8.7 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Curved spacetime | 3/5 | https://en.wikipedia.org/wiki/Curved_spacetime | reference | science, encyclopedia | 2026-05-05T11:15:02.570747+00:00 | kb-cron |
== Distortion of space == The
(
1
−
2
G
M
/
(
c
2
r
)
)
{\displaystyle (1-2GM/(c^{2}r))}
coefficient in front of
(
c
Δ
t
)
2
{\displaystyle (c\Delta t)^{2}}
describes the distortion of time in Newtonian gravitation, and this distortion completely accounts for all Newtonian gravitational effects. As expected, this correction factor is directly proportional to
G
{\displaystyle G}
and
M
{\displaystyle M}
, and because of the
r
{\displaystyle r}
in the denominator, the correction factor increases as one approaches the gravitating body, meaning that time is distorted. But general relativity is a theory of distorted space and distorted time, so if there are terms modifying the spatial components of the spacetime interval presented above, should not their effects be seen on, say, planetary and satellite orbits due to distortion correction factors applied to the spatial terms? The answer is that they are seen, but the effects are tiny. The reason is that planetary velocities are extremely small compared to the speed of light, so that for planets and satellites of the Solar System, the
(
c
Δ
t
)
2
{\displaystyle (c\Delta t)^{2}}
term dwarfs the spatial terms. Despite the minuteness of the spatial terms, the first indications that something was wrong with Newtonian gravitation were discovered over a century-and-a-half ago. In 1859, Urbain Le Verrier, in an analysis of available timed observations of transits of Mercury over the Sun's disk from 1697 to 1848, reported that known physics could not explain the orbit of Mercury, unless there possibly existed a planet or asteroid belt within the orbit of Mercury. The perihelion of Mercury's orbit exhibited an excess rate of precession over that which could be explained by the tugs of the other planets. The ability to detect and accurately measure the minute value of this anomalous precession (only 43 arc seconds per tropical century) is testimony to the sophistication of 19th century astrometry.
As the astronomer who had earlier discovered the existence of Neptune "at the tip of his pen" by analyzing irregularities in the orbit of Uranus, Le Verrier's announcement triggered a two-decades long period of "Vulcan-mania", as professional and amateur astronomers alike hunted for the hypothetical new planet. This search included several false sightings of Vulcan. It was ultimately established that no such planet or asteroid belt existed. In 1916, Einstein was to show that this anomalous precession of Mercury is explained by the spatial terms in the distortion of spacetime. distortion in the temporal term, being simply an expression of Newtonian gravitation, has no part in explaining this anomalous precession. The success of his calculation was a powerful indication to Einstein's peers that the general theory of relativity could be correct. The most spectacular of Einstein's predictions was his calculation that the distortion terms in the spatial components of the spacetime interval could be measured in the bending of light around a massive body. Light has a slope of ±1 on a spacetime diagram. Its movement in space is equal to its movement in time. For the weak field expression of the invariant interval, Einstein calculated an exactly equal but opposite sign distortion in its spatial components.
Δ
s
2
=
(
1
−
2
G
M
c
2
r
)
(
c
Δ
t
)
2
{\displaystyle \Delta s^{2}=\left(1-{\frac {2GM}{c^{2}r}}\right)(c\Delta t)^{2}}
−
(
1
+
2
G
M
c
2
r
)
[
(
Δ
x
)
2
+
(
Δ
y
)
2
+
(
Δ
z
)
2
]
{\displaystyle -\,\left(1+{\frac {2GM}{c^{2}r}}\right)\left[(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}\right]}
In Newton's gravitation, the
(
1
−
2
G
M
/
(
c
2
r
)
)
{\displaystyle (1-2GM/(c^{2}r))}
coefficient in front of
(
c
Δ
t
)
2
{\displaystyle (c\Delta t)^{2}}
predicts bending of light around a star. In general relativity, the
(
1
+
2
G
M
/
(
c
2
r
)
)
{\displaystyle (1+2GM/(c^{2}r))}
coefficient in front of
[
(
Δ
x
)
2
+
(
Δ
y
)
2
+
(
Δ
z
)
2
]
{\displaystyle \left[(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}\right]}
predicts a doubling of the total bending. The story of the 1919 Eddington eclipse expedition and Einstein's rise to fame is well told elsewhere.
== Sources of spacetime curvature ==
In Newton's theory of gravitation, the only source of gravitational force is mass. In contrast, general relativity identifies several sources of spacetime curvature in addition to mass. In the Einstein field equations, the sources of gravity are presented on the right-hand side in
T
μ
ν
,
{\displaystyle T_{\mu \nu },}
the stress–energy tensor. Fig. 5-5 classifies the various sources of gravity in the stress–energy tensor:
T
00
{\displaystyle T^{00}}
(red): The total mass–energy density, including any contributions to the potential energy from forces between the particles, as well as kinetic energy from random thermal motions.
T
0
i
{\displaystyle T^{0i}}
and
T
i
0
{\displaystyle T^{i0}}
(orange): These are momentum density terms. Even if there is no bulk motion, energy may be transmitted by heat conduction, and the conducted energy will carry momentum.