The Schwarzschild spacetime

Transcription

1 Chapter 9 The Schwarzschild spacetime One of the simplest solutions to the Einstein equations corresponds to a metric that describes the gravitational field exterior to a static, spherical, uncharged mass without angular momentum and isolated from all other mass Schwarzschild, 1916). The Schwarzschild solution is A solution of the vacuum Einstein equations G µν = R µν = 0. Only valid in the absence of matter and non-gravitational fields T µν = 0). Spherically symmetric and time independent. Thus, the Schwarzschild solution is valid outside spherical mass distributions, but the interior of a star will be described by a different metric that must be matched at the surface to the Schwarzschild one. 279

2 280 CHAPTER 9. THE SCHWARZSCHILD SPACETIME 9.1 The Form of the Metric Work in spherical coordinates r, θ, ϕ) and seek a time-independent solution assuming The angular part of the metric will be unchanged from its form in flat space because of the spherical symmetry. The parts of the metric describing dt and dr will be modified by functions that depend only on the radial coordinate r. Therefore, let us write the 4-D line element as ds 2 = Br)dt 2 + Ar)dr 2 }{{} Modified from flat space +r 2 dθ 2 + r 2 sin 2 θdϕ }{{} 2, Same as flat space where Ar) and Br) are unknown functions that may depend on r but not time. They may be determined by 1. Inserting this metric in the Einstein field equations for T µν = 0 vacuum Einstein equations). 2. Solving the resulting equations to determine the unknown functions Ar) and Br).

3 9.1. THE FORM OF THE METRIC 281 Substitute the metric form in vacuum Einstein R µν = 0 and carry out the following steps Exercise): 1. With the assumed form of the metric, Br) Ar) 0 0 g µν = 0 0 r r 2 sin 2 θ compute the non-vanishing connection coefficients Γ λ µν. Γ σ λ µ = 1 gµν 2 gνσ x λ + g λν x µ g ) µλ x ν 2. Use the connection coefficients to construct the Ricci tensor R µν. R µν = Γ λ µν,λ Γλ µλ,ν + Γλ µνγ σ λσ Γσ µλ Γλ νσ, Only need R µν, not full G µν since we will solve vacuum Einstein equations. 3. Solve the coupled set of equations for the functions Ar) and Br). The solution requires several steps but is remarkably simple: Br)=1 2M r Ar)=Br) 1, in G=c=1units), where M is the single parameter.

4 282 CHAPTER 9. THE SCHWARZSCHILD SPACETIME The line element is then ds 2 = 1 2M r ) dt M r ) 1 dr 2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2, where dτ 2 = ds 2. The corresponding metric tensor is 1 2M ) r g µν = 0 1 2M ) r. 0 0 r r 2 sin 2 θ which is diagonal but obviously not constant.

5 9.1. THE FORM OF THE METRIC 283 By comparing g 00 = 1 2GM ) rc }{{ 2 } Weak gravity earlier) g 00 = 1 2GM ) rc }{{ 2 } Schwarzschild G & c restored) we see that the parameter M mathematically the single free parameter of the solution arising as an integration constant) may be identified with the total mass that is the source of the gravitational curvature: Rest mass Contributions from mass energy densities and pressure Energy from spacetime curvature From the structure of the metric ds 2 = Br)dt 2 + Ar)dr 2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2, θ and ϕ have similar interpretations as for flat space. The coordinate radius r generally cannot be interpreted as a physical radius because Ar) 1. The coordinate time t generally cannot be interpreted as a physical clock time because Br) 1. The quantity r S 2M is called the Schwarzschild radius. It plays a central role in the description of the Schwarzschild spacetime.

6 284 CHAPTER 9. THE SCHWARZSCHILD SPACETIME g11 + g00 r = 2 M gµν r/m g11 Figure 9.1: The components g 00 and g 11 in the Schwarzschild metric. The line element metric) ds 2 = 1 2M ) dt M ) 1 dr 2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2 r r }{{}}{{} g 00 g 11 appears to contain two singularities see above figure) 1. A singularity at r= 0 from g 00 an essential singularity). 2. A singularity at r= r S = 2M from g 11 a coordinate singularity).

7 9.1. THE FORM OF THE METRIC 285 Coordinate Singularity: Place where a chosen set of coordinates does not describe the geometry properly. Example: At the North Pole the azimuthal angle ϕ takes a continuum of values 0 2π, so all those values correspond to a single point. But this has no physical significance. Coordinate singularities are not essential and can be removed by a different choice of coordinate system.

8 286 CHAPTER 9. THE SCHWARZSCHILD SPACETIME Measuring Distance and Time What is the physical meaning of the coordinatest,r,θ,ϕ)? We may assign a practical definition to the radial coordinate r by 1. Enclosing the origin of our Schwarzschild spacetime in a series of concentric spheres, 2. Measuring for each sphere a surface area conceptually by laying measuring rods end to end), 3. Assigning a radial coordinate r to that sphere using Area = 4πr 2. Then we can use distances and trigonometry to define the angular coordinate variables θ and ϕ. Finally we can define coordinate time t in terms of clocks attached to the concentric spheres. For Newtonian theory with its implicit assumption that events occur on a passive background of euclidean space and constantly flowing time, that s the whole story.

