Miêu tảNewton 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).
Ngày	21 tháng 5 năm 2016
Nguồn gốc	Tác phẩm được tạo bởi người tải lên - Mathematica Code
Tác giả	Yukterez (Simon Tyran, Vienna)
Phiên bản khác

In spherical coordinates and natural units of ${\displaystyle {\text{G=M=c=1}}}$, where lengths are measured in ${\displaystyle {\text{GM/c}}^{2}}$ and times in ${\displaystyle {\text{GM/c}}^{3}}$, the motion of a testparticle in the presence of a dominant mass is defined by

${\displaystyle {\ddot {\text{r}}}=-{\frac {1}{{\text{r}}^{2}}}+{\text{r}}\ {\dot {\phi }}^{2}\ ,\ \ {\ddot {\phi }}=-{\frac {2\ {\dot {\text{r}}}\ {\dot {\phi }}}{\text{r}}}}$

The initial conditions are

${\displaystyle {\dot {\text{r}}}_{0}=v_{0}\sin(\varphi _{0})\ ,\ \ {\dot {\phi }}_{0}={\frac {v_{0}}{{\text{r}}_{0}}}\cos(\varphi _{0})}$

The overdot stands for the time-derivative. ${\displaystyle \phi }$ is the angular coordinate, ${\displaystyle \varphi }$ the local elevation angle of the test particle, and ${\displaystyle v}$ it's velocity.

${\displaystyle {\rm {E}}}$ and ${\displaystyle {\rm {L}}}$, where the kinetic ${\displaystyle {\text{E}}_{\rm {kin}}=v^{2}/2}$ and potential ${\displaystyle {\text{E}}_{\rm {pot}}=-1/{\rm {r}}}$ component (all in units of ${\displaystyle {\rm {mc}}^{2}}$) give the total energy ${\displaystyle {\rm {E}}={\text{E}}_{\rm {kin}}+{\text{E}}_{\rm {pot}}}$, and the angular momentum, which is given by ${\displaystyle {\rm {L}}=v_{\perp }\ {\rm {r}}}$ (in units of ${\displaystyle {\rm {m}}}$) where ${\displaystyle v_{\perp }={\dot {\phi }}\ {\text{r}}}$ is the transverse and ${\displaystyle v_{\parallel }={\dot {\text{r}}}}$ the radial velocity component, are conserved quantities.

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

${\displaystyle {\ddot {\text{r}}}=-{\frac {1}{{\text{r}}^{2}}}+{\text{r}}\ {\dot {\phi }}^{2}-3\ {\dot {\phi }}^{2}\ ,\ \ {\ddot {\phi }}=-{\frac {2\ {\dot {\text{r}}}\ {\dot {\phi }}}{\text{r}}}}$

which is except for the ${\displaystyle -3\ {\dot {\phi }}^{2}}$ term identical with Newton, although the radial coordinate has a different meaning (see farther below). The time dilation is

${\displaystyle {\dot {\text{t}}}={\frac {\sqrt {1+{\dot {\text{r}}}^{2}\ {\text{r}}/({\text{r}}-2)+{\text{r}}^{2}\ {\dot {\phi }}^{2}}}{\sqrt {1-2/{\text{r}}}}}={\frac {1}{{\sqrt {1-2/{\text{r}}}}{\sqrt {1-v^{2}}}}}}$

The coordinates are differentiated by the test particle's proper time ${\displaystyle \tau }$, while ${\displaystyle {\text{t}}}$ is the coordinate time of the bookkeeper at infinity. So the total coordinate time ellapsed between the proper time interval

${\displaystyle \tau _{0}..\tau _{1}}$ is ${\displaystyle {\text{t}}=\int _{\tau _{0}}^{\tau _{1}}{\dot {\text{t}}}\,\mathrm {d} \tau }$

The local velocity ${\displaystyle v}$ (relative to the main mass) and the coordinate celerity are related by

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

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

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

${\displaystyle {\text{v}}_{\perp }=v_{\perp }{\sqrt {1-2/{\text{r}}}}}$ and ${\displaystyle {\text{v}}_{\parallel }=v_{\parallel }(1-2/{\text{r}})}$

The initial conditions in terms of the local physical velocity ${\displaystyle v}$ are therefore

${\displaystyle {\dot {\text{r}}}_{0}={\frac {v_{0}{\sqrt {1-2/{\text{r}}_{0}}}\sin(\varphi _{0})}{\sqrt {1-v_{0}^{2}}}}\ ,\ \ {\dot {\phi }}_{0}={\frac {v_{0}\cos(\varphi _{0})}{{\text{r}}_{0}{\sqrt {1-v_{0}^{2}}}}}}$

