Orbits in Black Hole Spacetimes - Math Problem Example

My 27 Dec 2007 Orbits in Black Hole Spacetimes This paper calculates the orbits of bodies and light rays in boththe Schwarzschild and the Reissner-Nordstrom spacetime metric. The Schwarzschild metric is a solution of the Einstein equations that describes the spacetime outside a spherical (nonrotating) star as well as the exterior of a nonrotating black hole. Given the metric, its geodesics (which in some sense are the straight lines on the spacetime) give all possible trajectories of light bodies without external forces. This provides an excellent approximation for the motion of the planets around the sun. The geodesics also give all possible light rays on the spacetime. Starting from the ordinary differential equations that geodesics obey, and which are discussed in the book Introducing Einstein’s Relativity by R. D’Inverno, this paper calculates: (i) the slow precession of the orbit of Mercury around the sun (see D’Inverno chapter 15.3), and (ii) the bending of light rays from distant stars as they pass close to the sun (DInverno chapter 15.4). This will involve solving the geodesic equations analytically to unveil effects are not predicted in Newton’s theory of gravity, which were the first two tests of general relativity (in 1915 and 1919). These calculations are taken further by extending to geodesics in Reissner-Nordstrom spacetime (the metric of a charged black hole). In the first section of this paper, the precession of Mercury’s orbit about the sun is calculated. The second section consists of the calculation of the bending of light rays from distant stars as they pass close to the sun. Sections three and four of this paper calculate the same orbits and light ray bending near a charged black hole, described by a Reissner-Nordstrom spacetime. 1. Advance of Perihelion of Mercury The planet Mercury orbits the Sun in a way that can be modeled by timelike geodesics of the Schwarzschild solution to Einstein’s equations. The Schwarzschild line element is (D’Inverno 188): where m = GM/c2. Using the variational method (see section 7.6 of D’Inverno), this leads to (D’Inverno 195): Timelike geodesics can be found from this equation by using the geodesic equation, which is obtained from the Euler-Lagrange equations: where u is an affine parameter and K is defined by: Using the three equations found from a = 0, 2 and 3, we get: Using the x-y plane, so that θ = π/2, then sin2 θ = 1, and the last equation becomes: In this case of θ = π/2, the time derivative of θ becomes 0: There are now four equations to find the four unknowns: t = t(τ) r = r(τ) θ = θ(τ) φ = φ (τ) Integrating the E-L equation for a = 3: this yields a constant: which is the conservation of angular momentum; the dot here is the derivative with respect to u, so that Similarly, integration of the E-L equation for a = 0: leads to Using this in the above equation for 2K eliminates the t dependence: Now we set u = 1/r, and take the derivative of this: Dividing by du and rearranging, we get: where we have used the result for the a=3 equation. Substituting this into the k2 equation, or Rearranging this yields: or after using or The above equation for (du/d)2 can be differentiated with respect to to obtain which is the general second order equation we must solve; the last term (3mu2) is the only term that is different from the Newtonian result. Using a prime notation, where this equation becomes where and we now use the perturbation method. The solution will have the form so that and Substituting this back into the first equation for u’’ gives or Equating the coefficients of different orders of ε to 0, the 0th order coefficients are and the general solution of this is where e is a constant, since and so that adding u0’’ and u0 agrees with the 0th order equation. Setting the first order coefficients to 0, we get Substituting in the solution for u0 gives where the half-angle relation 1/2 (cos 2 + 1)= cos2 was used. We try a solution of the form and take derivatives: Adding these results: Equating this with the equation involving the constant e above, Putting the values of these coefficients back into u1, yields so that to first order the general solution is choosing the term esin , which gets larger after each revolution, and neglecting the other terms, we have or so that the orbit has