fully covariant version of the above in the presence of matter:
Rµν = κTµν (2)
But we know this can’t be right, because the energy momentum tensor must be covari- µν 1 µν antly conserved, whereas the LHS is not: ∇µR = 2 g ∇µR by the Bianchi identities and does not vanish in general. In fact, (2) implies R = κT so that by using the µν Bianchi identities one sees that ∇µT = 0 iff ∇µT = 0, which is way too restrictive.
• A better guess for Einstein’s equations, which automatically yields energy-momentum conservation, is: 1 G = R − g R = κT (3) µν µν 2 µν µν
• In class we used dimensional arguments to show that κ must be proportional to G/c4: 3 2 T00 has dimensions of [ρ] = E/L , whereas G00 has dimensions 1/L . Thus [κ] = L/E.
3 Newton’s constant has units [G] = L2/mass2 ∗ (mass ∗ L/T 2) = L3/(Mass ∗ T 2). Since [E] = [mass][c2], this shows that [G/c4] = L3/(Mc2 ∗ T 2) ∗ (T 2/L2) = L/E as required.
• We can check more carefully the Newtonian limit and thereby determine whether it works and if so what the constant κ must be:
– First, using Eq.(138) in the previous section we see that in the Newtonian limit: 1 1 R ∼ − ∇~ 2h = ∇~ 2φ (4) 00 2 00 c2 – Tracing (3) we find 00 R = −κT ∼ −κg T00 = κT00 (5)
where we have assumed that in the non-relativistic limit the T00 = ρ is larger than the other components of the stress energy tensor.
– In this case Einstein’s equation implies that: 1 1 R = κT (1 + g ) = κT 00 00 2 00 2 00 1 1 → ∇~ 2φ = κc2ρ (6) c2 2
2 where we have used T00 = ρc , recalling that the energy-momentum tensor is the flow of four-momentum through a surface, i.e. in this case energy, not mass.
– which matches Eq.(124) of February summary providing we choose: 8πG κ = (7) c4 • Once we impose general covariance, don’t allow higher derivatives of the metric in the equations of motion and require correspondence with Newtonian physics the theory is uniquely determined with no new, tunable parameters!!! As we will see it therefore makes unambiguous testable predictions for corrections to Newtonian physics, such as the perihelion shift of Mercury and light bending around the sun.
• But first we will see how to derive the field equations from a variational principle. Given a coordinate invariant action, the energy momentum tensor can be derived (doesn’t have to be guessed), and the connection between the energy momentum conservation and symmetry (general covariance) becomes manifest.
4 B. Lagrangian Formulation of General Relativity
We now show how to derive Einstein’s equations from a variational principle. As always one starts with an action. In this case we require the action to have certain properties:
1. It must be invariant under coordinate transformations
2. It must, at this stage be at most second order in derivatives of the metric. Actually the condition is that the equations of motion resulting from varying this action have no more than second derivatives of the metric.In four spacetime dimensions these two statements are equivalent, but not in higher dimensions. Note also that higher order derivatives are possible in generalized theories of gravity, but we assume that they will not play a significant role until we get to much shorter length scales than we wish to consider at this stage.
3. It should separate into two parts:
(a) a matter action, which describes the dynamics of any non-gravitational fields. This part is distinguished by the fact that it cannot have more than first deriva- tives of the metric. Otherwise the equations of motion for the matter field could not be made to be indistinguishable from those of general relativity by going to a locally inertial frame (in which the first derivatives of the metric vanish. This would violate the strong equivalence principle. In general the matter action is constructed by taking the special relativity version and replacing all occurences of
the Minkowski metric by gµν and all partial derivatives by covariant derivatives.
(b) the gravitational action: assuming the previous conditions this is almost uniquely determined (in four dimensions) proportional the so-called Einstein-Hilbert ac- tion: Z 4 √ SEH = d x −gR(g) (8)
where R(g) is the Ricci scalar constructed from the metric gµν
Thus we start from the total action describing both geometry and matter:
1 I = S + S (9) 2κ EH M
5 where SM denotes a generic matter action constructed as per the rules given above.
For example, for a scalar field, the contribution to SM is: Z √ 1 1 S = d4x −g − ∇ φgµν∇ φ − m2φ2 − V (φ) (10) φ 2 µ ν 2 Note that for a scalar field the partial and covariant derivatives are identical. V (φ) is a potential energy term associated with self-interactions of the field. The signs are chosen to make sure that the kinetic energy term is positive, in keeping with the mechanics form of the lagrangian: L = KE − PE.
The other matter action we have dealt with, that of the electromagnetic field, is: 1 Z √ S = − d4x −gF gµνgαβF (11) EM 4 µα νβ
where Fµν is the electromagnetic field strength defined in terms of an electromagnetic
potential Aµ by:
Fµν = ∂µAν − ∂νAµ = ∇µAν − ∇νAµ (12)
Note that Fµν is by definition anti-symmetric (a two-form) and thus the Christoffel terms on the far right cancel so that it can be equivalently defined either by covariant or partial derivatives.