9 9.1. THE FORM OF THE METRIC 287 But in curved Schwarzschild spacetime The coordinates t, r, θ, ϕ) provide a global reference frame for an observer making measurements at an infinite distance from the gravitational source of the Schwarzschild spacetime. However, physical quantities measured by arbitary observers are not specified directly by these coordinates but rather Physical quantitites must be computed from the metric.

10 288 CHAPTER 9. THE SCHWARZSCHILD SPACETIME Proper and Coordinate Distances Consider distance measured in the radial direction. Set dt = dθ = dϕ = 0 in the line element to obtain an interval of radial distance ds 2 = ) dt 2 + ) 1 dr 2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2 1 2M 1 2M r r }{{} set t,θ,ϕ to constants dt=dθ=dϕ=0 dr ds=, 1 2GM rc 2 In this expression we term 1. ds the proper distance and 2. dr the coordinate distance. The physical interval in the radial direction measured by a local observer is given by the proper distance ds, not by dr. GM/rc 2 is a measure of the strength of gravity, so the proper distance and coordinate distance are equivalent only if gravity is negligibly weak, either because 1. The source M is weak, or 2. We are a very large coordinate distance r from the source.

11 9.1. THE FORM OF THE METRIC M/r ) -1/2 Curved space dr < ds ) C 1 Asymptotically flat space dr ~ ds) C 2 ds Asymptotically flat space dr ~ ds) C 4 C 3 dr Flat space dr = ds) Figure 9.2: Relationship between radial coordinate distance dr and proper distance ds in Schwarzschild spacetime. The relationship between the coordinate distance interval dr and the proper distance interval ds is illustrated further in Fig The circles C 1 and C 3 represent spheres having radius r in euclidean space. The circles C 2 and C 4 represent spheres having an infinitesimally larger radius r+ dr in euclidean space. In euclidean space the distance that would be measured between the spheres is dr But in the curved space the measured distance between the spheres is ds, which is larger than dr, by virtue of dr ds=, 1 2GM rc 2 Notice however that at large distances from the source of the gravitational field the Schwarzschild spacetime becomes flat and then dr ds.

12 290 CHAPTER 9. THE SCHWARZSCHILD SPACETIME Proper and Coordinate Times Likewise, to measure a time interval for a stationary clock at r set dr=dθ = dϕ = 0 in the line element and use ds 2 = dτ 2 c 2 to obtain ds 2 = ) dt 2 + ) 1 dr 2 + r 2 dθ 2 + r 2 sin 2 θdϕ 2 1 2M 1 2M r r }{{} set r,θ,ϕ to constants dr=dθ=dϕ=0 dτ = 1 2GM rc 2 In this expression dτ is termed the proper time and dt is termed the coordinate time. dt. The physical time interval measured by a local observer is given by the proper time dτ, not by the coordinate time dt. dt and dτ coincide only if the gravitational field is weak.

13 9.1. THE FORM OF THE METRIC 291 Thus we see that for the gravitational field outside a spherical mass distribution The coordinates r and t correspond directly to physical distance and time in Newtonian gravity. In general relativity the physical proper) distances and times must be computed from the metric and are not given directly by the coordinates. Only in regions of spacetime where gravity is very weak do we recover the Newtonian interpretation. This is as it should be: The goal of relativity is to make the laws of physics independent of the coordinate system in which they are formulated.

14 292 CHAPTER 9. THE SCHWARZSCHILD SPACETIME The coordinates in a physical theory are like street numbers. They provide a labeling that locates points in a space, but knowing the street numbers is not sufficient to determine distances. We can t answer the question of whether the distance between 36th Street and 37th street is the same as the distance between 40th Street and 41st Street until we know whether the streets are equally spaced. We must compute distances from a metric that gives a distance-measuring prescription. Streets that are always equally spaced correspond to a flat space. Streets with irregular spacing correspond to a position-dependent metric and thus to a curved space. For the flat space the difference in street number corresponds directly up to a scale) to a physical distance, but in the more general curved) case it does not.

15 9.1. THE FORM OF THE METRIC Embedding Diagrams It is sometimes useful to form a mental image of the structure for a curved space by embedding the space or a subset of its dimensions in 3-D euclidean space. Such embedding diagrams can be misleading, as illustrated well by the case of a cylinder embedded in 3-D euclidean space, which suggests that a cylinder is curved. But it isn t: The cylinder is intrinsically a flat 2-D surface: cut it and roll it out into a plane, or calculate its vanishing gaussian curvature. The cylinder has no intrinsic curvature; the appearance of curvature derives entirely from the embedding in 3-D space. This is termed extrinsic curvature. Nevertheless, the image of the cylinder embedded in 3D euclidean space is a useful representation of many properties associated with a cylinder.

16 294 CHAPTER 9. THE SCHWARZSCHILD SPACETIME We can embed only 2 dimensions of Schwarzschild spacetime in 3D euclidean space. Illustrate by choosing θ = π/2 and t = 0, to give a 2-D metric dl 2 = 1 2M ) 1 dr 2 + r 2 dϕ 2. r The metric of the 3-D embedding space is conveniently represented in cylindrical coordinates as dl 2 = dz 2 + dr 2 + r 2 dϕ 2 This can be written on z=zr) as dl 2 = ) dz 2 dr 2 + dr 2 + r 2 dϕ =[ 2 1+ dr ) ] dz 2 dr 2 + r 2 dϕ 2 dr Comparing [ ) ] dz 2 dl 2 = 1+ dr 2 +r 2 dϕ 2 dl 2 = 1 2M dr r ) 1 dr 2 +r 2 dϕ 2 implies that zr)=2 2Mr 2M), which defines an embedding surface zr) having a geometry that is the same as the Schwarzschild metric in the r ϕ) plane.

