Problem Set #4: 4.1, 4.3, 4.5 (Due Monday Nov. 18th) f = m i a (4.1) f = m g Φ (4.2) a = Φ. (4.4)

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1 Chapter 4 Gravitation Problem Set #4: 4.1, 4.3, 4.5 (Due Monday Nov. 18th) 4.1 Equivalence Principle The Newton s second law states that f = m i a (4.1) where m i is the inertial mass. The Newton s law of gravity states that f = m g Φ (4.2) where m g is the gravitational mass. The Weak Equivalence Principle states that m i = m g. (4.3) This means that for a freely falling particle the effect of the gravitational field is independent on its mass a = Φ. (4.4) This also means that by preforming local experiments it is impossible to distinguish the gravitational field from being in a uniformly accelerated reference frame. Of course all this was known long before Einstein, but what was newis the Special Theory of Relativity which can be included in the Einstein Equivalence Principle: Inasmallenoughregionofspace-time, the laws of physics reduce to those of special relativity; it is impossible to 54

2 CHAPTER 4. GRAVITATION 55 detect the existence of a gravitational field. An immediate and celebrated prediction of the Einstein Equivalence Principle is the gravitational redshift. Consider two observers at a distance z from each other moving with acceleration a so that the distance remains fixed. One observer emits a photon and it reaches the other observer after a time By that time the velocity of both boxes changed by t = z c. (4.5) v = a t = a z c (4.6) and due to non-relativistic Doppler effect the wavelength will be shifted by λ λ 0 = v c = az c 2. (4.7) According to the Einstein Equivalence Principle this effect should be indistinguishable when the acceleration is due to gravity, then the photon emitted from the ground should be shifted by λ λ and in terms of Newtonian potential λ = 1 Φdt = λ c = 1 c 2 z Φdz = = a gz c 2. (4.8) = Φ c 2. (4.9) This was checked experimentally by Pound and Rebka in Einstein Equivalence Principle also suggests that the gravitational field acts universally on massive and as we have seen on massless particles. This means that there are no gravitationally neutral objects with respect to which one can measure acceleration and, thus, the acceleration due to gravity is ill defined. (As we shall see even the gravitational mass of extended objects is also ill defined.) Therefore, to satisfy the principle we shall give up on the idea of extending inertial frames of Special Relativity throughout the entire space-time or otherwise the far away objects would accelerate in such

3 CHAPTER 4. GRAVITATION 56 frames. Of course we can still keep the notion of the local inertial frames where by local we mean a sufficiently small region of space-time volume. Thus it seems necessary (and as we will see also sufficient) to incorporate the ideas of differential geometry into the laws of gravitation. Evidently, the local inertial frames are nothing but local normal coordinates for which the Christoffel symbols vanish locally and all of the laws of Spatial Relativity are maintained; and the impossibility to define acceleration ofthefaraway objects is due to the fact that the parallel transport of vectors depends on the path for a curved space-time with non-vanishing Riemann tensor. In the normal coordinates a freely falling observer moves without acceleration d 2 x µ =0. (4.10) dλ2 This equation can be made covariant (or independent on coordinates) with help of Christoffel symbols d 2 x µ dλ 2 dx dx σ +Γµ σ dλ dλ =0. (4.11) which is just the geodesics equation that tells that the freely falling particle move along geodesics. We shall now check that (4.11) reducestothenewton slawofgravity (4.2) inthelimitof: small velocities dx i dτ dx0 dτ where τ is proper time and thus from (4.11) (4.12) d 2 x µ dx 0 dx 0 dτ 2 +Γµ 00 dτ dτ 0. (4.13) slowly changing metric and thus 0 g µν i g σ (4.14) Γ µ 00 = 1 2 gµλ ( 0 g λ0 + 0 g 0λ λ g 00 ) 1 2 gµλ λ g 00. (4.15) weak gravitational field g µν = η µν + h µν (4.16)

4 CHAPTER 4. GRAVITATION 57 where the perturbation h µν 1 (4.17) and thus from (4.16) and(4.14) Γ µ ηµi i h 00. (4.18) From (4.13) and(4.18) whose µ =0component or and µ = i component d 2 x µ dτ ηµi i h 00 dx 0 dτ ( ) dx 0 2 (4.19) dτ d 2 x 0 dτ 2 0 (4.20) const. (4.21) ( ) 2 dτ d 2 x i dx 0 dτ ih 00 (4.22) or by setting t = x 0, d 2 x i dt ih 00. (4.23) Note that (4.23) isthesameas(4.4) with or h 00 = 2Φ (4.24) g 00 = (1 + 2Φ). (4.25) Similarly one can adopt all of the laws of physics to respect the Einstein Equivalence Principle. This usually involves starting with normalcoordinates where the laws of physics are known and rewriting them in a covariant form it in terms of legitimate tensors (e.g. partial derivative are replaced by covariant etc.).