C. Variational Derivation of Einstein’s Equations
• Definition of stress energy tensor: 2 δS T ≡ −√ M (13) µν −g δgµν
• Action: c4 Z √ S = d4x −gR(g) + S (14) 16πG M • Einstein’s Equations: 8πG G = T (15) µν c4 µν • Know how to vary metric in action, for which the following relations are useful:
µα µα – g δgµβ = −gµβδg
√ 1 √ µν 1 √ µν – δ −g = 2 −gg δgµν = − 2 −ggµνδg
6 α 1 αβ – δ{µ ν} = 2 g (∇µδgβν + ∇νδgβµ − ∇βδgµν)
R 4 p µ R p µ R p µ – Σ d x |g|∇µV = Σ ∂µ |g|V = ∂Σ nµ |γ|V
D. Properties of Einstein’s Equations
• Degrees of Freedom: qualitative derivation of n(n + 1)/2 − 2n = n(n − 3)/2
• Self-coupled: gravitational binding energy contributions to inertial mass
• Attractive for “ordinary matter”
• Quantum Gravity: know constants that can be constructed from ~, G, C: r ~G c lpl = 3 = (16) c tpl
~c 2 Epl = = mplc (17) lpl
2 • Cosmological Constant Λ = 1/lΛ: know the problem, its theoretical quantum source and theoretical and experimental orders of magnitudes.
II. ENERGY CONDITIONS
These are physically motivated constraints on the general form of the stress energy tensor that are useful for proving various theorems and/or motivating conjectures. They are all a bit different, but the basic idea is to define what one considers “reasonable” matter/energy to look like in general relativity. Any or all of these can be violated once quantum mechanics is taken into account.
• Strong Energy Condition (SEC):
1 T tµtν ≥ T λ tσt (18) µν 2 λ σ
for all timelike tµ. This implies that dθ/dτ ≤ 0 for all timelike geodesics, where θ is their expansion. Thus it implies that strong energy condition restricts to mat- ter/energy for which gravity is purely attractive.
7 If we restrict consideration to a perfect, isotropic fluid:
T µν = (ρ + p)U µU ν + pgµν (19)
where U µ is tangent to the fluid world-lines, then the SEC implies that:
ρ + p ≥ 0 and ρ + 3p ≥ 0 (20)
Know how to prove the above!
• Weak Energy Condition (WEC):
µ ν Tµνt t ≥ 0 (21)
for all timelike tµ. For an isotropic perfect fluid this implies that
ρ ≥ 0 and ρ + p ≥ 0 (22)
ν If you think of Pµ = Tµνt as the four momentum observed by the observed whose four µ µ velocity is t , then this condition states that the energy density Pµt is non-negative.
• Dominant Energy Condition (DEC) :
µ ν µν α Tµνt t ≥ 0 and(T tν)(Tµαt ) ≤ 0 (23)
This requires WEC (non-negative energy density as seen by all timelike observers) and also that the four moment by non-spacelike. For an isotropic perfect fluid:
ρ ≥ |p| (24)
• Null Energy Condition (NEC): µ ν Tµνl l ≥ 0 (25)
for all null lµ. For isotropic perfect fluid:
ρ + p ≥ 0 (26)
8 III. SPHERICAL SYMMETRY
A. Derivation of solution and general properties
• Qualitative description of derivation:
– General form of spherically symmetric, static metric:
ds2 = −e2α(r,t)dt2 + eβr,tdr2 + r2dΩ(2) (27)
where dΩ(2) = dθ2 + sin2(θ)dφ2... (28)
– Spherical symmetry implies that there are only three independent Einstein equa- tions, one of which is redundant because of Bianchi identities.
– Birkhoff theorem: if you make the two functions depend on time, the equations tell you that the time dependence vanishes
• Singularities: two definitions
– A curvature invariant is singular
– Geodesic incompleteness
B. Geodesics
• Constants of motion
• Restrict to equatorial plane (how and why)
• Effective potential; derivation, interpretation
• Examples: for specific initial conditions calculate coordinate time and proper time elapsed, etc. (see assignments)
C. Experimental Tests
• Red shift
9 • Perihelion shift
• Light bending
D. Black Holes and Conformal Diagrams
• Know how to do coordinate transformations on metric: (t, r) → (t, r∗) → [(v, r∗)or(u, r∗)] → (u, v) → (u0, v0) → (T,R)
• Diagrams: know how to label and extract physical implications of all the properties in the following two diagrams***
• Wormholes: know how they arise as constant Kruskal slices in the above diagrams.
E. Charged Black Holes
• Assume that both the electromagnetic field and metric are spherically symmetric.
