Summary

DescriptionNewton versus Schwarzschild trajectories.gif	English: Comparison of a testparticle's trajectory in Newtonian and Schwarzschild spacetime in the strong gravitational field (r0=10rs=20GM/c²). The initial velocity in both cases is 126% of the circular orbital velocity. φ0 is the launching angle (0° is a horizontal shot, and 90° a radially upward shot). Since the metric is spherically symmetric the frame of reference can be rotated so that Φ is constant and the motion of the test-particle is confined to the r,θ-plane (or vice versa).
Date	21 May 2016
Source	Own work - Mathematica Code
Author	Yukterez (Simon Tyran, Vienna)
Other versions

Equations of motion

Newton

In spherical coordinates and natural units of ${\displaystyle {\text{G=M=c=1}}}$, where lengths are measured in ${\displaystyle {\text{GM/c}}^{2}}$ and times in ${\displaystyle {\text{GM/c}}^{3}}$, the motion of a testparticle in the presence of a dominant mass is defined by

${\displaystyle {\ddot {\text{r}}}=-{\frac {1}{{\text{r}}^{2}}}+{\text{r}}\ {\dot {\phi }}^{2}\ ,\ \ {\ddot {\phi }}=-{\frac {2\ {\dot {\text{r}}}\ {\dot {\phi }}}{\text{r}}}}$

The initial conditions are

${\displaystyle {\dot {\text{r}}}_{0}=v_{0}\sin(\varphi _{0})\ ,\ \ {\dot {\phi }}_{0}={\frac {v_{0}}{{\text{r}}_{0}}}\cos(\varphi _{0})}$

The overdot stands for the time-derivative. ${\displaystyle \phi }$ is the angular coordinate, ${\displaystyle \varphi }$ the local elevation angle of the test particle, and ${\displaystyle v}$ it's velocity.

${\displaystyle {\rm {E}}}$ and ${\displaystyle {\rm {L}}}$, where the kinetic ${\displaystyle {\text{E}}_{\rm {kin}}=v^{2}/2}$ and potential ${\displaystyle {\text{E}}_{\rm {pot}}=-1/{\rm {r}}}$ component (all in units of ${\displaystyle {\rm {mc}}^{2}}$) give the total energy ${\displaystyle {\rm {E}}={\text{E}}_{\rm {kin}}+{\text{E}}_{\rm {pot}}}$, and the angular momentum, which is given by ${\displaystyle {\rm {L}}=v_{\perp }\ {\rm {r}}}$ (in units of ${\displaystyle {\rm {m}}}$) where ${\displaystyle v_{\perp }={\dot {\phi }}\ {\text{r}}}$ is the transverse and ${\displaystyle v_{\parallel }={\dot {\text{r}}}}$ the radial velocity component, are conserved quantities.

Schwarzschild

The equations of motion ^[1] in Schwarzschild-coordinates are

${\displaystyle {\ddot {\text{r}}}=-{\frac {1}{{\text{r}}^{2}}}+{\text{r}}\ {\dot {\phi }}^{2}-3\ {\dot {\phi }}^{2}\ ,\ \ {\ddot {\phi }}=-{\frac {2\ {\dot {\text{r}}}\ {\dot {\phi }}}{\text{r}}}}$

which is except for the ${\displaystyle -3\ {\dot {\phi }}^{2}}$ term identical with Newton, although the radial coordinate has a different meaning (see farther below). The time dilation is

${\displaystyle {\dot {\text{t}}}={\frac {\sqrt {1+{\dot {\text{r}}}^{2}\ {\text{r}}/({\text{r}}-2)+{\text{r}}^{2}\ {\dot {\phi }}^{2}}}{\sqrt {1-2/{\text{r}}}}}={\frac {1}{{\sqrt {1-2/{\text{r}}}}{\sqrt {1-v^{2}}}}}}$

The coordinates are differentiated by the test particle's proper time ${\displaystyle \tau }$, while ${\displaystyle {\text{t}}}$ is the coordinate time of the bookkeeper at infinity. So the total coordinate time ellapsed between the proper time interval

${\displaystyle \tau _{0}..\tau _{1}}$ is ${\displaystyle {\text{t}}=\int _{\tau _{0}}^{\tau _{1}}{\dot {\text{t}}}\,\mathrm {d} \tau }$

The local velocity ${\displaystyle v}$ (relative to the main mass) and the coordinate celerity are related by

${\displaystyle {\dot {\phi }}={\frac {v_{\perp }}{{\text{r}}\ {\sqrt {1-v^{2}}}}}}$ for the input and ${\displaystyle v_{\perp }={\frac {{\dot {\text{r}}}\ {\dot {\phi }}}{{\dot {\text{t}}}\ {\sqrt {1-2/{\text{r}}}}}}}$ for the output of the transverse ${\displaystyle (\perp )}$ and

${\displaystyle {\dot {r}}=v_{\parallel }{\sqrt {\frac {1-2/{\text{r}}}{1-v^{2}}}}}$ or the other way around ${\displaystyle v_{\parallel }={\frac {\dot {\text{r}}}{{\dot {\text{t}}}\ (1-2/{\text{r}})}}}$ for the radial ${\displaystyle (\parallel )}$ component of motion.

