General Relativity combines gravitation with Special Relativity,
and explains gravitation as a curvature of space-time. Like any
relativistic effects, the amplitude of effects of General
Relativity is given by taking the square of a speed that occurs in
the experience and dividing it by c^{2}.

The speed to take here is the orbital speed r/c, whose square is
the gravitational potential. Thus the gravitational potential, to
be measured in m^{2}/s^{2}, must be divided by c^{2}
to obtain the dimensionless value it has in General Relativity.
And this dimensionless value gives the amplitude of some effects.

One effect is that the same mass on a planet has a higer value as
defined by a local observer on the planet, than by a remote
observer far away. Indeed the remote observer taking some given
mass m (a meteorite in space, with the value of its mass defined
by its gravitational effect on a third observer even more far
away), and dropping it on a planet, considers that the mass is
conserved (as this fall has no gravitational consequence on the
third observer). But on the surface, the meteorite is received
with some more kinetic energy corresponding to the gravitational
potential; and this energy is converted into mass (until it is
radiated back into space). Thus the mass measured on the surface
has about (1+(r/ct)^{2}) times the value it had far away.

In the case of the Earth, (r/ct)^{2}= (7911 /
299,800,000)^{2}= 6.963×10^{−10}.

Thus the total mass M_{e} of the Earth (as defined from
its graviational field far away) is smaller than the sum of masses
it contains. The difference is about 0.6 (r/ct)^{2}M_{e},
where 0.6 is some approximative coefficient corresponding to the
distribution of masses inside the Earth. We get 0.6* 6.963×10^{−10}
* 5.9736×10^{24} kg
= 2.5×10^{15}
kg, that is the mass of a cube with density 2.5 and size 10km. In
other words, roughly the mass of a mountain. Or, if you take this
quantity of a substance with the same density as the Earth (5.515
kg /mm.m^{2}), and spread it all over the Earth (4π(6371 km)^{2}), you
get a slice with width 0.89 mm.

In the case of the moon, (r/ct)^{2} = (1,737 km/c)^{2} ×
9.338×10^{−7}
s^{-2}=3.135 ×10^{−11}.
The mass difference is then 1.38×10^{12} kg, a little more than the
mass of 1 km^{3} of water.

As for the mass difference of the Earth-Moon system from the sum
of masses of the Earth and the Moon, it is just roughly equal to
the kinetic energy of the moon on its orbit, that is 4.08×10^{11}kg,
thus a little smaller because the moon's orbital velocity around
the earth is smaller than the escape velocity from the moon's
surface.

Another effect of general relativity is the difference of time
measurements between altitudes : the same long time interval
appears to be shorter as measured by a clock at the sea level,
than by a clock on a mountain. The relative difference is given by
the difference of graviational potential. This can be explained by
noticing that there are 2 ways for an amount of energy to change
its altitude, and this must respect a global energy conservation
law. One way consists in being stored as a small additional mass
in a falling object. When falling, this additional mass comes with
its own kinetic energy, giving a small change in the value of the
energy between altitudes. The other way is in the form of a number
of photons with a given energy proportional to their frequency.
They must seem to have a higher frequency as measured at sea level
than on the mountain. From the viewpoint of relativistic quantum physics,
these 2 ways of sending the amount of energy and the resulting
explanation of what causes a change in its value (either as the
kinetic energy due to the fall or as a change in the rythm of
time), are one and the same.

Concretely, this difference of gravitational potential is the
product of the altitude with the local value of the gravitational
acceleration (as this is a local law about the gravitational
potential, and general relativity does not distinguish between the
gravitational and the centrifugal force, thus implicitly
integrating the effect on time of the rotational speed of the
earth according to special relativity). Between the sea level and
an altitude of 2000m, the potential difference is 9.8×2000/(3×10^{8})^{2}
= 2.18×10^{-13}
, that is 6.9 seconds of difference between these clocks every
million year. Between a clock on the moon and a clock on the
Earth, the difference is roughly given by the difference of
gravitational potentials between the moon's and the Earth's
surface, that is 2 seconds per century.

The difference is bigger between different planets of the solar
system. Generally, a clock in a circular orbit with speed v is
slower (gives shorter values of long times) by a fraction 3/2
(v/c)^{2}, as 3/2 = 1 + 1/2 where the 1 is due to the
gravitational potential, and the 1/2 is due to the speed (only
concerning clocks moving in orbit). The orbital speed of the Earth
around the sun is 29.78 km/s, corresponding to the
gravitational potential of (29.78/299,792)^{2}= 9.87×10^{-9},
and thus a slow down by a fraction of 1.480×10^{-8}, that is 46.7
seconds per century. If we add to this the gravitational potential
of the Earth for a clock on the Earth's surface, the fraction of
slowdown is 1.480×10^{-8}+
6.96×10^{−10}
= 1.550×10^{-8},
that is 49 seconds per century.

