Christoffel symbols The non-vanishing
Christoffel symbols for the Schwarzschild-metric are: : \begin{align} \Gamma^t_{rt} = -\Gamma^r_{rr} &= \frac{r_\text{s}}{2r(r - r_\text{s})} \\[3pt] \Gamma^r_{tt} &= \frac{c^2 r_\text{s}(r - r_\text{s})}{2r^3} \\[3pt] \Gamma^r_{\phi\phi} &= (r_\text{s} - r)\sin^2(\theta) \\[3pt] \Gamma^r_{\theta\theta} &= r_\text{s} - r \\[3pt] \Gamma^\theta_{r\theta} = \Gamma^\phi_{r\phi} &= \frac{1}{r} \\[3pt] \Gamma^\theta_{\phi\phi} &= -\sin(\theta)\cos(\theta) \\[3pt] \Gamma^\phi_{\theta\phi} &= \cot(\theta) \end{align}
Geodesic equation According to Einstein's theory of general relativity, particles of negligible mass travel along
geodesics in the space-time. In flat space-time, far from a source of gravity, these geodesics correspond to straight lines; however, they may deviate from straight lines when the space-time is curved. The equation for the geodesic lines is : \frac{d^2x^{\lambda}}{d q^2} + \Gamma^{\lambda}_{\mu\nu} \frac{dx^{\mu}}{d q} \frac{dx^{\nu}}{dq} = 0 where Γ represents the
Christoffel symbol and the variable q parametrizes the particle's path through
space-time, its so-called
world line. The Christoffel symbol depends only on the
metric tensor g_{\mu\nu}, or rather on how it changes with position. The variable q is a constant multiple of the
proper time \tau for timelike orbits (which are traveled by massive particles), and is usually taken to be equal to it. For lightlike (or null) orbits (which are traveled by massless particles such as the
photon), the proper time is zero and, strictly speaking, cannot be used as the variable q. Nevertheless, lightlike orbits can be derived as the
ultrarelativistic limit of timelike orbits, that is, the limit as the particle mass
m goes to zero while holding its total
energy fixed. Therefore, to solve for the motion of a particle, the most straightforward way is to solve the geodesic equation, an approach adopted by Einstein and others. The Schwarzschild metric may be written as : c^{2}d\tau^{2} = w(r) c^2 dt^{2} - v(r) dr^{2} - r^{2} d\theta^{2} - r^{2} \sin^{2} \theta d\phi^{2} \, where the two functions w(r) = 1 - \frac{r_\text{s}}{r}and its reciprocal v(r)= \frac{1}{w(r)}are defined for brevity. From this metric, the Christoffel symbols \Gamma_{\mu\nu}^{\lambda}may be calculated, and the results substituted into the geodesic equations :\begin{align} 0 &= \frac{d^{2}\theta}{dq^{2}} + \frac{2}{r} \frac{d\theta}{dq} \frac{dr}{dq} - \sin \theta \cos \theta \left( \frac{d\phi}{dq} \right)^{2} \\[3pt] 0 &= \frac{d^{2}\phi}{dq^{2}} + \frac{2}{r} \frac{d\phi}{dq} \frac{dr}{dq} + 2 \cot \theta \frac{d\phi}{dq} \frac{d\theta}{dq} \\[3pt] 0 &= \frac{d^{2}t}{dq^{2}} + \frac{1}{w} \frac{dw}{dr} \frac{dt}{dq} \frac{dr}{dq} \\[3pt] 0 &= \frac{d^{2}r}{dq^{2}} - \frac{1}{v} \frac{dv}{dr} \left( \frac{dr}{dq} \right)^{2} - \frac{r}{v} \left( \frac{d\theta}{dq} \right)^{2} - \frac{r\sin^{2}\theta}{v} \left( \frac{d\phi}{dq} \right)^{2} + \frac{c^{2}}{2v} \frac{dw}{dr} \left( \frac{dt}{dq} \right)^{2} \end{align} It may be verified that \theta = \frac{\pi}{2} is a valid solution by substitution into the first of these four equations. By symmetry, the orbit must be planar, and we are free to arrange the coordinate frame so that the equatorial plane is the plane of the orbit. This \theta solution simplifies the second and fourth equations. To solve the second and third equations, it suffices to divide them by \frac{d\phi}{dq} and \frac{dt}{dq}, respectively. :\begin{align} 0 &= \frac{d}{dq} \left[ \ln \frac{d\phi}{dq} + \ln r^{2} \right] \\[3pt] 0 &= \frac{d}{dq} \left[ \ln \frac{dt}{dq} + \ln w \right], \end{align} which yields two constants of motion.
