General Relativity Physics Honours 2009 Prof. Geraint F. Lewis Rm 560, A29 Lecture Notes 3

Lecture Notes 3 Schwarzschild Geometry When faced with the field equations, Einstein felt that it analytic solutions may be impossible. In 1916, Karl Schwarzschild derived the spherically symmetric vacuum solution, which describes the spacetime outside of any spherical, stationary mass distribution; Ch. 9 This is in Geometrized units, where G=c=1. Note, that the geometrized mass has units of length and so the curved terms in the invariant above are dimensionless.

Lecture Notes 3 Schwarzschild Geometry An examination of the Schwarzschild metric reveals; Time Independence: The metric has the same form for all values of t. Hence we have a Killing vector; Spherical Symmetry: This implies further Killing vectors, including one due to the independence of  Weak Field Limit: When M/r is small, the Schwarzschild metric becomes the weak field metric we saw earlier.

Lecture Notes 3 Schwarzschild Geometry Something clearly goes wrong at r=0 (the central singularity) and r=2M (Schwarzschild radius or singularity). More on this later. Remember: the (t,r,,) in this expression are coordinates and r is not the distance from any centre! If we choose a t=constant & r=constant we see the resulting 2-dimensional surface is a sphere (in 3-dimensions). So we can simply relate the area to r, but not the volume!

Lecture Notes 3 Particle Orbits Massive particles follow time-like geodesics, but understanding their motion is aided by identifying conserved quantities. Given our two Killing vectors we obtain At large r the first conserved quantity is the energy per unit mass in flat space; While the second is the angular momentum per unit mass.

Lecture Notes 3 Particle Orbits The conservation of l implies particles orbit in a plane. Consider an time-like geodesic passing through the point =0, with d/d=0. The conservation of l ensures d/d=0 along the geodesic and so the particle remains in the plane =0. But the spherical symmetry implies this is true for all orbits. Hence we will consider equatorial orbits with =/2 and u  =0. Defining the 4-velocity of the particle to be

Lecture Notes 3 Particle Orbits Substituting in our conserved quantities we get where This result differs from the Newtonian picture (found in any classical mechanics text) with the addition of the r -3 term in the potential! At large r the orbits become more Newtonian. Remember, while orbits are closed in r -1 potentials, they are not in general potentials.

Lecture Notes 3 Particle Orbits Considering the relativistic and Newtonian potentials, we see that while they agree at large radii, they are markedly different at small radii. The Newtonian has a single minima, while the relativistic has a minimum and maximum;

Lecture Notes 3 Particle Orbits For l/M<12 1/2, the extrema in the effective potential vanish, and so the particle, even if it has an angular component to its 4-velocity, will fall to the origin. This is contrast to the Newtonian potential as the present of any angular velocity ensures that it will miss the origin.

Lecture Notes 3 Example Orbits These example orbits are for l/M=4.3, with differing values of total energy. The first presents two circular orbits, with the inner one being unstable. In the next one we see that a non- circular orbit can precess, while the latter two show motion that is not seen in Newtonian r -1 potentials.

Lecture Notes 3 Radial Plunge Orbits Radial plunge orbits have no angular momentum (l=0) and follow a strictly radial path. If we assume that particle is at rest at r=, then The equation of motion becomes Using the time-like Killing vector, the 4-velocity is

Lecture Notes 3 Radial Plunge Orbits It’s quite straightforward to solve for r(); Which can be integrated to give Where So, according to its own clock, a particle falls from a coordinate r to the origin of the coordinate system in a finite amount of proper time (note this result is the same as the Newtonian!).

Lecture Notes 3 Radial Plunge Orbits What about r(t), where t is the coordinate time. Integrating gives; In terms of coordinate time, the position of the particle asymptotes to r=2M as t, and so the particle never crosses the Schwarzschild radius!

Lecture Notes 3 Stable Circular Orbits These occur at the minimum of the effective potential. As l/M becomes smaller, then the location of the minimum moves inwards. When l/M=12 1/2, this orbit is the Innermost Stable Circular Orbit and occurs at; The angular velocity in terms of coordinate time is For a circular orbit, dr/d=0 and E = V eff and

Lecture Notes 3 Stable Circular Orbits With this, and noting the location of the stable circular orbits are minima of the effective potential, then and Remembering the 4-velocity is

Lecture Notes 3 Precession The orbit of a test mass around a massive object do not precess in Newtonian physics. As we have seen, this is not true in relativity. We can take two equations; And assuming =/2, then

Lecture Notes 3 Precession In Newtonian physics, the angle swept out between aphelion and perihelion (the closest & furthest distance) is . We can calculate the corresponding angle in relativity by integrating over this formula between the turning points on the relativistic orbit. The peri- and aphelion are those points in the orbit where dr/d=0 and so it is where And so

Lecture Notes 3 Precession: Newtonian Limit We can examine this in the Newtonian limit to see what we would expect for Solar System planets. Putting the speed of light back in we get; The first term can be related to the Newtonian energy through Again, relativity introduces an additional term into the expression (the final term in the integral).