See this introduction
to General Relativity on the case of cosmology for further
explanations.

The fundamental expression of general relativity is a relation
between mass density and space curvature. In the void, space-time
geometry satisfies the following relation : the curvature of a
"flat" spatial surface (not bent in the orthogonal direction)
equals the tidal effect on the orthogonal direction divided by c^{2},
with a positive sign (which attracts parallel lines) if this tidal
effect is repulsive.

Near a planet, the tidal effect is equal to 2T tearing apart in
the vertical direction, so that a "flat" horizontal surface has
curvature 2T/c^{2} ; it is attractive with amplitude T in
any horizontal direction, so that a "flat" vertical surface has
curvature -T/c^{2}.

When entering the surface of the planet, the horizonal curvature
keeps continuous values but the vertical ones change : in the case
of a planet with constant density, they become equal to the
horizontal curvature 2T/c^{2}; in any case the sum of 3
spatial curvatures is proportional to the density. The coefficient
of proportionality is thus 6T/ρc^{2} = 6×4πG/3c^{2}= 8πG/c^{2}.

Inside a theoretical planet with constant density, thus, we have
the uniform space curvature 2T/c^{2}=8πGρ/3c^{2}, with
coefficient the distance (per density in g/cm^{3})

8πG/3c^{2} = 6.221×10^{−24}
m^{-2}(g/m^{3})^{-1}.

For a density of 1 g/m^{3}, this curvature has radius

(6.221×10^{−24}
)^{-1/2}=400,930,000 km = 4.0093×10^{11}
m = (c/√2)×
1891.3 s = 1337 light-seconds = 22.29 light-minutes.

For the Earth's density, this radius of geometrical curvature is
569.5 light-seconds = 9.49 light-minutes = 170,720,000 km

to be compared to the Earth's mean distance to the Sun,
149,598,000 km = 8.317 light-minutes.

In practice, it means that a "planet" with the same density as
the Earth but as big as the Earth's orbit, would immediately
become a black hole with that size.

Let us describe the amplitude of the space curvature in the Earth - or rather, of a theoretical perfectly spherical planet with the same mass and size and without rotation.

Define:

- L = length of the equator
- r
= distance of a point of the equator to the center of the
planet

- k = roundness of the equator (in radian per meter)

If there was no effect we would have L/2π=r= 1/k. The effect is that
L/2π < 1/k, while r is
at least (3/4 L/2π +1/4k)
(= value if the density is uniform) and can be even bigger than
1/k (?) if the mass is concentrated in the center.

Precisely, the same reasoning as for cosmology says that (2π/L)^{2}=k^{2}+2T/c^{2}.
Thus 1/k - L/2π ≈
(L/2π)^{3} T/c^{2}.

Ifever you try to measure the distortion of space geometry from Euclidean geometry due to gravity, be careful that many usual ools to build and measure a picture may undergo bigger distortions than the effect you want to measure : vertical rulers dilate when hanged and shrink when put.

The above calculations taking densities as basic parameters and deducing characteristic times and lengths, seem to assume them as more or less fixed, and determining when black holes can appear. Such assumptions would be misleading.

First, the density of planets is not just determined by their composition, but compressing effects come into play, for example in the Earth's density of 5.5 that would otherwise be 4.4. As for Jupiter, "Theoretical models indicate that if Jupiter had much more mass than it does at present, the planet would shrink.(...) Jupiter is thought to have about as large a diameter as a planet of its composition and evolutionary history can achieve. The process of further shrinkage with increasing mass would continue until appreciable stellar ignition is achieved as in high-mass brown dwarfs around 50 Jupiter masses".

As for massive stars, they are much bigger, thus much less dense.

The general relativistic effects are defined by the gravitational potential that is the ratio of mass to radius. In particular, the radius of a black hole is proportional to its mass. Stars can be more massive but still as far from becoming black holes, if their radius increases at least as much as their mass. For example the star Rigel has a mass of 18 solar masses, but a radius of 74 times the Sun's radius. Thus while the Sun's density of 1.408 is just a little more than water (but its center is about a hundred times more dense), that of Rigel is much lower. As for Belelgeuse, its mass is 15-20 times the Sun but its radius is about 1000 times the solar radius, maybe as big as the orbit of Jupiter. On the other end, the dim red dwarf star Proxima Centauri, that is closest to the solar system, has a density about 40 times that of the Sun ; its radius is 98,000km, just a little more than that of Jupiter (66,854 km).

But how can densities vary so widely ? But first, why do they seem fixed in "ordinary conditions" ? We need to further investigate the properties of matter.

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