ОписаниеNewton 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).
Дата	21 мая 2016
Источник	Собственная работа - Mathematica Code
Автор	Yukterez (Simon Tyran, Vienna)
Другие версии

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

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

The initial conditions are

${\displaystyle {\dot{\text{r}}}_0=v_0\sin ({\phi }_0),{\dot{\varphi }}_0=\frac{v_0}{{\text{r}}_0}\cos ({\phi }_0)}$

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

${\displaystyle \mathrm{E}}$ and ${\displaystyle \mathrm{L}}$, where the kinetic ${\displaystyle {\text{E}}_{\mathrm{k}\mathrm{i}\mathrm{n}}=v^2/2}$ and potential ${\displaystyle {\text{E}}_{\mathrm{p}\mathrm{o}\mathrm{t}}=-1/\mathrm{r}}$ component (all in units of ${\displaystyle {\mathrm{m}\mathrm{c}}^2}$) give the total energy ${\displaystyle \mathrm{E}={\text{E}}_{\mathrm{k}\mathrm{i}\mathrm{n}}+{\text{E}}_{\mathrm{p}\mathrm{o}\mathrm{t}}}$, and the angular momentum, which is given by ${\displaystyle \mathrm{L}=v_{\perp }\mathrm{r}}$ (in units of ${\displaystyle \mathrm{m}}$) where ${\displaystyle v_{\perp }=\dot{\varphi }\text{r}}$ is the transverse and ${\displaystyle v_{\parallel }=\dot{\text{r}}}$ the radial velocity component, are conserved quantities.

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

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

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

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

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

${\displaystyle {\tau }_0\mathrm{..}{\tau }_1}$ 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

${\displaystyle \dot{\varphi }=\frac{v_{\perp }}{\text{r}\sqrt{1-v^2}}}$ for the input and ${\displaystyle v_{\perp }=\frac{\dot{\text{r}}\dot{\varphi }}{\dot{\text{t}}\sqrt{1-2/\text{r}}}}$ 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

${\displaystyle {\dot{\text{r}}}_0=\frac{v_0\sqrt{1-2/{\text{r}}_0}\sin ({\phi }_0)}{\sqrt{1-v_0^2}},{\dot{\varphi }}_0=\frac{v_0\cos ({\phi }_0)}{{\text{r}}_0\sqrt{1-v_0^2}}}$

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

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

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

${\displaystyle \text{E}=1+{\text{E}}_{\mathrm{k}\mathrm{i}\mathrm{n}}+{\text{E}}_{\mathrm{p}\mathrm{o}\mathrm{t}}=\dot{\text{t}}(1-\frac{2}{\text{r}})=\frac{\sqrt{1-2/\text{r}}}{\sqrt{1-v^2}}}$

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

${\displaystyle {\text{E}}_{\mathrm{k}\mathrm{i}\mathrm{n}}=\frac{1}{\sqrt{1-v^2}}-1}$ for the kinetic plus ${\displaystyle {\text{E}}_{\mathrm{p}\mathrm{o}\mathrm{t}}=\frac{\sqrt{1-2/\text{r}}-1}{\sqrt{1-v^2}}}$ 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

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

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

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

with

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

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

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

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