• There exists a Birkhoff theorem for gravity plus electromagnetism: all spherically symmetric solutions are also static. We will not prove Birkhoff’s theorem but assume for simplicity that the solution is static.
• Given the above, the metric and field strength must take the form (in adapted coor- dinates):
ds2 = −e2α(r)dt2 + e2β(r)dr2 + r2dΩ(2) (29) 0 −f(r) 0 0 f(r) 0 0 0 Fµν = (30) 0 0 0 g(r) 0 0 −g(r) 0 √ where Frt = f(r) = 0E(r), with E(r) being the radial component of the electro-
magnetic field strength. Fθφ = −g is proportional to the radial component of the magnetic field strength, which we set henceforth to zero.
• Action: c4 Z √ 1 Z √ I = d4x −gR(g) − d4x −gF µνF (31) 8πG 4 µν
10 where the field strength Fµν can be written as the curl of a vector potential:
Fµν = ∂µAν − ∂νAµ (32)
R 4 µν Note that d xF Fµν has units of action, i.e. Joule-seconds, as required.
• Equations of motion which you should know how to derive from the above action
11 via a variational principle, are: 8πG G = T (33) µν c4 µν 1 T = F F α − g F αβF (34) µν µα ν 4 µν αβ 1 √ ∇ F µν ≡ √ ∂ ( −gF µν) = 0 (35) ν −g ν
• Charged particle motion is described by the covariant Lorentz force equation: D2xµ dxα dxµ e dxν ≡ ∇ = F µ (36) dτ 2 dτ α dτ m ν dτ
• Solution (know how to derive and understand units):
2 2 −1 2 2GM GQ 2 2 2GM GQ 2 ds = − 1 − 2 + 4 2 c dt + 1 − 2 + 4 2 dr c r 4π0c r c r 4π0c r +r2dΩ(2) (37) 1 Q E(r) = 2 (38) 4π0 r • Properties:
– There is curvature singularity at r = 0.
µ µ – It is static: there is a timelike Killing vector K = δ0 in Schwarzschild-type coordinates, properly normalized at infinity.
2GM GQ2 – Killing/event horizons at 1 − 2 + 4 2 = 0. There are 0,1 or 2 horizons c r 4π0c r 2 2 depending on whether GQ /(4π0M ) is more than one, equal to one or less than one, respectively.
– Know the respective conformal diagrams.
IV. HORIZONS AND TRAPPED SURFACES
A. Stationary vs. Static
• A stationary spacetime is one that has a timelike Killing vector. Essentially this means that if you choose your coordinates carefully, the metric doesn’t depend on time. Note that the Schwarzschild solution is only stationary outside the horizon. The Reissner-Nordstrom solution and the Kerr solution (rotating black hole) are also stationary.
12 • A static spacetime is one that is stationary and also time reversal invariant, i.e. doesn’t change if t → −t. In terms of the metric this requires that any terms linear in dt be removable by a suitable coordinate transformation. The Schwarzschild and Reissner-Nordstrom solutions are static (note that in Painleve-Gullstrand coordinates there is a dtdr term in the metric which can be removed by going to Schwarzschild coorinates).
• All static spacetimes are stationary but not vice versa.
• A rotating black hole, which is described by the axially symmetric Kerr metric, has one rotational Killing vector R = ∂/∂φ (in appropriate coordinates) in addition to the timelike Killing vector K = ∂/∂t. It is not static because it has an off-diagonal term in the line element proportional to dφdt that cannot be removed by a coordinate transformation. This term spoils invariance under time reversal, as one might expect from a solution with non-zero angular momentum. Reversing time will flip the direc- tion of the angular momentum vector. The event horizon is a Killing horizon, but for the Killing vector that is a linear combination of the timelike and rotational Killing vectors: µ µ µ χ = K + ΩH R (39)
for some constant ΩH that depends on the angular momentum.
B. Event Horizons in General
• Definition: An event horizon is a surface Σ that is the boundary separating spacetime points that are connected to an asymptotic region (r = infinity) by timelike or null geodesics from those that aren’t causally connected to infinity.
• Properties:
– This definition is global (teleological) in that one needs to know the entire future of the spacetime to know which light rays can ultimately escape to infinity and which ones remain trapped forever.
– Σ is a made up of geodesics (called generators of Σ) that correspond to all signals from the interior that don’t quite escape to infinity. These signals must be null:
13 Clearly spacelike is not admissable as a signal, whereas if any were timelike (i.e. the path of some massive particle) than one could in principle escape by going a bit faster. Thus Σ is a null surface, a surface whose tangent vectors are null.
• In a general time dependent setting with few symmetries the event horizon, if it exists, is hard to locate.
