File:Newton versus Schwarzschild trajectories.gif

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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).
Source Own work - Mathematica Code
Author Yukterez (Simon Tyran, Vienna)
Other versions

Kerr orbit, a=0.9

Equations of motion[edit]


In spherical coordinates and natural units of , where lengths are measured in and times in , the motion of a testparticle in the presence of a dominant mass is defined by

The initial conditions are

The overdot stands for the time-derivative. is the angular coordinate, the local elevation angle of the test particle, and it's velocity.

and , where the kinetic and potential component (all in units of ) give the total energy , and the angular momentum, which is given by (in units of ) where is the transverse and the radial velocity component, are conserved quantities.


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

which is except for the term identical with Newton, although the radial coordinate has a different meaning (see farther below). The time dilation is

The coordinates are differentiated by the test particle's proper time , while is the coordinate time of the bookkeeper at infinity. So the total coordinate time ellapsed between the proper time interval


The local velocity (relative to the main mass) and the coordinate celerity are related by

for the input and for the output of the transverse and

or the other way around for the radial component of motion.

The shapiro-delayed velocity in the bookeeper's frame of reference is


The initial conditions in terms of the local physical velocity are therefore

The horizontal and vertical components differ by a factor of

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

and is not but

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

in units of and the total energy as the sum of rest-, kinetic- and potential energy

in units of , where is the test particle's restmass, are the constants of motion. The components of the total energy are

for the kinetic plus for the potential energy plus , the test particle's invariant rest mass.

The equations of motion in terms of and are

or, differentiated by the coordinate time


where in contrast to the overdot, which stands for , the overbar denotes .

For massless particles like photons in the formula for and is replaced with and the in the equations of motion set to , with as Planck's constant and for the photon's frequency.


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  1. Cole Miller for the Department of Astronomy, University of Maryland: ASTR 498, High Energy Astrophysics

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current19:36, 11 July 2018Thumbnail for version as of 19:36, 11 July 2018800 × 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 2017Thumbnail for version as of 08:31, 13 February 2017800 × 526 (2.17 MB)Yukterez (talk | contribs)reduced filesize by 1MB by reducing the colors
08:15, 13 February 2017Thumbnail for version as of 08:15, 13 February 2017800 × 526 (3.1 MB)Yukterez (talk | contribs)User created page with UploadWizard
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