17 9.1. THE FORM OF THE METRIC 295 Figure 9.3: An embedding diagram for the Schwarzschild r ϕ) plane. Fig. 9.3 illustrates the embedding function zr)=2 2Mr 2M) Fig. 9.3 is not what a black hole looks like, but it is a striking and useful visualization of the Schwarzschild geometry. Thus such embedding diagrams are a standard representation of black holes in popular-level discussion.

18 296 CHAPTER 9. THE SCHWARZSCHILD SPACETIME t Schwarzschild radius Source Light ray Emission of light with frequency ω 0 Distant observer Detection of light with frequency ω 0 r = 2M r = R 1 r ~ r Figure 9.4: A spacetime diagram for gravitational redshift in the Schwarzschild metric The Gravitational Redshift Let s now return to the gravitational redshift problem, which we treated earlier with a weak-field approximation. Consider emission of light from a radius R 1 that is then detected by a stationary observer at a radius r>> R 1 Fig. 9.4). For an observer with 4-velocity u, the energy measured for a photon with 4-momentum p is E = hω = p u, Observers stationary in space but not time so u i r)=0 u 0 0

19 9.1. THE FORM OF THE METRIC 297 Thus the 4-velocity normalization gives u u=g µν x) dxµ dx ν dτ dτ = g 00x)u 0 r)u 0 r)= 1. }{{} Solve for u 0 r) and we obtain u 0 r)= 1 = 1 2M g 00 r ) 1/2. Symmetry: Schwarzschild metric independent of time, which implies the existence of a Killing vector K µ =t,r,θ,ϕ)=1,0,0,0) associated with symmetry under time displacement. Thus, for a stationary observer at a distance r, u µ r)= 1 2M ) 1/2, 0, 0, 0) = 1 2M r r ) 1/2 K µ. The energy of the photon measured at r by a stationary observer is hωr)= p u= 1 2M r ) 1/2 K p) r

20 298 CHAPTER 9. THE SCHWARZSCHILD SPACETIME But K p is conserved along the photon geodesic K is a Killing vector) so K p is in fact independent of r. Therefore, hω 0 hωr 1 )= 1 2M ) 1/2 K p) R 1 hω hωr )= K p), and from hω / hω 0 we obtain immediately a gravitational redshift ω = ω 0 1 2M ) 1/2. R 1 We have made no weak-field assumptions so this result should be generally valid in Schwarzschild spacetime. For weak fields 2M/R 1 is small, the square root can be expanded, and the G and c factors restored to give ω ω 0 1 GM ) R 1 c 2 valid for weak fields) which is the result derived earlier using the equivalence principle. By viewing ω as defining clock ticks, the redshift may also be interpreted as a gravitational time dilation.

21 9.1. THE FORM OF THE METRIC Particle Orbits in the Schwarzschild Metric Symmetries of the Schwarzschild metric: 1. Time independence Killing vector K t =1,0,0,0) 2. No dependence on ϕ Killing vector K ϕ =0,0,0,1) 3. Additional Killing vectors associated with full rotational symmetry won t need in following). Conserved quantities associated with these Killing vectors: Physical interpretation: ε K t u= 1 2M r ) dt dτ l K ϕ u=r 2 sin 2 θ dϕ dτ. At low velocitiesl orbital angular momentum / unit rest mass) Since E = p 0 = mu 0 = mdt/dτ, Lim r ε = dt dτ = u0 = E m and ε energy / unit rest mass) at large distance. Also we have the velocity normalization constraint u u=g µν u µ u ν = 1.

22 300 CHAPTER 9. THE SCHWARZSCHILD SPACETIME Conservation of angular momentum confines the particle motion to a plane, which we conveniently take to be the equatorial plane with θ = π 2 implying that u2 u θ = 0 and dθ = 0. Then writing the velocity constraint out in the metric g µν u µ u ν = 1. ds 2 = 1 2M r gives ) dt M r ) 1 dr 2 + } r 2 {{ dθ} 2 +r 2 } sin{{ 2 θ} dϕ 2, =0 =1 1 2M r ) u 0 ) M r ) 1 u 1 ) 2 + r 2 u 3 ) 2 = 1. which we may rewrite using in the form ε = u µ dx 0 ) = dτ, dx1 dτ, dx2 dτ, dx3 dτ 1 2M ) dt l=r 2 sin 2 θ dϕ r dτ dτ. ε = 1 2 ) dr [ 1 2M dτ 2 r ) l 2 ) ] r

23 9.1. THE FORM OF THE METRIC 301 We can put this in the form E = 1 2 where we define a fictitious energy and an effective potential ) dr 2 +V effr), dτ E ε2 1 2 V eff r)= 1 2 [ 1 2M r ) l 2 ) ] r = M r + l2 2r 2 }{{} Newtonian Ml2 r 3 }{{} correction This is analogous to the energy integral of Newtonian mechanics with an effective potential V eff and a proper time interval dτ. The effective potential V eff r) is of Newtonian form except that The GR potential has an additional term proportional to r 3 that is not present in the Newtonian potential. The Schwarzschild coordinate r and the Newtonian coordinate r don t have the same physical interpretations.