Lagrangian approach Because test particles follow geodesics in a fixed metric, the orbits of those particles may be determined using the calculus of variations, also called the Lagrangian approach. Geodesics in
space-time are defined as curves for which small local variations in their coordinates (while holding their endpoints events fixed) make no significant change in their overall length
s. This may be expressed mathematically using the
calculus of variations : 0 = \delta s = \delta \int ds = \delta \int \sqrt{g_{\mu\nu} \frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\tau}} d\tau = \delta \int \sqrt{2T} d\tau where
τ is the
proper time,
s =
cτ is the arc-length in
space-time and
T is defined as : 2T = c^{2} = \left( \frac{ds}{d\tau} \right)^{2} = g_{\mu\nu} \frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\tau} = \left( 1 - \frac{r_\text{s}}{r} \right) c^{2} \left( \frac{dt}{d\tau} \right)^{2} - \frac{1}{1 - \frac{r_\text{s}}{r}} \left( \frac{dr}{d\tau} \right)^{2} - r^{2} \left( \frac{d\varphi}{d\tau} \right)^{2} in analogy with
kinetic energy. If the derivative with respect to proper time is represented by a dot for brevity : \dot{x}^{\mu} = \frac{dx^{\mu}}{d\tau}
T may be written as : 2T = c^{2} = \left( 1 - \frac{r_\text{s}}{r} \right) c^{2} \left( \dot{t} \right)^{2} - \frac{1}{1 - \frac{r_\text{s}}{r}} \left( \dot{r} \right)^{2} - r^{2} \left( \dot{\varphi} \right)^{2} Constant factors (such as
c or the square root of two) don't affect the answer to the variational problem; therefore, taking the variation inside the integral yields
Hamilton's principle : 0 = \delta \int \sqrt{2T} d\tau = \int \frac{\delta T}{\sqrt{2T}} d\tau = \frac{1}{c} \delta \int T d\tau. The solution of the variational problem is given by
Lagrange's equations : \frac{d}{d\tau} \left(\frac{\partial T}{\partial \dot{x}^{\sigma}} \right) = \frac{\partial T}{\partial x^{\sigma}}. When applied to
t and
φ, these equations reveal two
constants of motion :\begin{align} \frac{d}{d\tau} \left[ r^{2} \frac{d\varphi}{d\tau} \right] &= 0, \\ \frac{d}{d\tau} \left[ \left( 1 - \frac{r_\text{s}}{r} \right) \frac{dt}{d\tau} \right] &= 0, \end{align} which may be expressed in terms of two constant length-scales, a and b :\begin{align} r^{2} \frac{d\varphi}{d\tau} &= ac, \\ \left( 1 - \frac{r_\text{s}}{r} \right) \frac{dt}{d\tau} &= \frac{a}{b}. \end{align} As shown
above, substitution of these equations into the definition of the
Schwarzschild metric yields the equation for the orbit.