The shapiro-delayed velocity ${\displaystyle {\text{v}}}$ in the bookeeper's frame of reference is

${\displaystyle {\text{v}}_{\perp }=v_{\perp }{\sqrt {1-2/{\text{r}}}}}$ and ${\displaystyle {\text{v}}_{\parallel }=v_{\parallel }(1-2/{\text{r}})}$

The initial conditions in terms of the local physical velocity ${\displaystyle v}$ are therefore

${\displaystyle {\dot {\text{r}}}_{0}={\frac {v_{0}{\sqrt {1-2/{\text{r}}_{0}}}\sin(\varphi _{0})}{\sqrt {1-v_{0}^{2}}}}\ ,\ \ {\dot {\phi }}_{0}={\frac {v_{0}\cos(\varphi _{0})}{{\text{r}}_{0}{\sqrt {1-v_{0}^{2}}}}}}$

The horizontal and vertical components differ by a factor of ${\displaystyle {\sqrt {1-2/{\text{r}}}}}$

because additional to the gravitational time dilation there is also a radial length contraction of the same factor, which means that the physical distance between

${\displaystyle {\text{r}}_{1}}$ and ${\displaystyle {\text{r}}_{2}}$ is not ${\displaystyle {\text{r}}_{2}-{\text{r}}_{1}}$ but ${\displaystyle \int _{{\text{r}}_{1}}^{{\text{r}}_{2}}{\frac {\mathrm {d} {\text{r}}}{\sqrt {1-2/{\text{r}}}}}}$

due to the fact that space around a mass is not euclidean, and a shell of a given diameter contains more volume when a central mass is present than in the absence of a such.

The angular momentum

${\displaystyle {\text{L}}={\text{r}}^{2}\ {\dot {\phi }}=v_{\perp }\ {\rm {r}}/{\sqrt {1-v^{2}}}}$

in units of ${\displaystyle {\text{m}}}$ and the total energy as the sum of rest-, kinetic- and potential energy

${\displaystyle {\text{E}}=1+{\text{E}}_{\rm {kin}}+{\text{E}}_{\rm {pot}}={\dot {\text{t}}}\ \left(1-{\frac {2}{\text{r}}}\right)={\frac {\sqrt {1-2/{\text{r}}}}{\sqrt {1-v^{2}}}}}$

in units of ${\displaystyle {\text{mc}}^{2}}$, where ${\displaystyle {\text{m}}}$ is the test particle's restmass, are the constants of motion. The components of the total energy are

${\displaystyle {\text{E}}_{\rm {kin}}={\frac {1}{\sqrt {1-v^{2}}}}-1}$ for the kinetic plus ${\displaystyle {\text{E}}_{\rm {pot}}={\frac {{\sqrt {1-2/{\text{r}}}}-1}{\sqrt {1-v^{2}}}}}$ for the potential energy plus ${\displaystyle {\text{m}}=1}$, the test particle's invariant rest mass.

The equations of motion in terms of ${\displaystyle {\text{E}}}$ and ${\displaystyle {\text{L}}}$ are

${\displaystyle {\dot {\rm {r}}}=\xi \ ,\ \ {\dot {\phi }}={\frac {\text{L}}{{\text{m}}\ {\text{r}}^{2}}}\ ,\ \ {\dot {\text{t}}}={\frac {\text{E}}{{\text{m}}\ (1-2/{\text{r}})}}}$

or, differentiated by the coordinate time ${\displaystyle {\text{t}}}$

${\displaystyle {\bar {\text{r}}}=\xi \ ,\ \ {\bar {\phi }}={\frac {(1-2/{\text{r}})\ {\text{L}}}{{\text{E}}\ {\text{r}}^{2}}}\ ,\ \ {\bar {\tau }}=1/{\dot {\text{t}}}}$

with

${\displaystyle \xi =\pm {\sqrt {{\frac {{\text{E}}^{2}\ {\text{r}}^{4}}{{\text{L}}^{2}}}-\left(1-{\frac {2}{\text{r}}}\right)\left({\frac {{\text{m}}^{2}\ {\text{r}}^{4}}{{\text{L}}^{2}}}+{\text{r}}^{2}\right)}}}$

where in contrast to the overdot, which stands for ${\displaystyle {\dot {x}}={\rm {d}}x/{\rm {d}}{\tau }}$, the overbar denotes ${\displaystyle {\bar {x}}={\rm {d}}x/{\rm {d}}{\text{t}}}$.

For massless particles like photons ${\displaystyle {\rm {m}}/{\sqrt {1-v^{2}}}}$ in the formula for ${\displaystyle {\rm {E}}}$ and ${\displaystyle {\rm {L}}}$ is replaced with ${\displaystyle {\rm {h\ f}}}$ and the ${\displaystyle {\rm {m}}}$ in the equations of motion set to ${\displaystyle 1}$, with ${\displaystyle {\rm {h}}}$ as Planck's constant and ${\displaystyle {\rm {f}}}$ for the photon's frequency.

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References

Usage in Wikipedia Articles

en.wikipedia.org

• Schwarzschild metric
• Schwarzschild geodesics
• Two-body problem in general relativity

de.wikipedia.org

• Apsidendrehung
• Schwarzschild-Metrik

ru.wikipedia.org

• Метрика Шварцшильда
• Задача Кеплера в относительности

es.wikipedia.org

• La relatividad general

zh.wikipedia.org

• 史瓦西度規