24 302 CHAPTER 9. THE SCHWARZSCHILD SPACETIME Newtonian Veff Schwarzschild r/m Figure 9.5: Effective potentials for finite l in the Schwarzschild geometry and in Newtonian approximation. Figure 9.5 compares the Schwarzschild effective potential with an effective Newtonian potential. The Schwarzschild potential generally has one maximum and one minimum if l/m > 12. Note the very different behavior of Schwarzschild and Newtonian mechanics at the origin because of the correction term in V eff r)= M r + l2 2r 2 }{{} Newtonian Ml2 r 3 }{{} correction

25 9.1. THE FORM OF THE METRIC 303 Unstable equilibrium Stable V eff 0 Stable equilibrium Circular orbits Unstable V eff 0 Turning points Bound precessing orbits Turning point V eff 0 Scattering orbits V eff 0 Plunging orbits r/m Figure 9.6: Orbits in a Schwarzschild spacetime. Effective potential on left and corresponding classes of orbits on right.

26 304 CHAPTER 9. THE SCHWARZSCHILD SPACETIME Innermost Stable Circular Orbit The radial coordinate of the inner turning point for bound precessing orbits in the Schwarzschild metric is given by Exercise) r = l2 2M ) M 2 l Thus r has a minimum possible value when M l = The corresponding radius for the innermost stable circular orbit R ISCO is then R ISCO = 6M. The innermost stable circular orbit is important in determining how much gravitational energy can be extracted from matter accreting onto neutron stars and black holes.

27 9.1. THE FORM OF THE METRIC 305 r + r + r r r + r + r r δφ Figure 9.7: Precessing orbits in a Schwarzschild metric Precession of Orbits An orbit closes if the angle ϕ sweeps out exactly 2π in the passage between two successive inner or two successive outer radial turning points. In Newtonian gravity the central potential is 1/r closed elliptical orbits. In the Schwarzschild metric the effective potential deviates from 1/r and orbits precess: ϕ changes by more than 2π between successive radial turning points.

28 306 CHAPTER 9. THE SCHWARZSCHILD SPACETIME To investigate this precession quantitatively we require an expression for dϕ/dr. From the energy equation E = 1 2 ) dr 2 +V effr) dr dτ dτ =± 2E V eff r)), and from the conservation equation forl, l=r 2 sin 2 θ dϕ dτ dϕ dτ = l r 2 sin 2 θ. Combining, recalling that we are choosing an orbital plane θ = π 2, dϕ dr = dϕ/dτ dr/dτ =± =± l [ r 2 2E =± l r 2 l r 2 2E V eff r)) 1 2M )1+ l2 r r 2 [ ε 2 1 2M r where we have used E = 1 2 ε2 1). ) ] 1/2 + 1 )] 1/2 )1+ l2 r 2,

29 9.1. THE FORM OF THE METRIC 307 The change in ϕ per orbit, ϕ, can be obtained by integrating over one orbit, ϕ = r+ dϕ r r dr dr+ dϕ r+ r + dr dr= 2 dϕ r dr dr r+ dr = 2l r r 2 r+ dr = 2l r r 2 [ ε 2 1 2M r )1+ l2 r 2 c2 ε 2 1)+ 2GM }{{ r } Newtonian l2 r 2 )] 1/2 + 2GMl2 c 2 r 3 }{{} correction 1/2 where in the last step G and c have been reinserted through the substitutions M GM c 2 l l c, Evaluation of the integral requires some care because the integrand tends to at the integration limits: From one of our earlier expressions [ dr dτ =± ε 2 1 2M r )] 1/2 )1+ l2 r 2, which is the denominator of our integrand. But the limits are turning points of the radial motion and dr/dτ = 0 at r + or r.

30 308 CHAPTER 9. THE SCHWARZSCHILD SPACETIME In the Solar System and most other applications the values of ϕ are very small, so it is sufficient to keep only terms of order 1/c 2 beyond the Newtonian approximation. Expanding the integrand and evaluating the integral with due care Exercise) yields Precession angle=δϕ ϕ 2π ) GM 2 6π rad/orbit. cl This may be expressed in more familiar classical orbital parameters: In Newtonian mechanics L=mr 2 ω, where L is the angular momentum and ω the angular frequency. For Kepler orbits ) l 2 L 2 = = r 2dϕ m dτ ) 2 r 2dϕ dt ) 2 = GMa1 e 2 ), where e is the eccentricity and a is the semimajor axis. This permits us to write δϕ = 6πGM ac 2 1 e 2 ) = M M ) AU a ) 1 1 e 2 rad/orbit,

31 9.1. THE FORM OF THE METRIC 309 The form of δϕ = 6πGM ac 2 1 e 2 ) shows explicitly that the amount of relativistic precession is enhanced by large M for the central mass, tight orbits small values of a), large eccentricities e. The precession observed for most objects is small. Precession of Mercury s orbit in the Sun s gravitational field because of general relativistic effects is observed to be 43 arcseconds per century. The orbit of the Binary Pulsar precesses by about 4.2 degrees per year. The precise agreement of both of these observations with the predictions of general relativity is a strong test of the theory.