Hamiltonian approach A Lagrangian solution can be recast into an equivalent Hamiltonian form. In this case, the Hamiltonian H is given by : 2 H = c^{2} = \frac{p_{t}^{2}}{c^{2} \left( 1 - \frac{r_\text{s}}{r} \right)} - \left( 1 - \frac{r_\text{s}}{r} \right) p_{r}^{2} - \frac{p_{\theta}^{2}}{r^{2}} - \frac{p_{\varphi}^{2}}{r^{2}\sin^{2} \theta} Once again, the orbit may be restricted to \theta = \frac{\pi}{2}by symmetry. Since t and \varphi do not appear in the Hamiltonian, their conjugate momenta are constant; they may be expressed in terms of the speed of light c and two constant length-scales a and b :\begin{align} p_{\varphi} &= -ac \\ p_{\theta} &= 0 \\ p_{t} &= \frac{ac^{2}}{b} \end{align} The derivatives with respect to proper time are given by :\begin{align} \frac{dr}{d\tau} &= \frac{\partial H}{\partial p_{r}} = - \left(1 - \frac{r_\text{s}}{r} \right) p_{r} \\ \frac{d\varphi}{d\tau} &= \frac{\partial H}{\partial p_{\varphi}} = \frac{-p_{\varphi}}{r^{2}} = \frac{ac}{r^{2}} \\ \frac{dt}{d\tau} &= \frac{\partial H}{\partial p_{t}} = \frac{p_{t}}{c^{2} \left( 1 - \frac{r_\text{s}}{r} \right)} = \frac{a}{b \left( 1 - \frac{r_\text{s}}{r} \right)} \end{align} Dividing the first equation by the second yields the orbital equation : \frac{dr}{d\varphi} = - \frac{r^{2}}{ac} \left(1 - \frac{r_\text{s}}{r} \right) p_{r} The radial momentum
pr can be expressed in terms of
r using the constancy of the Hamiltonian H = \frac{c^{2}}{2}; this yields the fundamental orbital equation : \left( \frac{dr}{d\varphi} \right)^{2} = \frac{r^{4}}{b^{2}} - \left( 1 - \frac{r_\text{s}}{r} \right) \left( \frac{r^{4}}{a^{2}} + r^{2} \right)
Hamilton–Jacobi approach The orbital equation can be derived from the
Hamilton–Jacobi equation. The advantage of this approach is that it equates the motion of the particle with the propagation of a wave, and leads neatly into the derivation of the deflection of light by gravity in
general relativity, through
Fermat's principle. The basic idea is that, due to gravitational slowing of time, parts of a wave-front closer to a gravitating mass move more slowly than those further away, thus bending the direction of the wave-front's propagation. Using general covariance, the
Hamilton–Jacobi equation for a single particle of unit mass can be expressed in arbitrary coordinates as : g^{\mu\nu} \frac{\partial S}{\partial x^{\mu}} \frac{\partial S}{\partial x^{\nu}} = c^{2}. This is equivalent to the Hamiltonian formulation above, with the partial derivatives of the action taking the place of the generalized momenta. Using the
Schwarzschild metric gμν, this equation becomes : \frac{1}{c^{2} \left(1 - \frac{r_\text{s}}{r} \right)} \left( \frac{\partial S}{\partial t} \right)^{2} - \left( 1 - \frac{r_\text{s}}{r} \right) \left( \frac{\partial S}{\partial r} \right)^{2} - \frac{1}{r^{2}} \left( \frac{\partial S}{\partial \varphi} \right)^{2} = c^{2} where we again orient the spherical coordinate system with the plane of the orbit. The time
t and azimuthal angle
φ are cyclic coordinates, so that the solution for Hamilton's principal function
S can be written : S = -p_{t} t + p_{\varphi} \varphi + S_{r}(r) \, where p_t and p_{\varphi} are the constant generalized momenta. The
Hamilton–Jacobi equation gives an integral solution for the radial part S_r(r) : S_{r}(r) = \int^{r} \frac{dr}{1 - \frac{r_\text{s}}{r}} \sqrt{\frac{p_{t}^{2}}{c^{2}} - \left( 1 - \frac{r_\text{s}}{r} \right) \left( c^{2} + \frac{p_{\varphi}^{2}}{r^{2}} \right)}. Taking the derivative of Hamilton's principal function