5 CHAPTER 4. GRAVITATION Einstein Equations As was previously stated we want to derive a dynamical equation for the metric tensor which generalizes the Poisson equation where 2 = δ ij i j and is the mass density. Since we combine (4.24) and(4.26) into 2 Φ=4πG (4.26) T 00 = (4.27) 2 h 00 = 8πGT 00. (4.28) Of course, the next step should be to make this equations tensorial. The first obvious choice λ λ g µν T µν (4.29) is not going to work just because the left hand side is identically zero since λ g µν =0.Thesecondchoice R µν T µν (4.30) is a lot more reasonable, but does not work either just because theenergy and momentum conservations imply but µ T µν =0 (4.31) µ R µν 0 (4.32) for general space-times. An important exception is a vacuum Einstein equation for vanishing energy momentum tensor R µν = T µν =0. (4.33) There is however a geometric object the Einstein tensor G µν = R µν 1 2 Rg µν (4.34) with the right properties on arbitrary spaces. It is a symmetric (0, 2) tensor built of the second (and first and zeroth) derivatives of the metric and µ G µν =0. (4.35)

6 CHAPTER 4. GRAVITATION 59 Thus we can conjecture that G µν = R µν 1 2 Rg µν =8πGT µν (4.36) where the proportionality constant was determined from (4.28). This is the Einstein equation which is a rather complicated second-order non-linear equation with very few explicit solutions. The equation actually contains 10 coupled differential equation since both sides are symmetric (0, 2) tensors which is exactly the number of unknown parameters in ametricg µν. But because of the general covariance 4 of the 10 equations are unphysical (represented by four function x µ (x µ ))andweareleftwithonly6 physical (or coordinate independent) degrees of freedom. This agreeswiththenumber of dynamical equations in (4.36) whentheconstraintsduetoconservationof the energy and momentum (4.31) orthebianchiidentity(4.35) istakeninto account. The Einstein equation can also be derived using variational principle for the Hilbert action S H = d 4 x gr. (4.37) To obtain the Einstein equations with lower indices we will vary with respect to the inverse metric. Since R = g µν R µν (4.38) the variation of the acton will have three contributions: δs H = d 4 x ( gg µν δr µν + gr µν δg µν + Rδ g ). (4.39) But the first term is given by δr µλν = δ ( λ Γ µν νγ µλ +Γ λσ Γσ µν Γ νσ µλ) Γσ = ( ) ( ) = λ δγ µν ν δγ µλ + δγ λσ Γσ µν +Γ λσ δγσ µν δγ νσ Γσ µλ Γ νσ δγσ µλ = = [ ( ) λ δγ µν +Γ ] λσ δγσ µν Γ σ λµδγ σν Γ σ λνδγ µσ [ ( ν δγ µλ) +Γ νσ δγ σ µλ Γσ νµ δγ σλ Γσ νλ µσ] δγ = ( ) ( = λ δγ µν ν δγ ) µλ (4.40) Note that unlike Γ µν its variation δγ µν is a tensor as can be seen from the

7 CHAPTER 4. GRAVITATION 60 transformation law (3.6). Thus, d 4 x ( ) gg µν δr µν = d 4 x g ( ( ) ( )) g µν λ δγ λ µν g µν ν δγ λ µλ = = d 4 x g ( ( ( λ g µν δγµν) λ λ g µλ δγµν)) ν = = d 4 x ( g λ g µν δγ λ µν gµλ δγµν) ν =0 (4.41) is a boundary term which is set to zero by making the variations vanishat the boundary of space-time. Now the third term in (4.39) canbesimplifiedusingthefollowinguseful identity Tr(log (M)) = log (det (M)) (4.42) or Applying it to inverse metric gives us Tr ( M 1 δm ) = δ (det (M)) det (M) (4.43) M µν = g µν (4.44) Tr ( g µν δg νλ) = δ (g 1 ) g 1 g µν δg µν = g 1 δg. (4.45) δ g = 1 δg = 1 2 g 2 and by substituting in (4.39) weget δs H = d 4 x g ( g) g µν δg µν g = 1 2 ggµν δg µν (4.46) ( R µν 1 ) 2 Rg µν δg µν. (4.47) Therefore for the action to vanish for arbitrary variations of the metric the vacuum Einstein equations must be satisfied 1 δs =0 (4.48) g δg µν G µν = R µν 1 2 Rg µν =0. (4.49)

