A geodesic is the 4-dimensional equivalent of straight line in spacetime. Given a metric and the corresponding Christoffel symbols, the geodesic equation is:
\[\begin{equation} \frac{d^2x^\lambda}{d\tau^2} + \Gamma^\lambda_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} = 0 \end{equation}\]
Spacetime is curved if any of the \(\Gamma\) terms are non-zero. If so, there is always an acceleration. This is the equivalence principle. The acceleration can be zero if the other terms of the geodesic equation evaluate to zero.
In the absence of an external force, any object will follow a geodesic which is the shortest distance between two points in spacetime. It is important to note that the passage of time can change continuously while following a geodesic.
The Schwarzschild Christoffel symbols for the outer solution are:
\[\begin{align*} \Gamma^{t}_{rt} &= \Gamma^{t}_{tr} = \frac{r_s}{2r(r - r_s)}\\ \Gamma^r_{tt} &= \frac{GM(r - r_s)}{r^3}\\ \Gamma^r_{rr} &= \frac{-r_s}{2r(r - r_s)}\\ \Gamma^r_{\theta\theta} &= -(r - r_s)\\ \Gamma^r_{\phi\phi} &= -\sin^2\theta(r - r_s)\\ \Gamma^\theta_{\theta r} &= \Gamma^\theta_{r\theta} = \frac{1}{r}\\ \Gamma^\theta_{\phi\phi} &= -\sin\theta\cos\theta\\ \Gamma^\phi_{\phi r} &= \Gamma^\phi_{r\phi} = \frac{1}{r}\\ \Gamma^\phi_{\theta\phi} &= \Gamma^\phi_{\phi\theta} = \cot\theta \end{align*}\]
Using the Christoffel symbols from the Schwarzschild exterior solution, geodesics for objects moving under gravity can be determined. Expanding the geodesic equations using only non-zero \(\Gamma\) terms gives:
\[\begin{align*} \frac{d^2t}{d\tau^2} &= -(\Gamma^t_{tr} + \Gamma^1_{rt})\frac{dt}{d\tau}\frac{dr}{d\tau}&&\\ &= -\frac{r_s}{r(r - r_s)}\frac{dt}{d\tau}\frac{dr}{d\tau}\\ \frac{d^2r}{d\tau^2} &= - \Gamma^r_{tt}\bigg(\frac{dt}{d\tau}\bigg)^2 - \Gamma^r_{rr}\bigg(\frac{dr}{d\tau}\bigg)^2 - \Gamma^r_{\theta\theta}\bigg(\frac{d\theta}{d\tau}\bigg)^2 - \Gamma^r_{\phi\phi}\bigg(\frac{d\phi}{d\tau}\bigg)^2&&\\ &= -\frac{c^2r_s(r - r_s)}{r^3}\bigg(\frac{dt}{d\tau}\bigg)^2 + \frac{r_s}{2r(r - r_s)}\bigg(\frac{dr}{d\tau}\bigg)^2 + (r -r_s)\bigg(\frac{d\theta}{d\tau}\bigg)^2 + (r - r_s)\sin^2 \theta\bigg(\frac{d\phi}{d\tau}\bigg)^2\\ \frac{d^2\theta}{d\tau^2} &= -(\Gamma^\theta_{r\theta} + \Gamma^\theta_{\theta r})\frac{dr}{d\tau}\frac{d\theta}{d\tau} - \Gamma^\theta_{\phi\phi}\bigg(\frac{d\phi}{d\tau}\bigg)^2&&\\ &= -\frac{2}{r}\frac{dr}{d\tau}\frac{d\theta}{d\tau} + \sin\theta \cos\theta\bigg(\frac{d\phi}{d\tau}\bigg)^2\\ \frac{d^2\phi}{d\tau^2} &= -(\Gamma^\phi_{r\phi} + \Gamma^\phi_{\phi r})\frac{dr}{d\tau}\frac{d\phi}{d\tau} -(\Gamma^\phi_{\theta\phi} + \Gamma^\phi_{\phi\theta})\frac{d\theta}{d\tau}\frac{d\phi}{d\tau}&&\\ &= -\frac{2}{r}\frac{dr}{d\tau}\frac{d\phi}{d\tau} - 2\cot\theta\frac{d\theta}{d\tau}\frac{d\phi}{d\tau} \end{align*}\]