S with respect to the conserved momentum
pφ yields : \frac{\partial S}{\partial p_{\varphi}} = \varphi + \frac{\partial S_{r}}{\partial p_{\varphi}} = \mathrm{constant} which equals : \varphi - \int^{r} \frac{p_{\varphi} dr}{r^{2}\sqrt{\frac{p_{t}^{2}}{c^{2}} - \left( 1 - \frac{r_\text{s}}{r} \right) \left( c^{2} + \frac{p_{\varphi}^{2}}{r^{2}} \right)}} = \mathrm{constant} Taking an infinitesimal variation in φ and
r yields the fundamental orbital equation : \left( \frac{dr}{d\varphi} \right)^{2} = \frac{r^{4}}{b^{2}} - \left( 1 - \frac{r_\text{s}}{r} \right) \left( \frac{r^{4}}{a^{2}} + r^{2} \right). where the conserved length-scales
a and
b are defined by the conserved momenta by the equations :\begin{align} \frac{\partial S}{\partial \varphi} = p_{\varphi} &= -ac \\ \frac{\partial S}{\partial t} = p_{t} &= \frac{ac^{2}}{b} \end{align}
Hamilton's principle The
action integral for a particle affected only by gravity is : S = \int{ - m c^2 d\tau} = - m c \int{ c \frac{d\tau}{dq} dq} = - m c \int{ \sqrt{-g_{\mu\nu} \frac{dx^{\mu}}{dq} \frac{dx^{\nu}}{dq} } dq} where \tau is the
proper time and q is any smooth parameterization of the particle's world line. If one applies the
calculus of variations to this, one again gets the equations for a geodesic. To simplify the calculations, one first takes the variation of the square of the integrand. For the metric and coordinates of this case and assuming that the particle is moving in the equatorial plane \theta = \frac{\pi}{2}, that square is : \left(c \frac{d\tau}{dq}\right)^2 = - g_{\mu\nu} \frac{dx^{\mu}}{dq} \frac{dx^{\nu}}{dq} = \left( 1 - \frac{r_\text{s}}{r} \right) c^{2} \left( \frac{dt}{dq} \right)^{2} - \frac{1}{1 - \frac{r_\text{s}}{r}} \left( \frac{dr}{dq} \right)^{2} - r^{2} \left( \frac{d\varphi}{dq} \right)^{2} \,. Taking variation of this gives : \delta \left(c \frac{d\tau}{dq}\right)^2 = 2 c^{2} \frac{d\tau}{dq} \delta \frac{d\tau}{dq} = \delta \left[ \left( 1 - \frac{r_\text{s}}{r} \right) c^{2} \left( \frac{dt}{dq} \right)^{2} - \frac{1}{1 - \frac{r_\text{s}}{r}} \left( \frac{dr}{dq} \right)^{2} - r^{2} \left( \frac{d\varphi}{dq} \right)^{2} \right] \,.
Motion in longitude Vary with respect to longitude \varphi only to get : 2 c^{2} \frac{d\tau}{dq} \delta \frac{d\tau}{dq} = - 2 r^{2} \frac{d\varphi}{dq} \delta \frac{d\varphi}{dq} \,. Divide by 2 c \frac{d\tau}{dq} to get the variation of the integrand itself : c \, \delta \frac{d\tau}{dq} = - \frac{r^{2}}{c} \frac{d\varphi}{d\tau} \delta \frac{d\varphi}{dq} = - \frac{r^{2}}{c} \frac{d\varphi}{d\tau} \frac{d \delta \varphi}{dq} \,. Thus : 0 = \delta \int { c \frac{d\tau}{dq} dq } = \int { c \delta \frac{d\tau}{dq} dq } = \int { - \frac{r^{2}}{c} \frac{d\varphi}{d\tau} \frac{d \delta \varphi}{dq} dq } \,. Integrating by parts gives : 0 = - \frac{r^{2}}{c} \frac{d\varphi}{d\tau} \delta \varphi - \int { \frac{d}{dq} \left[ - \frac{r^{2}}{c} \frac{d\varphi}{d\tau} \right] \delta \varphi dq } \,. The variation of the longitude is assumed to be zero at the end points, so the first term disappears. The integral can be made nonzero by a perverse choice of \delta \varphi unless the other factor inside is zero everywhere. So the equation of motion is : \frac{d}{dq} \left[ - \frac{r^{2}}{c} \frac{d\varphi}{d\tau} \right] = 0 \,.