32 310 CHAPTER 9. THE SCHWARZSCHILD SPACETIME Escape Velocity in the Schwarzschild Metric Consider a stationary observer at a Schwarzschild radial coordinate R who launches a projectile radially with a velocity v such that the projectile reaches infinity with zero velocity. This defines the escape velocity in the Schwarzschild metric. The projectile follows a radial geodesic since there are no forces acting on it The energy per unit rest mass is ε and it is conserved time invariance of metric). At infinity ε = 1, since then the particle is at rest and the entire energy is rest mass energy. Thus ε = 1 at all times since it is conserved. If u obs is the 4-velocity of the stationary observer, the energy measured by the observer is E = p u obs = mu u obs = mg µν u µ u ν obs = mg 00 u 0 u 0 obs, where p=mu, with p the 4-momentum and m the rest mass, and the last step follows because the observer is stationary. But g 00 = 1 2M ) }{{ r u 0 obs } = 1 2M ) 1/2 R u 0 = 1 2M ) 1 }{{}}{{ r } From metric Stationary observer From ε =1 2M r )u0 = 1

33 9.1. THE FORM OF THE METRIC 311 Therefore, E = mg 00 u 0 u 0 obs = m 1 2M ) ) r 1 2M 1 ) r 1 2M 1/2 r ) 1/2 = m 1 2M R But in the observer s rest frame E = mγ = m1 v 2 ) 1/2 so comparison yields 2M/R=v 2 and thus Notice that v esc = 2M This, coincidentally, is the same result as for Newtonian theory. At the Schwarzschild radius R=r S = 2M, the escape velocity is equal to c. This is the first hint of an event horizon in the Schwarzschild spacetime. R

34 312 CHAPTER 9. THE SCHWARZSCHILD SPACETIME Radial Fall of a Test Particle in Schwarzschild Geometry It will be instructive for later discussion to consider the particular case of a radial plunge orbit that starts from infinity with zero kinetic energy ε = 1) and zero angular momentum l=0). First, let us find an expression for the proper time as a function of the coordinate r. From earlier expressions E = ε2 1 2 = 1 2 ) dr 2 M dτ r + l2 Ml2 2r2 r 3, which implies for l=0and ε = 1, ) dr 2M 1/2 dτ =±. r Choosing the negative sign infalling orbit) and integrating with initial condition τr = 0)=0 gives Exercise) τ 2M = 2 32M) 3/2 r 3/2 for the proper time τ to reach the origin as a function of the initial Schwarzschild coordinate r.

35 9.1. THE FORM OF THE METRIC 313 That was the proper time τ. To find an expression for the coordinate time t as a function of r, we note that ε = 1 and is conserved. Then from we have that ε = 1= 1 2M r ) dt dτ dr 2M dτ =± r ) 1/2 dt dr = dt/dτ dr/dτ = 1 2M r ) 1 ) 2M 1/2, r which may be integrated to give Exercise) ) t = t 0 2M 2 r ) 3/2+ r ) 1/2+ r/2m) 1/2 1 2 ln 3 2M 2M r/2m) 1/2. + 1

36 314 CHAPTER 9. THE SCHWARZSCHILD SPACETIME r/m Proper time τ Schwarzschild coordinate time t r S = 2M -Time/M Figure 9.8: Comparison of proper time and Schwarzschild coordinate time for a particle falling radially in the Schwarzschild geometry. The proper time τ to fall to the origin is finite. For the same trajectory an infinite amount of coordinate time t elapses to reach the Schwarzschild radius. The smooth trajectory of the proper time through r S suggests that the apparent singularity of the metric there is not real. Later we shall introduce alternative coordinates that explicitly remove the singularity at r=2m but not at r= 0).

37 9.1. THE FORM OF THE METRIC Light Ray Orbits Calculation of light ray orbits in the Schwarzschild metric largely parallels that of particle orbits, except that u u=g µν dx µ dλ dx ν dλ = 0, not 1!) where λ is an affine parameter. For motion in the equatorial plane θ = 2 π ), this becomes explicitly 1 2M r ) ) dt M dλ r ) 1 ) dr 2 ) + r 2 dϕ 2 = 0. dλ dλ By analogy with the corresponding arguments for particle motion ε K t u= 1 2M r l=k ϕ u=r 2 sin 2 θ dϕ dλ, are conserved along the orbits of light rays. ) dt dλ, With a proper choice of normalization for λ, ε conserved) may be interpreted as the photon energy l conserved) is the photon angular momentum at infinity.

38 316 CHAPTER 9. THE SCHWARZSCHILD SPACETIME + Veff 0 Plunging orbit + Veff 0 Circular orbit + Veff 0 Scattering orbit r Orbi ts Figure 9.9: Effective potential for photons and light ray orbits in a Schwarzschild metric. The dotted lines on the left side give the value of 1/b 2 for each orbit. By following steps analogous to the derivation for particle orbits the equation of motion is 1 b 2 = 1 ) dr 2 l 2 +V effr) dλ ) V eff r) 1 r 2 1 2M r b 2 l2 ε 2. The effective potential for photons and some classes of light ray orbits in the Schwarzschild geometry are illustrated in Fig. 9.9.

