Schwarzschild Blackhole
---------------------
Z Schwarzschild in 1916 developed one result of the new Einstein
physics, the modern blackhole. He showed that the ultimate state of
curvature of spacetime occurs when a mass gets so dense or so small
that in its vicinity no matter or energy can escape. This is the 20th
century reincarnation of Michell's work of the 1780s, but
Schwarzschild most likely did not know of that discovery.
We can not for the home astronomer go thru a detailed study of
Schwarzschild's work. We can show that, given the blackhole as
Schwarzschild found it, there are some amazing effects that Einstein
physics predicts for it.
Idealized Blackhole
-----------------
The Schwarzschild blackhole is a perfectly spherical stable
nonrotating body with no electric charge. In his day this was a sensible
first project for the new theory of general relativity. Today, late 20th
century, with the prospect of real blackholes forming from stars or
galaxies, this model is too simple. Every body that could become a
blackhole does rotate and potentially has a nonzero electric charge.
The presence of either rotation or charge vastly complicates the
blackhole model but makes for a far more realistic scenario. In 1918
Reissner and Nordstrom worked out the blackhole model for a nonrotating
charged blackhole. Kerr in 1963 described a rotating noncharged one. And
in 1972 Kerr and Newman sussed out the most general case of a blackhole
with both rotation and charge.
The table here lists the four possible blackholes
--------------------------- -------------------
| no rotation | with rotation
------------+--------------------+--------------
no charge | Schwarzschild | Kerr
with charge | Reissner=Nordstrom | Kerr-Newman
----------------------------------------------
By custom the names for the blackholes with electric charge are
not widely used. One speaks of a 'charged Kerr blackhole' rather than
a 'Kerr-Newman blackhole'. Do, never the less, know these names
because they turn up from time to time.
Spacetime with Mass
-----------------
This is not a proper derivation of the Schwarzschild spacetime system
but it shows how from what we know already there really can be a
disturbance caused purely by position relative to material bodies.
Recall that the [square of the] escape velocity within a gravity field
is given by, in Newton physics
Vesc^2 = 2 * gamma * m / r
Altho this is commonly thought of as the velocity required to escape from
the surface of a planet or such, it is valid for the escape from any point
within the gravity field, even a remote point. In the latter case r is
very large and so Vesc is quite small.
Michell and then Schwarzschild noted that this escape velocity can
equal the speed of light if the radius is sufficiently small or the body
sufficiently massive. Michell assumed the latter case of a star some 500
times larger (in size) than the Sun. Schwarzschild took the former path of
a compacted body.
The radius at which Vesc = c is
r = 2 * gamma * m / (Vesc^2)
r| = 2 * gamma * m / (c^2)
r| is that radius at which Vesc equals c. This is the
Schwarzschild raius. For the Sun r| is
r|[Sun] = 2 * gamma * m / (c^2)
= 2 * (6.67e-11n.m2/kg2) * (1.99e30kg / (3e8m/s)^2
= (2.95e3m)
-> 2.9km
Schwarzschild Surface or Radius
-----------------------------
Because of the spherical symmetry in the Schwarzschild model this r|
is the radius of a sphere around the body. The body itself has to be
within this sphere with empty space intervening so that we may in fact
stand at r| from it. In the usual astronomy situation the body extends
vastly farther out than r| preventing any possible approach to it. The
Earth is some 6,400Km in radius but r| is only 9mm. The solid bulk of the
Earth blocks any access to a point at r| from the center.
If we try to get in close by descending a well (for a solid planet) or
diving under the clouds or soup (for a liquid or gaseous one), we do
not experience the extreme effects of the Schwarzschild spacetime. For
as we go below the surface of the planet, we leave above us more and
more of the planet's mass. This mass is in a shell around us and this
shell produces no net gravity field at our location. The remaining
mass under us, in an ever smaller amount, actually produces less and
less of a field. In the end, when we reach the center of the planet,
there is no net gravity field at all. We float in weightless state!