• If the spacetime is asymptotically flat and static or stationary then one can always choose a spatial coordinate r that corresonds to the areal radius at infinity (not nec- essarily elsewhere). As one moves inward from infinity along any spatial slice (i.e. at any fixed time) the event horizon at that time will be the point where the outward going light ray tips over. I.e. at the value of r for which an outgoing null ray moves
instantaneously along the surface r = constant. Thus it is the point at which ∂µr is a null vector: µν g ∂µr∂νr = 0 (40)
ν ν Since we have been using r as a coordinate, and ∂µx = δ µ in general, the event
horizon in such coordinates will be at rH such that
grr| = 0 (41) rH
C. Trapped Surfaces
• Definition: A compact spacelike 2 dimensional surface on which all future directed light rays converge.
• In static spacetimes the boundary of the trapping region (i.e. the region containing the trapped surfaces) is the event horizon. This is not true in general.
• Spherical collapse of matter to form a black hole Know the following conformal diagram: The boundary of the trapping region, called the trapping horizon, on each time slice is given by the location where the lines of constant r become null. Since the normal to these lines is also null this condition is, as above
µν g ∂µr∂νr = 0 (42)
14 FIG. 1: Thin blue lines are lines of constant areal radius r. Dotted blue line is boundary of trapped region, inside which lines of constant r are spacelike instead of timelike.
D. Killing Horizons
• Recall definition of Killing vector, Kµ:
LK gµν = 0 ⇔ ∇(µKν) ≡ ∇µKν + ∇νKµ = 0 (43)
• Definition: A Killing horizon is a null hypersurface Σ along which a Killing vector µ µ χ , say, is null: χ χµ = 0.
µ µ • Recall Schwarzschild has a timelike Killing vector ∂/∂t, or K = δ0 in the exterior (asymptotic) region. At the horizon the norm of the Killing vector vanishes:
µ K Kµ = g00 = −(1 − 2M/r) = 0 at r = 2M (44)
so this is a Killing horizon as well as an event horizon.
15 • In general Killing horizons and event horizons are distinct concepts, but for stationary, asymptotically flat spacetimes every event horizon is also a Killing horizon.
µ • Since a Killing horizon is a surface defined by the condition f := χ χµ = 0, the following covariant vector is orthogonal to the Killing horizon, Σ:
α fµ = ∂µf = ∂µ(χ χα) (45)
• We can show that the Killing vector itself χµ is tangent to Σ. Proof:
µ µ α µ α χ fµ = χ ∇µ(χαχ ) = 2χ χ ∇µχα (46)
µ α which is zero by the symmetry of χ χ and the Killing equation ∇(µχα).
µ µ • Curious but important fact: χµ (indices lowered) is normal to χ (since χ is null) µ and hence perpendicular to tangents to Σ. Hence, χµ is orthogonal to χ and parallel
to fµ, since the normal to Σ must be unique up to rescalings.
E. Surface Gravity
• Definition The surface gravity κ of a Killing horizon Σ with Killing vector χµ:
1 f = χα∇ χ = −κχ (47) 2 µ µ α µ
• Note that the above is of the form of non-affinely parameterized Killing equation. Thus χµ on the Killing horizon is tangent to null geodesics lying on Σ.
• The definition (47) implies that the surface gravity κ changes if one rescales the Killing vector by a constant. This does not change either the killing equation or the location of the Killing horizon. In order to define a unique value for the surface gravity, it is standard to normalize the Killing vector in asymptotically flat spacetimes so that it has unit norm as r → ∞.
• There is an alternative expression for the surface gravity, which yields precisely the same value but is easier to calculate:
1 κ2 = − (∇ χ )(∇µχν) (48) 2 µ ν
16 • For any asymptotically flat, static, spherically symmetric black hole with metric of the form: ds2 = −f(r)dt2 + f −1(r)dr2 + r2dΩ(2) (49)
µ µ the Killing vector χ = δ0 is normalized to be one at infinity (g → 1 as r → ∞) and yields a surface gravity: 1 κ = ∂ f(r) (50) 2 r
• An important property of surface gravity is that it is constant on the Killing horizon (even for non-spherically symmetric horizons).
• Physical Interpretation of Surface Gravity:
– in a static asymptotically flat spacetime κ is the gravitational acceleration ag at the horizon as measured by an observer at infinity. Alternately, it is the force per unit mass that an observer at infinity would have to exert on an infinitely light and strong string in order to hold an object stationary at the horizon.
– Proof: to come,
V. THEOREMS AND CONJECTURES
A. Singularity Theorem
• Proven in the late sixties by Hawking and Penrose.
• Heuristic statement: Once a trapped surface forms there is no way to keep light cones from collapsing completely to a singularity, unless the spacetime has closed timelike curves or exotic energy/matter content.
• Rigorous statement :
Let [M, gµν] be a manifold with a generic metric satisfying Einstein’s equations with matter that satisfies the strong energy condition. If there exists a trapped surface then either there exists closed timelike curves or a singularity, signalled by incomplete timelike or null geodesics.
17 B. Cosmic Censorship Hypothesis
Generic (i.e. not finely tuned) initial data in asymptotically flat sapcetimes will only form singularities inside event horizons as long as the Dominant Energy Condition holds.