8 CHAPTER 4. GRAVITATION Energy-Momentum Tensor To obtain a non-vacuum equation on should also include matterlagrangian, S = 1 d 4 x gr.+ d 4 x g L M, (4.50) 16πG and the variational principle implies δs δg = 1 d 4 x g µν 16πG ( R µν 1 2 Rg µν For this to agree with Einstein equation ) + d 4 x g ( 1 δ ( ) gl M ) =0. g δg µν R µν 1 2 Rg µν =8πGT µν we define the energy momentum tensor as T µν = 2 1 δl M. (4.51) g δg µν For example, action for a scalar field ( S φ = dx 4 g 1 ) 2 gµν µ φ ν φ V (φ) (4.52) can be varied δs φ = dx [ 4 g 1 ( 2 µφ ν φ (δg µν )+ 1 ) 2 gµν µ φ ν φ V (φ) δ ( g )] = = dx [ 4 g 1 ( 2 µφ ν φ + 1 )( 2 gλ λ φ φ V (φ) 1 )] ggµν δg µν = 2 = dx 4 [ g 1 2 µφ ν φ g µνg λ λ φ φ + 1 ] 2 V (φ)g µν δg µν (4.53) which leads to the following energy momentum tensor T µν = 2 1 g δs δg µν = µφ ν φ 1 2 g µνg λ λ φ φ V (φ)g µν. (4.54) Note that the non-vacuum Einstein equation can be solved for an arbitrary manifold. Just specify the metric, calculate the Einstein tensor and demand that it is proportional to the energy momentum tensor. Of course, this does not tell us what should be the field content of the theory and

9 CHAPTER 4. GRAVITATION 62 whether the corresponding energy momentum tensor is realistic. Thus, it is useful to have some restrictions on T µν. Weak Energy Condition: for all time-like vectors t µ. Null Energy Condition: for all light-like vectors l µ. Dominant Energy Condition: T µν t µ t ν 0 T µν l µ l ν 0 T µν t µ t ν 0 for all time-like t µ and time-like or light-like T µν t µ. Strong Energy Condition: T µν t µ t ν 1 2 T λ λt σ t σ 0 for all time-like vectors t µ.thisconditionisusedtoprovefamoussingularity theorems, but is also violated in quantum systems, for example, subject to Casimir force. For a perfect fluid T µν =( + p)u µ U ν + pg µν where is the energy density, p is pressure and U µ is the four velocity. weak energy conditions implies 0 and + p 0, null energy condition implies + p 0, dominant energy condition implies p, and strong energy condition implies +3p 0 and + p 0. Consider time evolution of a small ball of small particles moving with four velocities U µ.thentheexpansion, rotation and shear of the ball can be defined as θ = λ U λ. (4.55) ω µν = [ν U µ] (4.56)

10 CHAPTER 4. GRAVITATION 63 σ µν = (ν U µ) 1 3 λu λ (g µν + U µ U ν ). (4.57) and the famous Raychaudhuri equation describes the time evolution of the expansion parameter, dθ dτ = 1 3 θ2 σ µν σ µν + ω µν ω µν R µν U µ U ν. (4.58) If the ball of particles is at rest U =(1, 0, 0, 0) with respect to a locally inertial system g µν = η µν and has vanishing expansion θ =0,rotationω =0 and shear σ =0,thentheRaychaudhuriandEinsteinequationsgive ( dθ dτ = R 00 = 8πG T 00 1 ) 2 Tg 00 = 4πG (T 00 + T 11 + T 22 + T 33 ) where we used the fact that (4.59) R = 8πGT. (4.60) If we denote T 00 =, T 11 = p x, T 22 = p y, T 33 = p z then, dθ dτ = 4πG ( + p x + p y + p z ). (4.61) Note that the gravity would tend to decrease the size of the ball if +3p 0. Thus, for a perfect fluid the strong energy condition implies that the gravity is attractive.