\[\begin{align*} \frac{d^2t}{d\tau^2} &= -\frac{r_s}{r(r - r_s)}\frac{dt}{d\tau}\frac{dr}{d\tau}\\ \frac{d^2r}{d\tau^2} &= -\frac{c^2r_s(r - r_s)}{r^3}\bigg(\frac{dt}{d\tau}\bigg)^2 + \frac{r_s}{2r(r - r_s)}\bigg(\frac{dr}{d\tau}\bigg)^2 + (r -r_s)\bigg(\frac{d\theta}{d\tau}\bigg)^2 + (r - r_s)\sin^2 \theta\bigg(\frac{d\phi}{d\tau}\bigg)^2\\ \frac{d^2\theta}{d\tau^2} &= -\frac{2}{r}\frac{dr}{d\tau}\frac{d\theta}{d\tau} + \sin\theta \cos\theta\bigg(\frac{d\phi}{d\tau}\bigg)^2\\ \frac{d^2\phi}{d\tau^2} &= -\frac{2}{r}\frac{dr}{d\tau}\frac{d\phi}{d\tau} - 2\cot\theta\frac{d\theta}{d\tau}\frac{d\phi}{d\tau} \end{align*}\]
Setting \(\theta = \frac{\pi}{2}\), \(\frac{d\theta}{d\tau} = 0\) is a valid solution. The coordinates can be transformed so that the motion lies in that plane. All movement in the plane is geodesic. This simplifies the equations:
\[\begin{align*} \frac{d^2t}{d\tau^2} &= -\frac{r_s}{r(r - r_s)}\frac{dt}{d\tau}\frac{dr}{d\tau}\\ \frac{d^2r}{d\tau^2} &= -\frac{c^2r_s(r - r_s)}{r^3}\bigg(\frac{dt}{d\tau}\bigg)^2 + \frac{r_s}{2r(r - r_s)}\bigg(\frac{dr}{d\tau}\bigg)^2 + (r - r_s)\bigg(\frac{d\phi}{d\tau}\bigg)^2\\ \frac{d^2\theta}{d\tau^2} &= 0\\ \frac{d^2\phi}{d\tau^2} &= -\frac{2}{r}\frac{dr}{d\tau}\frac{d\phi}{d\tau} \end{align*}\]
To simplify the \(t\) equation define the function \(w(r)\).
\[w = 1 - \frac{r_s}{r}\] \[\frac{dw}{dr} = \frac{r_s}{r^2}\] \[\frac{1}{w}\frac{dw}{dr} = \frac{r_s}{r(r - r_s)}\]
The geodesic equation becomes:
\[\begin{align*} \frac{d^2t}{d\tau^2} &= -\frac{1}{w}\frac{dw}{dr}\frac{dt}{d\tau}\frac{dr}{d\tau}\\ \end{align*}\]
The \(t\) term can now be integrated:
\[\ln\frac{dt}{d\tau} = - \ln{w} + \ln\eta\]
\[\frac{dt}{d\tau} = \frac{\eta}{w} = \eta\bigg(1 - \frac{r_s}{r}\bigg)^{-1}\]
Where \(\eta\) is a constant of integration. This constant is in fact the conserved total energy of an object in free fall following a geodesic in curved spacetime \(\eta = \frac{E}{mc^2}\).
\[\bigg(1 - \frac{r_s}{r}\bigg)\frac{dt}{d\tau} = \frac{E}{mc^2}\]
The consequences of this are quite profound. The ratio of the total energy \(E\) and the mass-energy \(mc^2\) of an object following a geodesic is constant. It is the rate of change of time that changes with distance.
