# 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)

Note: Due to technical limitations, thumbnails of high resolution GIF images such as this one will not be animated. The limit on Wikimedia Commons is width × height × number of frames ≤ 100 million.

## Captions

### Captions

orbit aroud a central mass, comparison Newton vs Einstein

## Summary

 Description 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 ${\text{G=M=c=1}}$ , 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 time-derivative. $\phi$ 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

The equations of motion  in Schwarzschild-coordinates 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

${\dot {\text{t}}}={\frac {\sqrt {1+{\dot {\text{r}}}^{2}\ {\text{r}}/({\text{r}}-2)+{\text{r}}^{2}\ {\dot {\phi }}^{2}}}{\sqrt {1-2/{\text{r}}}}}={\frac {1}{{\sqrt {1-2/{\text{r}}}}{\sqrt {1-v^{2}}}}}$ The coordinates are differentiated by the test particle's proper time $\tau$ , 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

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

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

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

${\text{v}}_{\perp }=v_{\perp }{\sqrt {1-2/{\text{r}}}}$ and ${\text{v}}_{\parallel }=v_{\parallel }(1-2/{\text{r}})$ The initial conditions in terms of the local physical velocity $v$ are therefore

${\dot {\text{r}}}_{0}={\frac {v_{0}{\sqrt {1-2/{\text{r}}_{0}}}\sin(\varphi _{0})}{\sqrt {1-v_{0}^{2}}}}\ ,\ \ {\dot {\phi }}_{0}={\frac {v_{0}\cos(\varphi _{0})}{{\text{r}}_{0}{\sqrt {1-v_{0}^{2}}}}}$ The horizontal and vertical components differ by a factor of ${\sqrt {1-2/{\text{r}}}}$ 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

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

${\text{L}}={\text{r}}^{2}\ {\dot {\phi }}=v_{\perp }\ {\rm {r}}/{\sqrt {1-v^{2}}}$ in units of ${\text{m}}$ and the total energy as the sum of rest-, kinetic- and potential energy

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

${\text{E}}_{\rm {kin}}={\frac {1}{\sqrt {1-v^{2}}}}-1$ for the kinetic plus ${\text{E}}_{\rm {pot}}={\frac {{\sqrt {1-2/{\text{r}}}}-1}{\sqrt {1-v^{2}}}}$ 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

${\dot {\rm {r}}}={\dot {\phi }}\ \xi \ ,\ \ {\dot {\phi }}={\frac {\text{L}}{{\text{m}}\ {\text{r}}^{2}}}\ ,\ \ {\dot {\text{t}}}={\frac {\text{E}}{{\text{m}}\ (1-2/{\text{r}})}}$ or, differentiated by the coordinate time ${\text{t}}$ ${\bar {\text{r}}}={\bar {\phi }}\ \xi \ ,\ \ {\bar {\phi }}={\frac {(1-2/{\text{r}})\ {\text{L}}}{{\text{E}}\ {\text{r}}^{2}}}\ ,\ \ {\bar {\tau }}=1/{\dot {\text{t}}}$ 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 ${\rm {m}}/{\sqrt {1-v^{2}}}$ 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

I, the copyright holder of this work, hereby publish it under the following license:   This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license. You are free: to share – to copy, distribute and transmit the work to remix – to adapt the work Under the following conditions: attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

## File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current18:47, 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

There are no pages that use this file.

## File usage on other wikis

The following other wikis use this file:

• Usage on de.wikipedia.org
• Usage on en.wikipedia.org

## Structured data

### Items portrayed in this file

#### some value

author name string: Yukterez (Simon Tyran, Vienna)