C. Hawking’s Area Theorem
Assuming cosmic censorship and the Weak Energy Theorem, the area of a future horizon in asymptotically flat spacetimes can never decrease. The area theorem is important in the context of black hole thermodynamics.
D. No Hair Theorem
• We did not discuss this one in class, but it states: All (stationary) black hole solutions in Einstein’s theory are characterized by only three externally observable parameters: mass, charge and angular momentum.
• It has only been proven in certain contexts and counter examples exist for cases where other types of fields (eg massless scalar) are added into the mix.
VI. BLACK HOLE THERMODYNAMICS
• Black holes horizons are characterized by:
– A finite number of externally observable parameters M, Q, J (no hair theorem)
– a Killing horizon, located at rh, determined by the equation f(rh; M, Q, J) = 0,
where for Killing horizons f = −g00
2 – Horizon area: Ah = 4πrh
1 – Surface gravity: κ = 2 ∂rf(r; M, Q, J)|rh
• In the 1960’s and 70’s it was shown that these quantities satisfy in general four rela- tions, dubbed “the four laws of black hole mechanics”:
1. Zeroth law: κ is constant on the horizon, which is only a trivial statement in the case of spherically symmetric horizons.
18 2. First law: If you change the mass, charge of a black hole f must remains zero on the horizon. The change in horizon area is related to the other changes by (ignoring angular momentum for simplicity:
κ ∆M = ∆A + V ∆Q (51) 8π h
2 where V = Q/(4π0rh) is the electromagnetic potential at the horizon.
3. Second law: ∆Ah ≥ 0 , i.e horizon area can never decrease.
4. Third Law: κ = 0 cannot be reached from a black hole with non-zero surface gravity by a finite number of processes (eg throwing electric charge into a neutral black hole until an extremal black hole results).
• Connection to Thermodynamics: Note the strong resemblance of these four laws to the four laws of thermodynamics once the correspondance is assumed:
E = Mc2 Energy κ ∝ T Temperature (52)
Ah ∝ S Entropy (53)
• In the early 70’s Jacob Bekenstein conjectured that this might be more than an anal- ogy, namely that black hole horizons actually have entropy. This was treated with scepticism, at best, since entropy requires many, many microscopic degrees of free- dom, whereas spherically symmetric black holes have no internal degrees of freedom. They are characterized by a very small number of externally observable parameters.
• In 1976, in an attempt to prove Bekenstein wrong, Hawking calculated the quantum effects of a black hole horizon on a field just outside the horizon. Amazingly, he found that the horizon boundary conditions caused thermal radiation to be observed far from the horizon. The temperature of this radiation for a Schwarzschild black hole is:
~c3 Th = (54) 8πGMkB
where kB is the Boltzmann constant. This is a beautiful relationship that combines quantum mechanics, general relativity and thermodynamics.
19 • Given that the surface gravity of a Scharzschild black hole is 1/(4M) this is com- pletely consistent with the thermodynamics interpretation of the four laws of black hole mechanics with:
~κ Th = (55) 2πkB k A κ A S = b = (56) 4 ~G 4 lpl2 • Thus the black hole entropy is one quarter the area of the horizon expressed in Planck units! This is a very general result, one that continues to defy explanation, although it seems clear that the resolution lies in the underlying quantum theory of gravity. Note
that since in statistical mechanics S = kB ln(ΩE) where ΩE is the number of distinct microstates with the given energy, the entropy, and associated number of microstates for a solar mass black hole is huge. (Check for yourself.)
VII. COSMOLOGY: THEORY
A. Symmetry Considerations
• Observations tell us that the Universe looks pretty much the same in all directions on large enough distance scales.
• Copernican Principle: states that we don’t occupy a special place in the Universe; i.e. it is unlikely that we are at the center of the Universe. Thus, if the universe looks isotropic from here, it must look isotropic from everywhere.
• The Copernican principle plus the observed isotropy imply that the universe is spatially homogeneous as well as isotropic.
• Spatial homogeneity + isotropy imply six Killing vectors so that three space is maxi- mally symmetric.
• Comoving coordinates are those for which the constant time slices are homogeneous and isotropic. In terms of these coordinates the metric describing the geometry of spacetime takes the following simple form:
ds2 = −dt2 + R2(t)dσ2 (57)
20 where 2 i j dσ = γijdu du , i, j = 1, 2, 3 (58)
is the line element of a maximally symmetric three space, whose most general metric will be derived below.
B. FRW Metric
• Note: Special relativity and later general relativity were constructed in order to elim- inate the need for any preferred frames of reference in order to describe physics. The remarkable fact that the universe is manifestly homogeneous and isotropic introduces just such a preferred frame: that of a comoving observer.