To get the extreme spacetime effects there must be enough of the
planet's (star's, quasar's, whatever's) mass already inside the
Schwarzschild radius to begin with. We can not merely get within that
radius with all the mass above it.
The surface of r| radius is not in any way a solid barrier. We would
not 'crash' or 'plow' thru it. It is a geometric figure in vacuum around
the central body. You can pass thru it with no resistance. You can really
fall to and thru the Schwarzschild surface and continue falling below it
to the central body, exactly like falling to and thru an isobar of air
pressure in the Earth's atmosphere.
The Schwarzschild radius or surface is also also called the event
horizon.
Schwarzschild Spacetime Equation
------------------------------
We keep the same notation as before for the motional and stational
frames. So,
delT[m/s] = delT[m/m] / beta
= delT[m/m] / (1 -(V[m/s] / c)^2)^(1/2)
V{m/s]^2 = 2 * gamma*m / r[m/s]
c^2 = 2 * gamma * m / r|
(V[m/s]/c)^2 = (2 * gamma * m / r{m/s]) / (2 * gamma * m / r|)
= (1 / r[m/s]) / (1 / r|)
= r| / r[m/s]
This we substitute into the expression for beta
delT[m/s] = delT[m/m] / (1 - (r| / r[m/s]))^(1/2)
+---------------------------------------------------+
| SCHWARZSCHILD TIME DILATION |
| |
| delT[m/s] = delT[m/m] / (1 - (r| / r[m/s]))^(1/2) |
+---------------------------------------------------+
There seems to be no special name for the denominator of this
spacetime formula, comparable to beta. It could perhaps be called the
'Schwarzschild factor', On a couple occasions it is denoted by epsilon
but this is not a standard notation.
The Safe Frame
------------
The Schwarzschild dilation relates the time of an observer in the m
frame near the attrahent body to that of an observer on the s frame
remote from that body. As experienced by the s platform the time on
the m platform runs slower, the m observer's second is felt to be
longer than a second at the s observer.
Note carefully that we are now comparing observers at rest in the
gravity field, not observers in motion. This shift of meaning of the
m and s frames can not be seen in the way we got to the dilation
equation because we did not use the proper derivation. But it is
important to understand that in this treatment we are not dealing with
relative motion but relative location.
Hence we can not say m means 'motional' and s means 'stational'.
Happily we can keep these subscripts but with different meanings. m now
stands for 'movable' and s now stands for 'safe'. So we have the movable
and the safe platforms.
The usual explanations of relativity or blackholes speak of a 'remote'
or 'far away' observer. This can make it seem that one must be far from
the blackhole to appreciate its behavior. It may seem, too, overall that
relativity effects occur only at places very far from us. Neither notion
is true; both are quite misleading.
The 'remote' observer must be only so far off that the blackhole
effects on him are negligible compared to those on the close-in observer.
While only a place infinity far from a blackhole is entirely free from
its distortion of spacetime, it is remarkable how close you can be to a
blackhole and still have minuscule spacetime distortion. That's why we
now say the s frame is the safe frame, the platform safe from the
blackhole.
How safe is safe? This is up to you. Let's look at the case of the
Sun if it were to somehow collapse into a blackhole. For the Sun r| is
2.9Km. Let's require that the safe observer suffer a spacetime warp, tine
dilation, of no more than 1/1,000,000. Then we have
delT[m/s] = delT[m/m] / (1 - (r| / r[m/s]))^(1/2)
delT[m/s] / delT[m/m] = 1 / (1 - (r| / r[m/s]))^(1/2)
delT[m/m] / delT[m/s] = (1 - (r| / r[m/s]))^(1/2)
(delT[m/m] / delT[m/s])^2 = 1- (r| / r[m/s])
(delT[m/m] / delT[m/s])^2) - 1 = -r| / r[m/s]
1 - (delT[m/m] / delT[m/s])^2) = r| / r[m/s]
r| / r[m/s] = 1 - (delT[m/m] / delT[m/s])^2)
= 1 - (1 / (1.000001))^2
= 1 - (0.9999990)^2
= 1 - (0.9999980)
= (0.0000020)
r[m/s] / r| = 500,000
That is, for the s frame to experience only a 1 part in one million
dilation of time, compared to no dilation if it were at infinity, it may
be as close as 500,000 times the Schwarzschild radius from the
attrahent mass. In the case of the Sun, this is (500,000)*(2.9km) =
1,450,000km.