a period 2/(1-); for small , The precession of the orbit is found the difference of this and 2π, or 2πε , which is the angle amount that the axis of the orbit rotates after each orbit is traced out by the planet Mercury. This angle of precession is equal to: To find h, which is the angular momentum per unit mass, by squaring the equation found above: where the last equality is from Newtonian mechanics (Hartle 204) and e is the eccentricity of the ellipse. The precession becomes Using Kepler’s law of periods (Nave): where a is the semi-major axis, and T is the orbital period. Rearranging, so that Using the data for Mercury (Halliday and Resnick 345): a =5.79E10m , T=.241 yr, and the eccentricity (Halliday and Resnick A6): e = .206, =4.9839E8* 1.0055E-15 = 5.0114E-7 rad We multiply this by Mercury’s 415 orbits per century (Ales): precession = 5.0114E-7 rad *415 orbits/century =2.0797E-4 rad/century *1/(pi/10800)rad/min of arc*60 sec of arc/min of arc = 42.89 sec of arc/century where we have used the conversion factor 1 minute of arc = 1MOA = (pi/10800) rad (Wikipedia). The value of the precession, 2πε, is therefore predicted to be 42.98 seconds of arc per century. 2. Bending of Light by the Sun The light rays of distant stars are bent by the Sun, and this is described by a Schwarzschild spacetime outside the sun. This calculation is similar to the calculation of the previous section, except that a null geodesic is used here. For a null geodesic, and the differentiation (dots) are with respect to an affine parameter. Using the x-y plane so that =/2, the sin = 1 and By using the same equations for t-dot and -dot found in the previous section, and we can substitute these into the equation for 2K in order to find a general equation to solve. This method results in, for null geodesics (Foster and Nightingale 144), where F is a constant. Differentiating this: On simplifying, this becomes the second-order differential equation : or which is the general second-order differential equation we must solve; it is actually a more simplified case of the previous section. In the special relativistic limit, m= 0, and this equation becomes: The general solution of this equation is: for D = constant, since and so that adding the first and second derivatives of u with respect to yields zero as expected. This general solution for u is the equation for straight line motion. The light ray in Schwarzschild spacetime is a perturbation of this classical result, so that our solution will have the form where u0 is the classical result and we take 0 = 0. Taking derivatives, From the general solution, Neglecting higher terms of order (mu)2, and using u0’’ = -u0, From this a solution for u1 is found: since To check this solution, as expected. The general solution is formed from the solutions for u0 and u1: where C is a constant of integration. To find the angle of deflection (= ) of a light ray near the Sun, 1 is defined as the angle between the observer and the Sun, and 2 is the angle between the star that emitted the light rays and the Sun; summing the angles 1 and 2 gives the angle of deflection . The two values of (which is the angle to the light ray, taking the Sun as the origin) are taken to be -1 and + 2 , when essentially r goes to infinity and u goes to 0. After using small angle formulae for 1 and 2 : sin (-1 ) ~ -1 and sin ( + 2 )~ -2 , cos (-1 )~ 1 and cos ( + 2 )~ -1 cos2 (-1 )~ 1 and cos2 ( + 2 )~ 1 two equations are found: for = -1 : and for = + 2 : Adding these two equations eliminates the constant C: so that Making the substitution m = GM/c2, and using D = b = impact parameter, The smallest value of b is for light rays just grazing the Sun, so that b is effectively the radius of the Sun (b = 6.96E8m) (Hartle 211).This deflection can be found using data from the Sun (Halliday and Resnick A5): M = 1.99E30 kg and the gravitational constant is G = 6.67 E-11m3/s2*kg, so that = 4 (6.67 E-11m3/s2*kg)(1.99E30 kg)/[(3E8m/s)26.96E8m] = 8.4759E-6 rad*(10800*60/pi)seconds of arc/rad = 1.75 seconds of arc so that light rays grazing the Sun are predicted to be deflected by 1.75 seconds of arc. 3. Precession of an Orbit Around a Charged Black Hole In this section, we consider a body in orbit around a charged black hole, and calculate the precession of the orbit using a Reissner-Nordstrom spacetime. The Reissner-Nordstrom