User:Prokaryotic Caspase Homolog/sandbox anomalous precession

Anomalous perihelion precession of Mercury

Movement along geodesics

Figure 6–8. Calculus of variations

According to Newton's laws of motion, a planet orbiting the Sun would move in a straight line except for being pulled off course by the Sun's gravity. According to general relativity, there is no such thing as gravitational force. Rather, as discussed in section Basic propositions, a planet orbiting the Sun continuously follows the local "nearest thing to a straight line", which is to say, it follows a geodesic path.

Finding the equation of a geodesic requires knowing something about the calculus of variations, which is outside the scope of the typical undergraduate math curriculum, so we will not go into details of the analysis.

Determining the straightest path between two points resembles the task of finding the maximum or minimum of a function. In ordinary calculus, given the function $y=f(x),\,$ an "extremum" or "stationary point" may be found wherever the derivative of the function is zero.

In the calculus of variations, we seek to minimize the value of the functional between the start and end points. In the example shown in Fig. 6–8, this is by finding the function for which

$\delta \int _{A}^{B}ds=0$

where $\delta$ is the variation and the integral of $ds$ is the world-line.

Skipping the details of the derivation, the general formula for the equation of a geodesic is

$${\frac {d^{2}x^{\sigma }}{ds^{2}}}+\Gamma _{\alpha \beta }^{\sigma }{\frac {dx^{\alpha }}{ds}}{\frac {dx^{\beta }}{ds}}=0$$ (R1)

valid for all dimensionalities and shapes of space(time). As a geometric expression, the derivative is with respect to the line element, whereas classical theory involves time derivatives.

Let us consider a flat, three dimensional Euclidean space using Cartesian coordinates. For such a space,

$g_{11}=g_{22}=g_{33}=1\,$ and

$g_{\mu \nu }=0\,$ for $\mu \neq \nu$

The derivatives of the $g\,{\text{'s}}$ in the Christoffel symbol (K1) are all zero, so (R1) becomes

$${\frac {d^{2}x^{\sigma }}{ds^{2}}}=0\quad \quad (n=3)$$ (R2)

After replacing $ds$ by the proper time $dt$ (the time along the timelike world line, i.e. the time experienced by the moving object) and expanding R2, we get

$${\frac {d^{2}x^{1}}{dt^{2}}}=0,\quad {\frac {d^{2}x^{2}}{dt^{2}}}=0,\quad {\frac {d^{2}x^{3}}{dt^{2}}}=0$$ (R3)

which is to say, an object freely moving in Euclidean three-space travels with unaccelerated motion along a straight line.

Orbital motion: Stability of the orbital plane

Equation (R1) is a general expression for the geodesic. To apply it to the gravitational field around the Sun, the $g\,{\text{'s}}$ in the Christoffel symbols must be replaced with those specific to the Schwarzschild metric.

Equations (Q4) present the values of $\Gamma _{\alpha \beta }^{\sigma }$ in terms of $\lambda ,\,\nu ,\,r,\,\theta$ while (Q7) allows simplification of the expression to terms of $\nu ,\,r,\,\theta .\,$ Since $e^{\nu }=\gamma$ and (Q9) allows us to express $\gamma$ in terms of $r$, we can thus express $\Gamma _{\alpha \beta }^{\sigma }$ in terms of $r$ and $\theta .$

Remember that (R1) is actually four equations. In particular, $x^{\sigma }$ for $\sigma =2$ corresponds to $\theta$ in Fig. 6-7. Suppose we launched an object into orbit around the Sun with $\theta =\pi /2$ and an initial velocity in the $xy$ plane? How would the object subsequently behave? Equation (R1) for $x^{2}\equiv \theta$ becomes

$${\frac {d^{2}\theta }{ds^{2}}}+\Gamma _{\alpha \beta }^{2}{\frac {dx^{\alpha }}{ds}}{\frac {dx^{\beta }}{ds}}=0$$ (R4)

From (Q7), we know that the non-zero Christoffel symbols for $\sigma =2$ are

$\Gamma _{12}^{2}=\Gamma _{21}^{2}={\frac {1}{r}}$

and

$\Gamma _{33}^{2}=-\sin \theta \cdot \cos \theta$

so that in summing (R4) over all values of $\alpha$ and $\beta ,$ we get

$${\frac {d^{2}\theta }{ds^{2}}}+{\frac {2}{r}}{\frac {dr}{ds}}{\frac {d\theta }{ds}}-\sin \theta \cdot \cos \theta \left({\frac {d\phi }{ds}}\right)^{2}=0$$ (R5)

Since we stipulated an initial $\theta =\pi /2$ and an initial velocity in the $xy$ plane, $\cos \theta =0$ and $d\theta /ds=0$ so that (R5) becomes

$${\frac {d^{2}\theta }{ds^{2}}}=0$$ (R6)

In other words, a planet launched into orbit around the Sun remains in orbit around the same plane in which it was launched, the same as in Newtonian physics.