This is but twice the Sun's [uncollapsed] radius! Hardly a vast
distance, even within the solar system, let alone in interstellar or
intergalactic space.
Thus, it is not at all necessary to recede to immense distances, as
implied by many blackhole texts, to appreciate the workings of the
blackhole. It is only necessary to recede far enough away to let your
own time dilation be sufficiently small. Hence the concept of a 'safe'
frame.
No Change of Gravity Regime
-------------------------
An other common misapprehension about blackholes is that they after
creation reach out and suck in everything around them. This is ridiculous.
For an observer well away from the mass the gravity field at his location
is not altered by the creation of the blackhole. What happens is that
the bulk object is replaced by a point with the same mass.
Note that this is, in a twist of weird fate, the realization of
Newton's proof that a sphericly symmetrical mass can be replaced by a
point mass with no alteration in its regime of gravity.
Where the immense dilation comes from is the ability in the case of
the blackhole to approach right up to the Schwarzschild radius. This
comes simply from the mass of the body having somehow compacted itself
within that radius, leaving open space for the approach.
So if the Sun were in some magic manner convert to a blackhole, the
gravity field at Earth, and elsewhere in the solar system, would remain
the same. The orbits of the planets, comets, and others remain unaltered.
There would be new phaenomena near the collapsed Sun. Before the
compaction you can not approach too near to the Sun for the intense heat.
You would vaporize long before reaching the photosphere. The ambient
temperature of your spaceship would pass 2,000K at about 9 solar radii
away. Only certain exotic ceramics and carbon alloys may resist
vaporation.
When the Sun is a blackhole, there is no radiation from it and so
the surrounding temperature is pretty much absolute zero. (It's
actually 3K, the temperature of the bigbang residuum afterglow.) You
could survive a close encounter with the Sun and get to within a few
hundred kilometers of its Schwarzschild surface. The time dilation
effects show up, the divergence from the safe observer being a few
thousandths, readily measured even by a quartz wristwatch.
Time Dilation versus Distance
---------------------------
The calculation above is summarized in the following table of time
dilation versus distance from the blackhole. The dilation is in units
of (delT[m/m]/delT[m/s])-1.
-----------------------------------------------
r[m/s]/r| dilation Sun km solar system
--------- -------- ------ ------------
1.001 30.639 2.9+ inside Sun's core
1.003 17.285 2.91 |
1.01 9.050 2.93 |
1.03 4.859 2.99 |
1.1 2.317 3.19 |
1.3 1.082 3.77 |
2.0 0.414 5.8 |
3.0 0.225 8.7 |
10 0.054 29 |
30 0.017 87 |
1.0E2 2.038E-3 2.9E2 |
3.0E2 1.071E-3 8.7E2 |
1.0E3 5.004E-4 2.9D3 |
3.0E3 1.667E-4 8.7E3 \|/
1.0E4 5.000E-5 2.9E4 in Sun's outer layers
3.0E4 1.667E-5 8.7E4 |
1.0E5 5.000E-6 2.9E5 \|/
3.0E5 1.667E-6 8.7E5 to prominences & inner corona
1.0E6 5.000E-7 2.9E6 to outer corona visible in eclipses
3.0E6 1.667E-7 8.7E6 about closest survivable approach
1.0E7 5.000E-8 2.9E7 about 1/2 to Mercury
3.0E7 1.667E-8 8.7E7 about to Venus
1.0E8 5.000E-9 2.9E8 between Earth and Mars
3.0E8 1.667E-9 8.7E8 in asteroid belt
1.0E9 5.000E-10 2.9E9 about to Saturn
---------------------------------------------------
Nonsymmetrical Nonreciprocal Relation
-----------------------------------
We went thru a simple substitution from the special to the general
relativity spacetime equation. Because this is not the proper
derivation of the Schwarzschild equation a crucial feature of it was
not noticed thru the substitution process.
