File:Newton versus Schwarzschild trajectories.gif
Newton_versus_Schwarzschild_trajectories.gif (777 × 514 pixels, file size: 7.97 MB, MIME type: image/gif, looped, 250 frames, 15 s)
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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 testparticle is confined to the r,θplane (or vice versa). 
Date  
Source  Own work  Mathematica Code 
Author  Yukterez (Simon Tyran, Vienna) 
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Equations of motion[edit]
Newton[edit]
In spherical coordinates and natural units of ${\text{G=M=c=1}}$, where lengths are measured in ${\text{GM/c}}^{2}$ and times in ${\text{GM/c}}^{3}$, the motion of a testparticle in the presence of a dominant mass is defined by
${\ddot {\text{r}}}={\frac {1}{{\text{r}}^{2}}}+{\text{r}}\ {\dot {\phi }}^{2}\ ,\ \ {\ddot {\phi }}={\frac {2\ {\dot {\text{r}}}\ {\dot {\phi }}}{\text{r}}}$
The initial conditions are
${\dot {\text{r}}}_{0}=v_{0}\sin(\varphi _{0})\ ,\ \ {\dot {\phi }}_{0}={\frac {v_{0}}{{\text{r}}_{0}}}\cos(\varphi _{0})$
The overdot stands for the timederivative. $\phi$ is the angular coordinate, $\varphi$ the local elevation angle of the test particle, and $v$ it's velocity.
${\rm {E}}$ and ${\rm {L}}$, where the kinetic ${\text{E}}_{\rm {kin}}=v^{2}/2$ and potential ${\text{E}}_{\rm {pot}}=1/{\rm {r}}$ component (all in units of ${\rm {mc}}^{2}$) give the total energy ${\rm {E}}={\text{E}}_{\rm {kin}}+{\text{E}}_{\rm {pot}}$, and the angular momentum, which is given by ${\rm {L}}=v_{\perp }\ {\rm {r}}$ (in units of ${\rm {m}}$) where $v_{\perp }={\dot {\phi }}\ {\text{r}}$ is the transverse and $v_{\parallel }={\dot {\text{r}}}$ the radial velocity component, are conserved quantities.
Schwarzschild[edit]
The equations of motion ^{[1]} in Schwarzschildcoordinates are
${\ddot {\text{r}}}={\frac {1}{{\text{r}}^{2}}}+{\text{r}}\ {\dot {\phi }}^{2}3\ {\dot {\phi }}^{2}\ ,\ \ {\ddot {\phi }}={\frac {2\ {\dot {\text{r}}}\ {\dot {\phi }}}{\text{r}}}$
which is except for the $3\ {\dot {\phi }}^{2}$ term identical with Newton, although the radial coordinate has a different meaning (see farther below). The time dilation is
${\dot {\text{t}}}={\frac {\sqrt {1+{\dot {\text{r}}}^{2}\ {\text{r}}/({\text{r}}2)+{\text{r}}^{2}\ {\dot {\phi }}^{2}}}{\sqrt {12/{\text{r}}}}}={\frac {1}{{\sqrt {12/{\text{r}}}}{\sqrt {1v^{2}}}}}$
The coordinates are differentiated by the test particle's proper time $\tau$, while ${\text{t}}$ is the coordinate time of the bookkeeper at infinity. So the total coordinate time ellapsed between the proper time interval
$\tau _{0}..\tau _{1}$ is ${\text{t}}=\int _{\tau _{0}}^{\tau _{1}}{\dot {\text{t}}}\,\mathrm {d} \tau$
The local velocity $v$ (relative to the main mass) and the coordinate celerity are related by
${\dot {\phi }}={\frac {v_{\perp }}{{\text{r}}\ {\sqrt {1v^{2}}}}}$ for the input and $v_{\perp }={\frac {{\dot {\text{r}}}\ {\dot {\phi }}}{{\dot {\text{t}}}\ {\sqrt {12/{\text{r}}}}}}$ for the output of the transverse $(\perp )$ and
${\dot {r}}=v_{\parallel }{\sqrt {\frac {12/{\text{r}}}{1v^{2}}}}$ or the other way around $v_{\parallel }={\frac {\dot {\text{r}}}{{\dot {\text{t}}}\ (12/{\text{r}})}}$ for the radial $(\parallel )$ component of motion.
