File:Newton versus Schwarzschild trajectories.gif
Newton_versus_Schwarzschild_trajectories.gif (800 × 526 pixels, file size: 2.17 MB, MIME type: image/gif, looped, 500 frames, 15 s)
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Summary[edit]
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  21 May 2016 
Source  Own work  Mathematica Code 
Author  Yukterez (Simon Tyran, Vienna) 
Other versions 
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 ${\text{m}}=1$, 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  18:47, 30 September 2021  800 × 526 (2.17 MB)  Yukterez (talk  contribs)  revert vandalism  
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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