Lecture Notes 3 Precession: Newtonian Limit Without this final term, the angle swept out per orbit is always  = 2. Including it results in a shift of to first order in 1/c 2 (See assignment). Remember that l is related to the conservation of angular momentum and so we can rewrite the above expression in terms of the semimajor axis a and eccentricity  so

Lecture Notes 3 Light Ray Orbits The equivalence principle tells us that light rays should be influenced as they pass through a gravitational field. We can use the same geodesic formulism to study this. Firstly, we still have two conserved quantities due to the symmetries of the Schwarzschild metric. Ch. 9.4 Note that the derivatives are with respect to the affine parameter, not the proper time. Secondly we have the normalization of the 4-velocity

Lecture Notes 3 Light Ray Orbits So; again assuming =/2. Using the conserved quantities And so we can write;

Lecture Notes 3 Light Ray Orbits This has the same form as the massive particle orbits if we assume an effective potential of the form; And treating b -2 as an energy term. What is the physical meaning of b? Consider orbits that start at r>>2M; Also for large r, then And so d=b and b is the impact parameter of the orbit!

Lecture Notes 3 Light Ray Orbits The effective potential has a peak, and so unstable circular orbit, at Considering light rays starting from infinity, those with b -1 less than this scatter back to infinity, while those with more than this exceed the potential barrier and fall into the centre.

Lecture Notes 3 Escaping to Infinity Consider a source at r<3M emitting light in all directions. Some light will escape to infinity, while some will fall into the black hole. What is the critical angle at which light barely escapes? Example 9.2 We need to consider the light ray as seen in the orthonormal basis. Again, working in a plane where =/2, then;

Lecture Notes 3 Escaping to Infinity As the metric is diagonal, we can simply define the orthogonal basis vectors as; Hence, the photon 4-velocity in the orthonormal frame is (you should convince yourself that the 4-velocity of the photon in the orthonormal frame is null!)

Lecture Notes 3 Escaping to Infinity Each angle corresponds to a different value of b -1 and an examination of the potential shows that rays which escape to infinity have “energies” greater than the potential barrier. Hence the critical angle occurs at b 2 = 27M 2 and

Lecture Notes 3 Deflection of Light How much is light deflected by a massive, spherical object? From our conserved and geodesic equations we have; And so

Lecture Notes 3 Deflection of Light From infinity, the photon travels to a radius r 1, before heading out again. This radius occurs at And the angle swept out is; The smaller the impact parameter, the larger the deflection angle, to the point where the photon enters a circular orbit or falls into the centre.

Lecture Notes 3 Deflection of Light In solving for the deflection angle, introduce a new variable; Note, if M=0 then the resulting integral is , no deflection. Considering a light ray grazing the surface of the Sun; We can write the deflection angle as

Lecture Notes 3 Deflection of Light We can expand this expression out in the lowest order terms of 2M/b and get Remember, w 1 is the root of the denominator. The result is that the deflection is given by

Lecture Notes 3 Shapiro Time Delay The Shapiro time delay is apparent when photons are sent on a return path near a massive object. In the Solar System, this involves “bouncing” radar off a reflector (space ship or planet) located on the other side of the Sun, and seeing how long the signal takes to return. The result is different to what you would expect in flat (special relativisitic) spacetime.

Lecture Notes 3 Shapiro Time Delay As with the deflection of light, we can write And the total time taken for the trip is where and

Lecture Notes 3 Shapiro Time Delay As with the deflection of light, we can find the weak field limit of this integral which would apply in the Solar System The first term in this expression is simply the expected Newtonian time delay, and the other terms are a relativistic correction (but what is wrong with the above?). For photons grazing the Solar surface we get

Lecture Notes 3 Solar System Tests Chapter 10 discusses Solar System tests of general relativity, including measurements of the Parameterized-Post-Newtonian (PPN) parameters; these add higher terms to the metric and ‘extend’ relativity. For Einstein’s theory of relativity, these parameters must be exactly unity. While interesting, the contents of this chapter will not be examinable. However, you should read through the material. We will summarize the solar system tests.

Lecture Notes 3 Perihelion Shift of Mercury Mercury is the closest planet to the Sun, with a semi-major axis of 58 x 10 6 km and eccentricity of The orbit of Mercury has been known to precess for quite a while. The vast majority of the precession is due to Newtonian effects. However, a residual precession of ± 0.04 “/century could not be explained. The prediction from Einstein’s analysis of the orbit in the weak- field limit predicts;

Lecture Notes 3 Gravitational Lensing Einstein’s first prediction was that light would be deflected as it passed by massive objects. He calculated that a light ray grazing the Sun would be deflected by Made in 1916, this prediction could not be tested until the end of WWI. Eddington organized two expeditions to observed an eclipse in With the Moon blocking out the Sun, the positions of stars could be measured, agreeing (roughly) with Einstein’s prediction. Now measured to an accuracy of ~1%.

Lecture Notes 3 Gravitational Redshift The final test proposed by Einstein in 1916 was the gravitational redshift. This was finally measured by the Pound-Rebka experiment in 1959 by firing gamma rays up and down a 22m tower at Harvard. Measuring a frequency change of 1 part in 10 15, their measurement agreed with t Einstein prediction with an uncertainty of 10%. Five years later, the accuracy was improved to a 1% agreement and now measurements can accurately agree to less than a percent accuracy.

Lecture Notes 3 Shapiro Time Delay The Shapiro delay has also been measured using space probes, including Mariner in 1970 and Viking in The expected delay is of order 100s of microseconds over a total journey time of ~hrs, but atomic clocks are accurate to 1 part in A recent measurement using the Cassini space probe found the agreement to be (Bertotti et al 2003)