• Know how to derive most general maximally symmetric three metric. It starts from the observation that the curvature and Ricci tensors of a maximally symmetric space satisfy:
R R = (γ γ − γ γ ) (59) ijkl 6 ik jl il kj R R = γ (60) ij 3 ij
where the Ricci scalar R is constant. This leads to:
1 dσ2 = dr2 + r2dΩ(2) k = −1, 0, 1 (61) 1 − kr2 √ • Simple coordinate transformation χ = dr/ 1 − kr2 (know) transforms above into:
2 2 2 (2) dσ = dχ + Sk(χ)dΩ (62)
Sk(χ) = sinh(χ) k = −1 = χ k = 0
= sin(χ) k = +1 (63) (64)
k = −1, 0, 1 correspond to hyperbolic, flat and spherical geometries. The first two are open (range of the coordinate χ is −∞ < χ < ∞) whereas the last (i.e. the three sphere) is compact (χ is an angle). All the coordinates are angles (periodic).
21 R(t) • Dimensionless scale factor a(t) = , where R0 is arbitrary but normally taken to be R0 the current radius of the universe.
• In terms of the dimensionless scale factor the FRW metric is:
dr2 ds2 = −dt2 + a(t)2dσ2 = −dt2 + a(t)2 + dΩ(2) (65) 1 − κr2
2 −2 where κ = k/R0 now has dimensions of L and we have absorbed a factor of R0 into
the “radial” variable i.e. r → R0r, which now has dimensions of length.
C. Hubble Parameter
• Definition: a˙ H(t) = (66) a
• Current measured value:
km/s km/s H = 70 ± 10 = 100h (67) 0 Mpc Mpc
where one megaparsec is 1Mpc = 3.3 × 106ly
−1 9.8 9 • Hubble time: tH = H0 = h × 10 years
a/¨ a˙ aa¨ • Deceleration parameter: q = a/a˙ = − a˙ 2
D. Matter Content
• Choose simple model for matter content of universe: perfect fluid. If universe is maximally symmetric the fluid must also be homogeneous, isotropic and co-moving. That is it must be stationary with respect to the co-moving spatial coordinates χ, θ, φ. Its four velocity, in co-moving coordinates, must therefore be
dxµ U µ = = δµ (68) dτ 0
µ ν This is correctly normalized U gµνU = g00 = −1.
22 • The stress energy tensor of this co-moving fluid is:
µ ν Tµν = (ρ(t) + p(t))U U + p(t)gµν ρ 0 = (69) 0 pgij
• Trace is T = −ρ + 3p
µ • Energy-momentum conservation ∇µT ν = 0 implies:
−ρ˙ + 3(ρ + p)(˙a/a) = 0 (70)
• Equation of state Assume for simplicity p = wρ, w = constant. This can be integrated to give ρ(t) ∝ a3(1+w) (71)
• The null dominant energy condition requires |w|2 ≤ 1.
• Types of Matter
−3 – pressureless dust: w = 0 → ρm ∝ a
µ – Radiation: Maxwell’s equations imply that Tµ = 0 for radiation, so that w = −4 1/3 → ρem ∝ a
– Vacuum energy (cosmological constant): Tµν = Λgµν. This requires w = −1 →
ρΛ = constat
– At the current epoch ρm seems to dominate the energy content of the universe,
but as we move forward in time and the universe expands, eventually ρΛ will
dominate. Conversely, going backwards in time, since a was a lot smaller, ρem dominated.
E. The Friedmann Equation
• In terms of the scale factor, density and pressure, Einstein’s equations read: a¨ G = −3 = 4πG(ρ + 3p) (72) 00 a a¨ a˙ 2 κ G = + 2 + 2 = 4πG(ρ − p) (73) ij a a2 a2
23 • Adding 3 times the second to the first yields the Friedmann equation: a˙ 2 8πG κ H2 := = ρ − (74) a2 3 a2
• Eq.(72) is called the “second Friedmann equation”.
• Counting
– Because of the symmetry, there are only two independent components of Gµν as indicated above.
– There are three functions a(t), ρ(t) and p(t), but:
– p(t) is determined in terms of ρ(t) by the equation of state.
– As we have seen above conservation of energy-momentum determines ρ in terms of a. This leaves one free function (the scale factor) and two equations.
– The Bianchi identies imply that G00 and Gij are not independent so that: – We need only solve the Friedmann equation for the scale factor and we are done!
• Critical Density: The value of ρ at which κ vanishes (and hence flips from -ve to +ve): 3H2 ρ = (75) crit 8πG • Density Parameter: 8πG ρ Ω := 2 ρ = (76) 3H ρcrit In terms of the density parameter the Friedmann equation is: κ Ω − 1 = (77) H2a2 so:
ρ < ρcrit ⇔ Ω < 1 ⇔ κ < 0 ⇔ Open (78)
ρ = ρcrit ⇔ Ω = 1 ⇔ κ = 0 ⇔ Flat (79)
ρ > ρcrit ⇔ Ω > 1 ⇔ κ > 0 ⇔ closed (80) (81)
• Measurement of ρ0 and H0 can determine the large scale structure of the Universe.