The shapirodelayed velocity ${\text{v}}$ in the bookeeper's frame of reference is
${\text{v}}_{\perp }=v_{\perp }{\sqrt {12/{\text{r}}}}$ and ${\text{v}}_{\parallel }=v_{\parallel }(12/{\text{r}})$
The initial conditions in terms of the local physical velocity $v$ are therefore
${\dot {\text{r}}}_{0}={\frac {v_{0}{\sqrt {12/{\text{r}}_{0}}}\sin(\varphi _{0})}{\sqrt {1v_{0}^{2}}}}\ ,\ \ {\dot {\phi }}_{0}={\frac {v_{0}\cos(\varphi _{0})}{{\text{r}}_{0}{\sqrt {1v_{0}^{2}}}}}$
The horizontal and vertical components differ by a factor of ${\sqrt {12/{\text{r}}}}$
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
${\text{r}}_{1}$ and ${\text{r}}_{2}$ is not ${\text{r}}_{2}{\text{r}}_{1}$ but $\int _{{\text{r}}_{1}}^{{\text{r}}_{2}}{\frac {\mathrm {d} {\text{r}}}{\sqrt {12/{\text{r}}}}}$
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
${\text{L}}={\text{r}}^{2}\ {\dot {\phi }}=v_{\perp }\ {\rm {r}}/{\sqrt {1v^{2}}}$
in units of ${\text{m}}$ and the total energy as the sum of rest, kinetic and potential energy
${\text{E}}=1+{\text{E}}_{\rm {kin}}+{\text{E}}_{\rm {pot}}={\dot {\text{t}}}\ \left(1{\frac {2}{\text{r}}}\right)={\frac {\sqrt {12/{\text{r}}}}{\sqrt {1v^{2}}}}$
in units of ${\text{mc}}^{2}$, where ${\text{m}}$ is the test particle's restmass, are the constants of motion. The components of the total energy are
${\text{E}}_{\rm {kin}}={\frac {1}{\sqrt {1v^{2}}}}1$ for the kinetic plus ${\text{E}}_{\rm {pot}}={\frac {{\sqrt {12/{\text{r}}}}1}{\sqrt {1v^{2}}}}$ for the potential energy plus ${\text{m}}$, the test particle's invariant rest mass.
The equations of motion in terms of ${\text{E}}$ and ${\text{L}}$ are
${\dot {\rm {r}}}={\dot {\phi }}\ \xi \ ,\ \ {\dot {\phi }}={\frac {\text{L}}{{\text{m}}\ {\text{r}}^{2}}}\ ,\ \ {\dot {\text{t}}}={\frac {\text{E}}{{\text{m}}\ (12/{\text{r}})}}$
or, differentiated by the coordinate time ${\text{t}}$
${\bar {\text{r}}}={\bar {\phi }}\ \xi \ ,\ \ {\bar {\phi }}={\frac {(12/{\text{r}})\ {\text{L}}}{{\text{E}}\ {\text{r}}^{2}}}\ ,\ \ {\bar {\tau }}=1/{\dot {\text{t}}}$
with
$\xi =\pm {\sqrt {{\frac {{\text{E}}^{2}\ {\text{r}}^{4}}{{\text{L}}^{2}}}\left(1{\frac {2}{\text{r}}}\right)\left({\frac {{\text{m}}^{2}\ {\text{r}}^{4}}{{\text{L}}^{2}}}+{\text{r}}^{2}\right)}}$
where in contrast to the overdot, which stands for ${\dot {x}}={\rm {d}}x/{\rm {d}}{\tau }$, the overbar denotes ${\bar {x}}={\rm {d}}x/{\rm {d}}{\text{t}}$.
For massless particles like photons ${\rm {m}}/{\sqrt {1v^{2}}}$ in the formula for ${\rm {E}}$ and ${\rm {L}}$ is replaced with ${\rm {h\ f}}$ and the ${\rm {m}}$ in the equations of motion set to $1$, with ${\rm {h}}$ as Planck's constant and ${\rm {f}}$ for the photon's frequency.
Licensing[edit]

This file is licensed under the Creative Commons AttributionShare Alike 4.0 International license.  
https://creativecommons.org/licenses/bysa/4.0 CC BYSA 4.0 Creative Commons AttributionShare Alike 4.0 truetrue 
References[edit]
 ↑ Cole Miller for the Department of Astronomy, University of Maryland: ASTR 498, High Energy Astrophysics
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Date/Time  Thumbnail  Dimensions  User  Comment  

current  15:03, 14 March 2020  777 × 514 (7.97 MB)  Bürgerentscheid (talk  contribs)  frames reduced and slightly resized to fit 100 MP limit  
19:36, 11 July 2018  800 × 526 (2.17 MB)  Yukterez (talk  contribs)  choosing dt/dτ instead of dτ/dt for the time dilation factor to fit existing conventions  
08:31, 13 February 2017  800 × 526 (2.17 MB)  Yukterez (talk  contribs)  reduced filesize by 1MB by reducing the colors  
08:15, 13 February 2017  800 × 526 (3.1 MB)  Yukterez (talk  contribs)  User created page with UploadWizard 
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