39 9.1. THE FORM OF THE METRIC 317 r 1 φ φ r b δφ Figure 9.10: Deflection of light by an angle δ ϕ in a Schwarzschild metric Deflection of Light in a Gravitational Field Proceeding in a manner similar to that for the calculation of the precession angle for orbits of massive objects, we may calculate the deflection dϕ/dr for a light ray in the Schwarzschild metric. δϕ = 4GM M c 2 b = M ) R b ) radians. For a photon grazing the surface of the Sun, M = 1M and b=1r, which gives δϕ 1.75 arcseconds. Observation of this deflection during a total solar eclipse catapulted Einstein to worldwide fame almost overnight in the early 1920s.

40 318 CHAPTER 9. THE SCHWARZSCHILD SPACETIME 9.2 Shapiro time delay of light Light passing near a gravitating body follows a curved path and the time for light to travel between two points depends on this curvature. The deviation in travel time between that in the curved spacetime and the travel time if there were no curvature is termed the Shapiro time delay. This does not mean that the speed of light varies. The local speed of light is always c, but the observed elapsed time for light to go between two points in spacetime depends on the metric. Thus, measurement of this time delay is a test of general relativity.

41 9.2. SHAPIRO TIME DELAY OF LIGHT 319 To determine the time delay of light over a given path it is necessary to evaluate the integral of dt/dr. Proceeding in a similar manner as the earlier discussion of light deflection, we may use ε = 1 2M r ) dt dλ to write dt dr = dt/dλ dr/dλ =±ε 1 2M r =± 1 1 2M b r 1 b 2 = 1 l 2 dr dλ ) 2 +V effr) ) 1 [ )] 1 1/2 l 2 b 2 V eff ) 1 ) 1 1/2 b 2 V effr). This may be integrated to give the time for light to travel between specified points. In a typical Shapiro-delay experiment, radar waves are bounced off a planet and the time to go and return is measured for paths that pass very close to the surface of the Sun, or the delay in transmitting signals from space probes to Earth is measured as the signals pass near the Sun. The results of such experiments are consistent with the equation given above, thereby providing further confidence in the validity of general relativity.

42 320 CHAPTER 9. THE SCHWARZSCHILD SPACETIME 9.3 Gyroscopes in curved spacetime Consider the behavior of gyroscopes in free fall. These will follow a timelike geodesic, with a 4-velocity uτ) governed by the geodesic equation du λ dτ + Γλ µν uµ u ν = 0. In addition the gyroscope will have a spacelike spin 4- vector s µ =0, s) in this frame. In the freely-falling local inertial frame the 4-velocity components of the gyroscope are u=1,0,0,0), so s u= 0, which is a tensor equation and thus true in all frames. In flat spacetime or in a local inertial frame the spin is constant and ds µ /dτ = 0. In curved spacetime the appropriate covariant generalization gives an equation analogous to the geodesic equation, ds µ dτ + Γµ αβ sα u β = 0. This equation describes how the components of the gyroscopic spin s µ change along a geodesic and preserves the scalar product s u on the geodesic. As in classical mechanics the magnitude of the total spin is a constant of motion but the direction can precess in angle.

43 9.3. GYROSCOPES IN CURVED SPACETIME 321 Let us now use the equation ds µ dτ + Γµ αβ sα u β = 0. to investigate two predictions of general relativity that lead to precession of the spin vector for freely-falling gyroscopes in gravitational fields, geodetic precession and dragging of inertial frames.

44 322 CHAPTER 9. THE SCHWARZSCHILD SPACETIME 9.4 Geodetic precession Consider a gyroscope in a circular orbit around a non-rotating gravitating sphere of mass M. An observer comoving with the gyroscope in orbital free fall will find that the gyroscope precesses, even if the source of the field is not rotating. This is called geodetic precession because the precession is observed along a geodesic. Assume that Spacetime is described by the Schwarzschild metric, The radius for the orbit is R, and The spin points initially in the direction of a distant star. For an observer at rest in the gyroscope s frame The spin has only spatial components. By symmetry it must remain in the same plane. Choosing θ = 2 π to define this plane in Schwarzschild coordinates t, r, θ, ϕ), any precession occurs in the ϕ direction. For the 4-velocity u 0,u 1,u 2,u 3 ) u t,u r,u θ,u ϕ ) the component in the ϕ direction is u ϕ = dϕ dτ = dϕ dt dt dτ = Ωut Ω dϕ = dt M R 3 u t dt dτ, where Ω is the classical orbital angular velocity.

