File:Newton versus Schwarzschild trajectories.gif
Newton_versus_Schwarzschild_trajectories.gif (800 × 526 píxels, mida del fitxer: 2,17 Mo, tipus MIME: image/gif, en bucle, 500 fotogrames, 15 s)
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[modifica]DescripcióNewton 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). |
Data | |
Font | Treball propi - Mathematica Code |
Autor | Yukterez (Simon Tyran, Vienna) |
Altres versions |
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Equations of motion
[modifica]Newton
[modifica]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.
Schwarzschild
[modifica]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
is
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
and
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
with
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.
Llicència
[modifica]![w:ca:Creative Commons](https://upload.wikimedia.org/wikipedia/commons/thumb/7/79/CC_some_rights_reserved.svg/90px-CC_some_rights_reserved.svg.png)
![reconeixement](https://upload.wikimedia.org/wikipedia/commons/thumb/1/11/Cc-by_new_white.svg/24px-Cc-by_new_white.svg.png)
![compartir igual](https://upload.wikimedia.org/wikipedia/commons/thumb/d/df/Cc-sa_white.svg/24px-Cc-sa_white.svg.png)
- Sou lliure de:
- compartir – copiar, distribuir i comunicar públicament l'obra
- adaptar – fer-ne obres derivades
- Amb les condicions següents:
- reconeixement – Heu de donar la informació adequada sobre l'autor, proporcionar un enllaç a la llicència i indicar si s'han realitzat canvis. Podeu fer-ho amb qualsevol mitjà raonable, però de cap manera no suggereixi que l'autor us dóna suport o aprova l'ús que en feu.
- compartir igual – Si modifiqueu, transformeu, o generareu amb el material, haureu de distribuir les vostres contribucions sota una llicència similar o una de compatible com l'original
References
[modifica]- ↑ Cole Miller for the Department of Astronomy, University of Maryland: ASTR 498, High Energy Astrophysics
Usage in Wikipedia Articles
[modifica]en.wikipedia.org
de.wikipedia.org
ru.wikipedia.org
es.wikipedia.org
zh.wikipedia.org
Historial del fitxer
Cliqueu una data/hora per veure el fitxer tal com era aleshores.
Data/hora | Miniatura | Dimensions | Usuari/a | Comentari | |
---|---|---|---|---|---|
actual | 18:47, 30 set 2021 | ![]() | 800 × 526 (2,17 Mo) | Yukterez (discussió | contribucions) | revert vandalism |
15:03, 14 març 2020 | ![]() | 777 × 514 (7,97 Mo) | Bürgerentscheid (discussió | contribucions) | frames reduced and slightly resized to fit 100 MP limit | |
19:36, 11 jul 2018 | ![]() | 800 × 526 (2,17 Mo) | Yukterez (discussió | contribucions) | choosing dt/dτ instead of dτ/dt for the time dilation factor to fit existing conventions | |
08:31, 13 feb 2017 | ![]() | 800 × 526 (2,17 Mo) | Yukterez (discussió | contribucions) | reduced filesize by 1MB by reducing the colors | |
08:15, 13 feb 2017 | ![]() | 800 × 526 (3,1 Mo) | Yukterez (discussió | contribucions) | User created page with UploadWizard |
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