The horizontal and vertical components differ by a factor of ${\displaystyle {\sqrt {1-2/{\text{r}}}}}$

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

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

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

${\displaystyle {\text{L}}={\text{r}}^{2}\ {\dot {\phi }}=v_{\perp }\ {\rm {r}}/{\sqrt {1-v^{2}}}}$

in units of ${\displaystyle {\text{m}}}$ and the total energy as the sum of rest-, kinetic- and potential energy

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

in units of ${\displaystyle {\text{mc}}^{2}}$, where ${\displaystyle {\text{m}}}$ is the test particle's restmass, are the constants of motion. The components of the total energy are

${\displaystyle {\text{E}}_{\rm {kin}}={\frac {1}{\sqrt {1-v^{2}}}}-1}$ for the kinetic plus ${\displaystyle {\text{E}}_{\rm {pot}}={\frac {{\sqrt {1-2/{\text{r}}}}-1}{\sqrt {1-v^{2}}}}}$ for the potential energy plus ${\displaystyle {\text{m}}=1}$, the test particle's invariant rest mass.

The equations of motion in terms of ${\displaystyle {\text{E}}}$ and ${\displaystyle {\text{L}}}$ are

${\displaystyle {\dot {\rm {r}}}=\xi \ ,\ \ {\dot {\phi }}={\frac {\text{L}}{{\text{m}}\ {\text{r}}^{2}}}\ ,\ \ {\dot {\text{t}}}={\frac {\text{E}}{{\text{m}}\ (1-2/{\text{r}})}}}$

or, differentiated by the coordinate time ${\displaystyle {\text{t}}}$

${\displaystyle {\bar {\text{r}}}=\xi \ ,\ \ {\bar {\phi }}={\frac {(1-2/{\text{r}})\ {\text{L}}}{{\text{E}}\ {\text{r}}^{2}}}\ ,\ \ {\bar {\tau }}=1/{\dot {\text{t}}}}$

with

${\displaystyle \xi =\pm {\sqrt {{\frac {{\text{E}}^{2}\ {\text{r}}^{4}}{{\text{L}}^{2}}}-\left(1-{\frac {2}{\text{r}}}\right)\left({\frac {{\text{m}}^{2}\ {\text{r}}^{4}}{{\text{L}}^{2}}}+{\text{r}}^{2}\right)}}}$

where in contrast to the overdot, which stands for ${\displaystyle {\dot {x}}={\rm {d}}x/{\rm {d}}{\tau }}$, the overbar denotes ${\displaystyle {\bar {x}}={\rm {d}}x/{\rm {d}}{\text{t}}}$.

For massless particles like photons ${\displaystyle {\rm {m}}/{\sqrt {1-v^{2}}}}$ in the formula for ${\displaystyle {\rm {E}}}$ and ${\displaystyle {\rm {L}}}$ is replaced with ${\displaystyle {\rm {h\ f}}}$ and the ${\displaystyle {\rm {m}}}$ in the equations of motion set to ${\displaystyle 1}$, with ${\displaystyle {\rm {h}}}$ as Planck's constant and ${\displaystyle {\rm {f}}}$ for the photon's frequency.

Tôi, người giữ bản quyền tác phẩm này, từ đây phát hành nó theo giấy phép sau:

Tập tin này được phát hành theo Giấy phép Creative Commons Ghi công–Chia sẻ tương tự 4.0 Quốc tế.

Bạn được phép:

• chia sẻ – sao chép, phân phối và chuyển giao tác phẩm
• pha trộn – để chuyển thể tác phẩm

Theo các điều kiện sau:

• ghi công – Bạn phải ghi lại tác giả và nguồn, liên kết đến giấy phép, và các thay đổi đã được thực hiện, nếu có. Bạn có thể làm các điều trên bằng bất kỳ cách hợp lý nào, miễn sao không ám chỉ rằng người cho giấy phép ủng hộ bạn hay việc sử dụng của bạn.
• chia sẻ tương tự – Nếu bạn biến tấu, biến đổi, hoặc làm tác phẩm khác dựa trên tác phẩm này, bạn chỉ được phép phân phối tác phẩm mới theo giấy phép y hệt hoặc tương thích với tác phẩm gốc.

https://creativecommons.org/licenses/by-sa/4.0CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 truetrue

en.wikipedia.org

• Schwarzschild metric
• Schwarzschild geodesics
• Two-body problem in general relativity

de.wikipedia.org

• Apsidendrehung
• Schwarzschild-Metrik

ru.wikipedia.org

• Метрика Шварцшильда
• Задача Кеплера в относительности

es.wikipedia.org

• La relatividad general

zh.wikipedia.org

• 史瓦西度規