45 9.4. GEODETIC PRECESSION 323 The time evolution for the spin componentss t,s r,s θ,s ϕ ) will be governed by ds µ dτ + Γµ αβ sα u β = 0. evaluated in the Schwarzschild basis. Choose initially s θ = 0. It will remain zero because of symmetry. The component s t is related to the other spin components by the requirement that s u = 0 along the geodesic, implying that Exercise) s t = R2 Ω 1 2M/R sϕ. The remaining spin components s r and s ϕ require solution of the two equations ds r dτ + Γr αβ sα u β = 0, 9.1) ds ϕ dτ + Γϕ αβ sα u β = ) As you are asked to demonstrate Exercise), the solutions are s ϕ t)= s 0 1 2M Ω )sinωt) s r t)=s 0 1 2M R ωr R cosωt), where s 0 =s s) 1/2 is the invariant magnitude of the spin and ω 1 3M ) 1/2 Ω. R

46 324 CHAPTER 9. THE SCHWARZSCHILD SPACETIME World tube of planet or star t = P = 2π/Ω Time World line of satellite t = 0 Space Figure 9.11: Geodetic world line for gyroscope in orbit around a spherical mass. Imagine a gyroscope on a satellite in a circular Earth orbit. Assume that the spin of the gyroscope starts off at t = 0 pointing in the radial direction. The world lines for the gyroscope and Earth are illustrated in Fig

47 9.4. GEODETIC PRECESSION 325 World tube of planet or star t = P = 2π/Ω Time World line of satellite t = 0 Space After one complete orbit with a period t = P=2π/Ω, s r t = P)=s 0 1 2M/R) 1/2 cosωp) = s 0 1 2M/R) 1/2 cos 2π ω ). Ω In the absence of geodetic spin precession so that ω = Ω), the angle ϕ would change by 2π for one orbit, so the additional spin precession angle for each orbit is ϕ = 2π 2π ω Ω = 2π 1 ω ) = 2π Ω in the direction of the orbital motion, where [ 1 1 3M R ) 1/2 ], ω 1 3M R ) 1/2 Ω. was used in the last step.

48 326 CHAPTER 9. THE SCHWARZSCHILD SPACETIME World tube of planet or star t = P = 2π/Ω Time World line of satellite t = 0 Space For any object in the Solar System M/R=GM/Rc 2 is very small, so The square root can be expanded to give ) 1/2 ϕ = 2π 1 1 3M R 3πM R }{{} expand = 3πGM c 2 R rad orbit 1, for the geodetic precession per orbit for gyroscopes on a satellite in Earth orbit. Since the radial direction is perpendicular to the direction of orbital motion, this expression also gives the precession that would be measured by an observer comoving with the gyroscope in orbit. This geodetic precession effect is small, but cumulative for successive orbits.

49 9.4. GEODETIC PRECESSION km polar orbit Frame-dragging 39 mas/yr Earth Distant guide star IM Pegasi) Geodetic precession mas/yr Figure 9.12: Geodetic precession and frame dragging for a gyroscope on Gravity Probe B. Precession angles are greatly exaggerated. The star IM Pegasi was chosen as the directional reference because it was approximately in the desired direction for the gyroscopic spin axis and its proper motion on the celestial sphere was known precisely. Gravity Probe B GP-B) tested geodetic precession for gyroscopes aboard a satellite in an almost circular orbit averaging 642 km above the surface of the Earth. Figure 9.12 illustrates. The expected geodetic precession per orbit is arcsec orbit 1, corresponding to a predicted geodetic precession rate of ϕ/ t = 6.6 arcsec yr 1. The geodetic precession rate measured by GP-B was within 0.07% of this value.

50 328 CHAPTER 9. THE SCHWARZSCHILD SPACETIME 9.5 Gyroscopes in rotating spacetimes The Schwarzschild solution gives a spacetime valid outside any spherical, static, non-rotating body. But astronomical bodies are typically spinning. Thus it is important physically to ask about solutions of the Einstein equations for rotating spacetimes. Some objects are spinning slowly, which suggests that their exterior metric might be approximated by an expansion about the Schwarzschild spherical spacetime. On the other hand, in later chapters black holes will be encountered that are spinning with an angular momentum comparable to the maximum allowed by the laws of physics. These cannot in any sense be understood in terms of perturbations of the Schwarzschild metric. In the remainder of this chapter the simpler topic of very slowly rotating spherical spacetimes will be taken up; in later chapters the more complex issue of stronglydeformed metrics implying potentially large angular momentum will be addressed.

51 9.5. GYROSCOPES IN ROTATING SPACETIMES Slow rotation in the Schwarzschild metric As an example of a slowly-rotating astronomical body, consider the Sun. It rotates differentially with a period of about a month, somewhat faster at the equator than at the poles. The spacetime outside the Sun is described by the Schwarzschild solution provided that 1. it is a vacuum, 2. the gravitational effect of all other bodies in the Universe can be neglected, 3. the Sun is static with no spin, and 4. the Sun is spherical. To a very good approximation these conditions are satisfied, so the Schwarzschild metric is almost but not quite) valid outside the Sun. Let us now parse the not quite part of the preceding statement.

52 330 CHAPTER 9. THE SCHWARZSCHILD SPACETIME Take the exterior of the Sun to be a vacuum and ignore the effect of all other masses. Thus the deviation from the Schwarzschild metric outside the Sun is caused only by its 1) angular momentum and by its 2) deviation from spherical symmetry. Deviations from spherical symmetry are very small and caused classically by centripetal effects of its rotation. If the Schwarzschild metric were expanded in powers of the angular momentum J, Centripetal forces vary as ω 2 and so come in only at the level of the J 2 term in the expansion. Thus to first order in J the Sun would be expected to remain spherical as it rotates. However, recall that general relativity has many formal similarities with electromagnetism and that electromagnetic forces can arise from motion of charge. These are magnetic effects. In general relativity mass acts as gravitational charge. This suggests that gravitational forces that is, curvature of spacetime) may arise from motion of mass mass currents ), in addition to arising from the mass itself. This is the case, and forces arising from mass currents are called gravitomagnetic effects in general relativity.