The current best data suggests that Ω0 ∼ 1, so that the universe is very close to flat.
24 F. Time evolution of the scale factor and density
• The universe contains more than one type of matter, in which case the energy- momentum tensor can be approximated by the same form as in Eq.(69) but with: X ρ(t) = ρ(i)(t) i X p(t) = p(i)(t) (82) i
• We assume equations of state for each component of the form:
p(i)(t) = w(i)ρ(i)(t) (83)
• We now assume that the various types of matter do not interact directly (only through gravity), so that they obey separate conservation laws which can each be integrated to yield
−3(1+w(i)) −n(i) ρ(i)(t) = ρ(i) 0a (t) = ρ(i) 0a (84)
with n(i) := 3(1 + w(i)).
• The Friedmann equation can then be written: 8πG X κ H2 = ρ − 3 (i) a2 i 8πG X κ = ρ a−n(i) − (85) 3 (i) 0 a2 i
• The above suggests defining for convenience a ’density’ associated with the curvature term in the Friedmann equation: 3κ ρ := − (86) c 8πGa2
−2 ρc is the curvature contribution to the energy density. Since ρc ∝ a we can also
define an effective equation of state, with p = wcρ, wc = −1/3.
• Friedmann equation in its most useful form: 8πG X H2 = ρ (87) 3 (i) (i),c X or equivalently 1 = Ω(i) (88) (i),c
25 In the above, the subscript c in the summation indicates that it should include the contribution from the curvature.
• We now have four different contributions to the Friedman equation, summarized below:
Type ω(i) n(i) Matter 0 3 Radiation 1/3 4 (89) Vacuum −1 0 Curvature −1/3 2
• Integration of the Friedmann Equation For simplicity we consider only ordinary matter, vacuum energy and the non-zero curvature. Multiplying (88) by a2, dividing
by H0 :=a/a ˙ and setting a0 = 1 without loss of generality yields:
2 a˙ −1 2 2 = Ωm0a + ΩΛ0a + Ωc0 (90) H0 where we have defined the current critical densities:
8πG ρ(i)0 Ω(i)c = 2 (91) 3 H0
Note that setting t = t0 in (90) implies that
Ωm0 + ΩΛ0 + Ωc0 = 1 (92)
Finally, after a bit of algebra (90) yields: Z Z −1 2 −1/2 H0 dt = da Ωm0a + ΩΛ0a + Ωc0 (93)
By evaluating the above for different values of Ω(i) one obtains the curves in Fig. (8.3) of Carroll. (See assignment 18).
VIII. COSMOLOGY: OBSERVATIONS
A. Redshifts
• Killing tensor Kµν of a metric gµν is a symmetric tensor that satisfies:
∇(αKµν) = ∇αKµν + ∇νKαµ + ∇µKνα = 0 (94)
26 • You have verified in your assignment that the following is a Killing tensor for the FRW metric (90): 2 Kµν = a (t)(gµν + UµUν) (95)
µ µ where as usual U = δ0 is the four velocity of a co-moving observer.
• The existence of a Killing tensor implies that the following quantity is conserved along geodesics with four velocity V µ = dxµ/dλ:
2 µ ν K = V V Kµν (96)
For null geodesics: 2 2 µ K = a (t)V Uµ (97)
• As always the energy and hence frequency of a photon with four velocity V µ as mea- sured by an observer with four velocity U µ is
µ ω = −V Uµ (98)
Thus if we consider the frequency ωem as measured by a co-moving observer at tem
when the photon is first emitted and compare it to the frequency ωobs as measured by
a co-moving observer who detects it a some later time tobs we find: ω κ/a(t ) a(t ) obs = obs = em (99) ωem κ/a(tem) a(tobs)
• Redshift If tobs = t0 is now, so that a(tobs) = a0 = 1, then we can express the redshift as:
λ − λ 1/ω − 1/ω z := obs em = obs em λem 1/ωem ω = em − 1 ωobs a 1 − a(t ) = 0 − 1 = em (100) a(tem) a(tem) 1 → a(t) = (101) 1 + z
27 B. Proper Distance
• Proper Distance: the proper distance between us and a distant galaxy at fixed cosmological time t is obtained by integrating the line element along a path at fixed t, θ, φ Z Z dp = ds = a(t)R0 dχ = a(t)R0χ (102)
assuming we have chosen coordinates such the χ axis lies on the line between us and the galaxy is at χ = 0.
• If both the earth and the distance galaxy are approximately co-moving then χ doesn’t
change as the universe expands so that setting t = t0 gives:
˙ dp = R0χa˙|t0 = R0χa0H0 = dpH0 (103)
which gives: ˙ vp := dp = H0dp (104)
This states that the proper velocity is proportional to the proper distance.