Orbital motion: Modified Keplerian ellipses

Starting with (R1), we explore the behavior of the other variables of the geodesic equation applied to the Schwarzschild metric:

For $\sigma =1,$ (R1) becomes

${\frac {d^{2}x^{1}}{ds^{2}}}+\Gamma _{11}^{1}\left({\frac {dx^{1}}{ds}}\right)^{2}+\Gamma _{22}^{1}\left({\frac {dx^{2}}{ds}}\right)^{2}$ $+\;\Gamma _{33}^{1}\left({\frac {dx^{3}}{ds}}\right)^{2}+\Gamma _{44}^{1}\left({\frac {dx^{4}}{ds}}\right)^{2}=0$

or

${\frac {d^{2}r}{ds^{2}}}+{\tfrac {1}{2}}\lambda '\left({\frac {dr}{ds}}\right)^{2}-re^{-\lambda }\left({\frac {d\theta }{ds}}\right)^{2}$ $-\;r\cdot \sin ^{2}\theta \cdot e^{-\lambda }\left({\frac {d\phi }{ds}}\right)^{2}+{\tfrac {1}{2}}e^{\nu -\lambda }\nu '\left({\frac {dt}{ds}}\right)^{2}=0$

Since we have stipulated that $\theta =\pi /2,\;$ $d\theta /ds=0\,$ and $\,\sin \theta =1,\,$ the above equation therefore becomes

$${\frac {d^{2}r}{ds^{2}}}+{\tfrac {1}{2}}\lambda '\left({\frac {dr}{ds}}\right)^{2}-re^{-\lambda }\left({\frac {d\phi }{ds}}\right)^{2}+{\tfrac {1}{2}}e^{\nu -\lambda }\nu '\left({\frac {dt}{ds}}\right)^{2}=0$$ (R7)

Likewise, for $\sigma =3\,$ and $\,\sigma =4,\,$ we get

$${\frac {d^{2}\phi }{ds^{2}}}+{\frac {2}{r}}{\frac {dr}{ds}}{\frac {d\phi }{ds}}=0$$ (R8)

$${\frac {d^{2}t}{ds^{2}}}+\nu '{\frac {dr}{ds}}{\frac {dt}{ds}}=0$$ (R9)

(Q10), (R7), (R8), and (R9) may be combined to get:

$$\left.{\begin{aligned}&{\frac {d^{2}u}{d\phi ^{2}}}+u={\frac {m}{h^{2}}}+3mu^{2}\\&r^{2}{\frac {d\phi }{ds}}=h\end{aligned}}\right\}$$ (R10)

where $m$ and $h$ are constants of integration and $u=1/r.$

The equations above are those of an object in orbit around a central mass. The second of the two equations is essentially a statement of the conservation of angular momentum. The first of the two equations is expressed in this form so that it may be compared with the Binet equation, devised by Jacques Binet in the 1800s while exploring the shapes of orbits under alternative force laws.

For an inverse square law, the Binet equation predicts, in agreement with Newton, that orbits are conic sections. Given a Newtonian inverse square law, the equations of motion are:

$$\left.{\begin{aligned}&{\frac {d^{2}u}{d\phi ^{2}}}+u={\frac {m}{h^{2}}}\\&r^{2}{\frac {d\phi }{dt}}=h\end{aligned}}\right\}$$ (R11)

where $m$ is the mass of the Sun, $r$ is the orbital radius, and $d\phi /dt$ is the angular velocity of the planet.

The relativistic equations for orbital motion (R10) are observed to be nearly identical to the Newtonian equations (R11) except for the presence of $3mu^{2}$ in the relativistic equations and the use of $ds$ rather than $dt.$

The Binet equation provides the physical meaning of $m,$ which we had introduced as an arbitrary constant of integration in the derivation of the Schwarzschild metric in (Q9).

Orbital motion: Anomalous precession

Fibure 6–9. Perihelion precession

The presence of the term $3mu^{2}$ in (R10) means that the orbit does not form a closed loop, but rather shifts slightly with each revolution, as illustrated (in much exaggerated form) in Fig. 6–9.

Now in fact, there are a number of effects in the Solar System that cause the perihelia of planets to deviate from closed Keplerian ellipses even in the absence of relativity. Newtonian theory predicts closed ellipses only for an isolated two-body system. The presence of other planets perturb each others' orbits, so that Mercury's orbit, for instance, would precess by slightly over 532 arcsec/century due to these Newtonian effects.

In 1859, Urbain Le Verrier, after extensive extensive analysis of historical data on timed transits of Mercury over the Sun's disk from 1697 to 1848, concluded that there was a significant excess deviation of Mercury's orbit from the precession predicted by these Newtonian effects amounting to 38 arcseconds/century (This estimate was later refined to 43 arcseconds/century by Simon Newcomb in 1882). Over the next half-century, extensive observations definitively ruled out the hypothetical planet Vulcan proposed by Le Verrier as orbiting between Mercury and the Sun that might account for this discrepancy.

Starting from (R10), the excess angular advance of Mercury's perihelion per orbit may be calculated:

$$\Delta \phi _{orbit}={\frac {6\pi m}{a(1-e^{2})}}={\frac {6\pi GM/c^{2}}{a(1-e^{2})}}\;,$$ (R12)

The first equality is in relativistic units, while the second equality is in MKS units. In the second equality, we replace $m,$ the geometric mass (units of length) with M, the mass in kilograms.

$G$ is the gravitational constant (6.672 × 10^-11 m³/kg-s²)

$M$ is the mass of the Sun (1.99 × 10³⁰ kg)

$c$ is the speed of light (2.998 × 10⁸ m/s)

$a$ is Mercury's perihelion (5.791 × 10¹⁰ m)

$e$ is Mercury's orbital eccentricity (0.20563)

We find that

$\Delta \phi _{orbit}=5.021\times 10^{-7}{\text{radian}}$

which works out to 43 arcsec/century.