metric is (D’Inverno 240): which reduces to the Schwarzschild metric when q = 0. Using the variational method, we obtain: And from the Euler-Lagrange equations, three more equations are obtained (again, for a = 0,2,3): The last two equations (for a = 2 and a = 3) are unchanged, since the - and -dependence of the metric was not changed. Because of this, many of the results of section 1 are unchanged here. For = /2 we again get: so that The main change in this section is that the a = 0 E-L equation now yields: Now when we eliminate the t-dependence in the 2K equation we obtain: If we again set u = 1/r, or Rearranging this yields: or Since d/du = hu2, this becomes or so that the result is the same as that for section 1 except for the two extra terms in q. Differentiating with respect to gives which after simplifying becomes Using the prime notation, this is . The boxed equation above is the general second-order differential equation that we must solve. We want to find solutions for using the perturbation method. where the small parameters are and = 2q2/h2 The general solution of where is found to be By taking derivatives of u0, it can be checked that this solves the 0th order equation above. Now we want to treat the extra term, -2q2u3 as a perturbation of the solution found in section 1 (with the general solution slightly modified as shown above). Taking derivatives of u, Putting these back into the boxed equation above, Neglecting higher terms and using u0’’ +Au0 = m/h2, so that there are 2 equations to solve to find u1 and v1: terms : terms: The solutions are and so that the total solution is choosing the term esin and the e cos term, which gets larger after each revolution, and neglecting the other - and -terms, we have which will change the period from where the orbit has a period 2/(1-); for small , to a new orbital period T where so that the precession is affected by the perturbation of the -terms. 4. Bending of Light Rays Near a Charged Black Hole In this section we recalculate the bending of light rays (calculated in section 2 with a Schwarzschild metric) by using a Reissner-Nordstrom spacetime metric instead of the Schwarzschild spacetime metric. Since light rays follow a null geodesic, 2K is again set to 0: and the differentiation (dots) are with respect to an affine parameter. Using the x-y plane so that =/2, the sin = 1 and Using the same relations for t-dot and -dot found in the previous section for R-N metrics, namely These are substituted into the equation for 2K: Using the equation found previously for r-dot: Using u = 1/r: Now differentiating this with respect to , so that Using the prime notation, this becomes which is the general second-order differential equation that we need to solve in this case. The general solution results from considering and leads to as in previous sections. It remains to find the specific solution, where using the perturbation method. The small parameters are = m/D and = q2/D2 Taking derivatives of u, Putting these back into the boxed equation above, Neglecting higher terms and using u0’’ = -u0, so that there are 2 equations to solve to find u1 and v1: terms : terms: The solutions of these are and so that the total solution is As before, letting the two values of to be -1 and + 2 , when essentially r goes to infinity and u goes to 0. After using small angle formulae for 1 and 2 : sin (-1 ) ~ -1 and sin ( + 2 )~ -2 , cos (-1 )~ 1 and cos ( + 2 )~ -1 cos2 (-1 )~ 1 and cos2 ( + 2 )~ 1 two equations are found: for = -1 : and for = + 2 : Adding the above 2 equations: The electric charge of the Reissner-Nordstrom black hole decreases the light ray deflection of the Schwarzschild case, as can be seen by the second term above. Works Cited Ales, Jorge. “Precession of Mercury Perihelion.” 2002. Cosmological Model of the Living Universe. 27 Dec 2007 D’Inverno, Ray. Introducing Einstein’s Relativity. Oxford: Clarendon Press, 1995. Foster, J. and Nightingale, J.D. A Short Course in General Relativity. 2nd Ed. New York: Springer, 1995. Halliday, David and Resnick, Robert. Fundamentals of Physics. 3rd Ed. Extended. New York: John Wiley and Sons, 1988. Hartle, James B. Gravity: An Introduction to Einstein’s General Relativity. San Francisco: Addison Wesley, 2003. Nave, C.R. “Kepler’s Laws.” HyperPhysics. 27 Dec 2007 Wikipedia. “Minute of arc.” 27 Dec 2007 Read More