−1 • For distances that are small compared to the cosmological distance d0 = cH0 we can ignore higher order effects to get
1 a0 − aem ∆a Zem = − 1 = = ∼ H0δt (105) aem aem a
Moreover we can approximate the time of flight δt ∼ d/c and since: δdp/dp = δap/ap we get: δd δd δt v d z = = = p = H p (106) em d δt d c 0 c
which is the famous linear redshift Hubble law. One can measure H0 by plotting observed distance vs red-shift for relatively nearby objects.
• The linear relationship should only hold for small z <<.
• To get a rough idea of the scales involved, with H0 = 70km/s/Mpc, z = 1 corresponds to a distance of about 4000 Mpc, or 10 billion light years. By comparison, distance to the center of our galaxy is about 27,000 light years and the distance to the nearest galaxy is about 2.5 million light years.
28 C. Luminosity Distance
• Preamble: For larger distances, it is difficult (impossible) to measure the proper distance between us and a distant object. We have to wait for the light to get to us, and the universe will have expanded in the meantime. Luminosity distance easier to measure. Depends on local quantity, named the observed energy flux from a distant object of known luminosity.
• Luminosity L: amount of energy per unit time given off in radiation by an object.
• Flux F : amount of energy per unit time per unit area that falls on a detector on earth from a particular distant object.
• Standard candle: object whose instrinsic luminosity is known. Type Ia Supernova: explosion of white dwarf that has accreted too much mass and is approaching the Chandresekhar limit. The good news: Luminosity independent of history, excellent standard candle. Bad news: they are rare and (like most explosions) short-lived.
• Definition: Luminosity distance: 1 F 2 = (107) 4πdl L which in flat, non-expanding spacetime simply states that the total luminosity is spread 2 out over a sphere of area 4πdl . For very distant objects in an expanding universe we have to be more careful.
D. Derivation of Luminosity vs Redshift Relation
• In an expanding FRW spacetime the flux of a spherical wavefront of radiation is reduced not only by the fact that it is spread over a larger area, but also by a redshift and time-dilation effect. The former reduces the energy of each photon by a factor of (1+z) the latter reduces the number of photons per unit time that hit a surface by a factor of (1+z). In order to calculate the luminosity distance we have to take these two effects into account. In particular, if F is the measured flux then L 1 F = (108) A (1 + z)2
29 which takes into account both the size of the two sphere over which the luminosity is distributed and the red-shift/time-dilation effects.
• We now calculate the invariant area A over which the luminosity is distributed as a function of the red-shift of the source relative to the observer. What we want is area of the two-sphere centered on the source (located at χ = 0 without loss of generality) on which the observer is located. From the FRW metric (88) we see that this is just:
2 2 2 2 2 A = 4πa (t)R0Sk(χ) = 4πR0Sk(χ) (109)
where we have assumed that we are detecting the radiation now, when a(t0) = 1. So now the luminosity distance is given by 1 F 1 2 = = 2 2 2 (110) 4πdl L 4πR0Sk(χ)(1 + z) so that:
dl = R0Sk(χ)(1 + z) (111)
We now need to relate χ to z and we are done.
• Look at null “radial” (i.e. dθ = dφ = 0) geodesics in the FRW spacetime:
2 2 2 2 2 ds = 0 = −dt + a (t)R0dχ (112)
This implies that , Z Z 1 Z da χ = dχ = R−1 dt = R−1 (113) 0 a 0 a2H(a) where we used H(a) =a/a ˙ . Now using a = (1 + z)−1 we get Z z −1 dz χ(z) = R0 (114) 0 H(a(z)) But
X ni 1/2 H = H0 Ω(i)0(1 + z) (115) (i)c so Z z 0 −1 −1 dz dl = (1 + z)R0Sk R0 H0 0 (116) 0 E(z ) where
0 ni 1/2 E(z ) = Ω(i)0(1 + z) (117)
30 • When k = 0, Sk(χ) = χ so R0 cancels, whereas when k 6= 0 you can use Ωc0 = 2 2 2 2 2 2 2 −κ /(H0 a0) = k /(R0H0 a0) to eliminate R0.
• Eq.(116) is a key relationship in observational cosmology. It predicts the relationship between the luminosity distance and the red-shift. So if you can find a large number of distance objects with known intrinsic luminosities and spectra, and measure the
flux and red-shift you can determine the parameters H0 and Ω(i) by fitting the data to the theoretical curves. This is what was essentially done, using type Ia supernovae as standard candles. The resulting best, including closer objects where the relationship
is close to linear, is k = 1, Ωm0 = 0.3 and ΩΛ0 = 0.7.
• If you plot dl vs z using Eq.(116) you will get a curve that fits the data points in Fig. 1 of the Nobel prize winning experimental work described in the paper found at http://arxiv.org/pdf/astro-ph/9812133v1.pdf .
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