Motion in time Vary with respect to time t only to get : 2 c^{2} \frac{d\tau}{dq} \delta \frac{d\tau}{dq} = 2 \left( 1 - \frac{r_\text{s}}{r} \right) c^{2} \frac{dt}{dq} \delta \frac{dt}{dq} \,. Divide by 2 c \frac{d\tau}{dq} to get the variation of the integrand itself : c \delta \frac{d\tau}{dq} = c \left( 1 - \frac{r_\text{s}}{r} \right) \frac{dt}{d\tau} \delta \frac{dt}{dq} = c \left( 1 - \frac{r_\text{s}}{r} \right) \frac{dt}{d\tau} \frac{d \delta t}{dq} \,. Thus : 0 = \delta \int { c \frac{d\tau}{dq} dq } = \int { c \left( 1 - \frac{r_\text{s}}{r} \right) \frac{dt}{d\tau} \frac{d \delta t}{dq} dq } \,. Integrating by parts gives : 0 = c \left( 1 - \frac{r_\text{s}}{r} \right) \frac{dt}{d\tau} \delta t - \int { \frac{d}{dq} \left[ c \left( 1 - \frac{r_\text{s}}{r} \right) \frac{dt}{d\tau} \right] \delta t dq } \,. So the equation of motion is : \frac{d}{dq} \left[ c \left( 1 - \frac{r_\text{s}}{r} \right) \frac{dt}{d\tau} \right] = 0 \,.
Conserved momenta Integrate these equations of motion to determine the constants of integration getting :\begin{align} L = p_{\phi} &= m r^{2} \frac{d\varphi}{d\tau}\,, \\ E = - p_{t} &= m c^2 \left( 1 - \frac{r_\text{s}}{r} \right) \frac{dt}{d\tau}\,. \end{align} These two equations for the constants of motion L (angular momentum) and E (energy) can be combined to form one equation that is true even for
photons and other massless particles for which the
proper time along a geodesic is zero. : \frac{d\varphi}{dt} = \left( 1 - \frac{r_\text{s}}{r} \right) \frac{L \, c^2}{E \, r^2} \,.
Radial motion Substituting : \frac{d\varphi}{d\tau} = \frac{L}{m \, r^2} \, and : \frac{dt}{d\tau} = \frac{E}{\left( 1 - \frac{r_\text{s}}{r} \right) m \, c^2} \, into the metric equation (and using \theta = \frac{\pi}{2}) gives : c^{2} = \frac{1}{1 - \frac{r_\text{s}}{r}} \, \frac{E^2}{m^2 c^2} - \frac{1}{1 - \frac{r_\text{s}}{r}} \left( \frac{dr}{d\tau} \right)^{2} - \frac{1}{r^{2}} \, \frac{L^2}{m^2} \,, from which one can derive : {\left( \frac{dr}{d\tau} \right)}^{2} = \frac{E^2}{m^2 c^2} - \left( 1 - \frac{r_\text{s}}{r} \right) \left( c^{2} + \frac{L^2}{m^2 r^2} \right) \,, which is the equation of motion for r. The dependence of r on \varphi can be found by dividing this by :{\left( \frac{d\varphi}{d\tau} \right)}^2 = \frac{L^2}{m^2 r^4} to get : {\left( \frac{dr}{d\varphi} \right)}^{2} = \frac{E^2 r^4}{L^2 c^2} - \left( 1 - \frac{r_\text{s}}{r} \right) \left( \frac{m^2 c^{2} r^4}{L^2} + r^2 \right) \, which is true even for particles without mass. If length scales are defined by : a = \frac{L}{m \, c} and : b = \frac{L \, c}{E} \,, then the dependence of r on \varphi simplifies to :{\left( \frac{dr}{d\varphi} \right)}^{2} = \frac{r^4}{b^2} - \left( 1 - \frac{r_\text{s}}{r} \right) \left( \frac{r^4}{a^2} + r^2 \right) \,. ==See also==