53 9.5. GYROSCOPES IN ROTATING SPACETIMES 331 Assuming a slowly-rotating spherical field, expansion about the Schwarzschild metric to first order in J gives a metric ds 2 = [1 2Mr )] [ 1r 4J sin +O 2 dt 2 2 )] θ 1 +O r r 2 dϕ dt [ M )] 1 ) r +O r 2 dr 2 + r 2 dθ 2 + sin 2 θdϕ 2 ). Upon restoring factors of G and c, this may be written as ds 2 = ds 2 0 4GJ c 3 r 2 sin2 θrdϕ)cdt)+o J 2), where ds 2 0 is the contribution from the unperturbed Schwarzschild metric and O J 2) indicates that terms of order J 2 and higher have been discarded. For Newtonian theory J Mrv, where v is the linear velocity. Therefore, for the coefficient of the term proportional to J, which shows that GJ c 3 r 2 GMrv c 3 r 2 = v c GM c 2 r, The effect of mass motion on curvature is of order v/c relative to the primary effect caused by the mass itself, which is proportional to GM/c 2 r.

54 332 CHAPTER 9. THE SCHWARZSCHILD SPACETIME Gyroscope u z s J Planet y x Figure 9.13: A gyroscope in free fall on the rotation axis of a spherical planet in slow rotation. The initial 4-spin s of the gyroscope is perpendicular to the 4-velocity u, and the angular momentum of the planet is J Dragging of inertial frames Consider the following thought experiment: Imagine a spherical body in slow rotation, with a metric ds 2 = ds 2 0 4GJ c 3 r 2 sin2 θrdϕ)cdt)+o J 2), valid approximately in the spacetime surrounding it. Now imagine dropping a gyroscope from above the North Pole with an initial spin perpendicular to the rotation axis. Figure 9.13 illustrates.

55 9.5. GYROSCOPES IN ROTATING SPACETIMES 333 z Gyroscope s u J Planet y x Since spherical coordinates are singular on the z axis, it is convenient to work in cartesian coordinates, as indicated in the figure above. In cartesian coordinates the Schwarzschild metric takes the form ) ds 2 = ds 2 4GJ xdy ydx 0 cartesian) c 3 r 2cdt), r where ds 2 0 cartesian) is the unperturbed Schwarzschild metric. Only terms up to order 1/c 3 need be retained. Terms involving the mass M in the Schwarzschild metric will contribute at order 1/c 5, Thus for low angular momentum the unperturbed Schwarzschild metric can be replaced by its M 0 limit, which is just the flat Minkowski metric. Thus, it is sufficient to work with the approximate metric ds 2 = cdt) 2 + dx 2 + dy 2 + dz }{{} 2 4GJ ) xdy ydx c 3 r 2cdt). r Minkowski metric }{{} 1st order gravitomagnetic

56 334 CHAPTER 9. THE SCHWARZSCHILD SPACETIME Taking the spin to lie in the x y plane, initially so clearly u µ =u t,0,0,u z ) s µ =0,s x,s y,0), s u=0, and by symmetry arguments the spin will remain in the x y plane. The spin of the freely-falling gyroscope will be governed by ds µ dτ + Γµ αβ sα u β = 0. To leading order in 1/c the only contribution to the summation in the second term of this equation will be from terms with Γ 1 02 Γx ty = 2GJ c 2 z 3 Γ 2 01 Γy tx = 2GJ c 2 z 3 where r=zsince the gyroscope lies on the z axis. Then ds µ dτ + Γµ αβ sα u β = 0. yields the equations ds x dτ = Γx yt sy u t = 2GJ c 2 z 3 sy u t ds y dτ = Γy xts x u t = 2GJ c 2 z 3 sx u t. Utilizing u t = dt/dτ, this may be written ds x dt = 2GJ c 2 z 3 sy ds y dt = 2GJ c 2 z 3 sx, and the corresponding angular rate of precession in the x y plane is Ω LT = 2GJ c 2 z 3.

57 9.5. GYROSCOPES IN ROTATING SPACETIMES 335 This result was obtained in a Lorentz frame for which the source of the spherical field is at rest and the gyroscope is falling. However, it is valid also in the frame of the gyroscope because Lorentz boosts along the z axis do not affect the transverse spin components s x and s y, and There is no time dilation to leading order in c. This gyroscopic precession effect is called the Lense Thirring effect or frame dragging. The frame-dragging effect should be distinguished from geodetic precession, which is much larger and occurs even if the source for the gravitational field is not rotating. Gravity Probe B measured the frame dragging effect for Earth s slowly-rotating field and found a value within 5% of the prediction of general relativity.

58 336 CHAPTER 9. THE SCHWARZSCHILD SPACETIME 642 km polar orbit Frame-dragging 39 mas/yr Earth Distant guide star IM Pegasi) Geodetic precession mas/yr The following table compares the predictions of general relativity with measurements from Gravity Probe B for geodetic precession and frame dragging in a polar Earth orbit. Effect GP-B measurement General relativity Geodetic precession ±18.3 mas yr mas yr 1 Frame dragging 37.2±7.2 mas yr mas yr 1 Thus GP-B found geodetic precession within 0.07% and frame dragging within 5% of GR predictions.