The energy \(E\) has three components:
Any changes to the radius cause corresponding changes to kinetic energy.
The \(\phi\) term can be integrated.
\[\ln\frac{d\phi}{d\tau} = -2\ln r + \ln h\]
\[r^2\frac{d\phi}{d\tau} = h\]
Where \(h\) is a constant of integration. It is the specific angular momentum \(h = \frac{L}{\mu}\), where \(\mu = \frac{Mm}{M + m}\) is the reduced mass.
Consider an object in a circular orbit. The orbit is at a fixed radius \(r\), \(\frac{dr}{d\tau} = 0, \frac{d^2r}{d\tau^2} = 0\). The angular velocity is constant \(\frac{d\phi}{d\tau} = \omega\).
The \(r\) geodesic becomes:
\[\frac{d^2r}{d\tau^2} = -\frac{c^2r_s(r - r_s)}{r^3}\bigg(\frac{dt}{d\tau}\bigg)^2 + (r - r_s)\omega^2 = 0\]
Substituting the calculated value for \(\frac{dt}{d\tau}\) gives:
\[\frac{GM\eta^2}{r^(r - r_s)} = (r - r_s)\omega^2\]
Given that \(\eta \approx 1, r_s \approx 0\) provided that the mass and velocity aren’t too large, then the solution agrees with Newton’s Law.
\[\frac{GM}{r^2} = r\omega^2\]
Consider an object falling vertically downwards. There is no angular motion \(\frac{d\phi}{d\tau} = 0\).
The \(r\) geodesic becomes:
\[\frac{d^2r}{d\tau^2} = -\frac{c^2r_s(r - r_s)}{r^3}\bigg(\frac{dt}{d\tau}\bigg)^2 + \frac{r_s}{2r(r - r_s)}\bigg(\frac{dr}{d\tau}\bigg)^2\]
Substituting the calculated value for \(\frac{dt}{d\tau}\) gives:
\[\frac{d^2r}{d\tau^2} = -\frac{GM\eta^2}{r^(r - r_s)} + \frac{r_s}{2r(r - r_s)}\bigg(\frac{dr}{d\tau}\bigg)^2\]
Given that \(\eta \approx 1, r_s \approx 0\) provided that the mass and velocity aren’t too large, then the solution agrees with Newton’s Law.
\[\frac{d^2r}{d\tau^2} = -\frac{GM}{r^2}\]
The final term in the equation show a reduction in acceleration as velocity increases.
The Schwarzschild metric defines curved spacetime with non-zero Christoffel symbols. A geodesic is the path that an object must follow from its current position in the absense of an external force. Non-zero Christoffel symbols mean that each of the four geodesic components describe an acceleration. The acceleration can be zero if the other terms are zero or cancel out.
The Schwarzschild metric describes a stretching of the time dimension and a corresponding contraction of the radial space dimension. Time slows when the time dimension is stretched as each time interval gets further apart.
The time geodesic component is a statement of the conservation of energy. An object’s energy remains the same at all points on a geodesic. Proper time passes slower as distance to the star or planet decreases.
Angular momentum around the star or planet is also conserved.
Any stable orbit around a massive body is a geodesic.
An object falling towards a massive body follows a geodesic. An object has to fall as the geodesic has an acceleration component towards the massive body. Although the object’s velocity increases as it accelerates, its total energy remains the same. The velocity is not Newton’s absolute velocity \(v = \frac{dr}{dt}\), it is the velocity in proper time \(\frac{dr}{d\tau}\).
So, what is weight? An object following a geodesic is in free fall and experiences no weight. That includes both orbiting and falling objects.
An object on the surface of a planet is on a geodesic with an acceleration pulling it down. The object can’t pass through the surface of the planet due to the electron degeneracy pressure caused by the Pauli Exclusion Principle. This pressure creates an upward force to counter the geodesic acceleration.
Weight is not the force of gravity pulling you down. It is the upward force exerted by the ground you stand on to counter the downward geodesic acceleration and stop you falling through it!