The situation of the m and s platforms is not symmetrical or
reciprocal at all. We can not, like in special relativity, flip the m
and s subscripts. The movable frame is really near to the attrahent
mass and the safe one is really far from it. They do experience
different spacetime warpings.
The s frame is presumed far enough away from the mass to neglect
its own spacetime warping. In a real situation, both frames suffer
this distortion. To get the time dilation between two points within
the gravity field, we must examine each separately and compare them.
That is, for two movable platforms m1 and m2, we have
delT[m1/s] = delT[m1/m1] / (1 - (r| / r[m1/s])^(1/2)
delT[m2/s] = delT[m2/m2] / (1 - (r| / r[m2/s])^(1/2)
Locally within each m frame the time runs normally, just as it does in
special relativity, so delT[m1/m1] = delT[m2/m2].
delT[m1/m1] = delT[m1/s] * (1 - (r| / r[m1/s])^(1/2)
delT[m2/m2] = delT[m2/s] * (1 - (r| / r[m2/s])^(1/2)
delT[m1/s] * (1 - (r| / r[m1/s])^(1/2) =
delT[m2/s] * (1 - (r| / r[m2/s])^(1/2)
delT[m1/s] / delT[m2/s] = (1 - (r| / r[m2/s])^(1/2)
/ (1 - (r| / r[m1/s])^(1/2)
delT[m1/s] / delT[m2/s] = ((1 - (r| / r[m2/s])
/ (1 -(r| / r[m1/s]))^(1/2)
+------------------------------------------------------+
| TIME DILATION BETWEEN POINTS WITHIN GRAVITY FIELD |
| |
| delT[m1/s] / delT[m2/s] = |
| ((1 - (r| / r[m2/s]) / (1 - (r| / r[m1/s]))^(1/2) |
+------------------------------------------------------+
This relation is obviously not at all symmetrical between the two
points because r[m1/s] is not equal to r[m2/s]. The right hand side
can not equal unity (same dilation for the two, thus proving
reciprocity).
Tests of Scwarzschild Spacetime
-----------------------------
In the early days of relativity, astronomers mostly ignored it as
being of little use to them. Everything in the heavens behaved quite
well according to Newton physics. There were no new effects that may
call for Einstein physics to explain them.
EInstein was hindered in his teaching of the new relativity theory
because there were no obvious ways in Earth, in the 1910s period, to
demonstrate it. He tried hard to come up with tests that had visible
and convincing effects. He found three using the Sun as the central
mass.
The three tests for many decades were the best proofs of
relativity in astronomy. Relativity was just starting to show yp in
he newly emerging atomic physics but this was outside the
astronomers's work. Today these three are called the 'charter' or
'classical' tests. They are the redshift in the solar spectral lines,
the excess migration of Mercury's orbital perihelion, and deflection
of starlight near the Sun's limb.
All three produce very subtile effects simply because the mass of
the Sun was barely large enough to show up the effects in a convincing
way. Yet at the time the Sun was the most massive body we could study
close up. The stars, known to be commonly much more massive than the
Sun, were too far away for detailed examination.
Redshift of Spectral Lines
------------------------
This effect follows directly from the Schwarzschild equation
because the time dilation is just the dilation of wavelength of the
sunlight, Which is to say, the 'clock ticks' are the successive
wavecrests of the light.
delT[m/s] = delT[m/m] / (1 - r| / r[m/s]))^(1/2)
lambda[m/s] = lambda[m/m] / (1 - r| / r[m/s]))^(1/2)
lambda[m/s] / lambda[m/m] = 1 / (1 - r| / r[m/s]))^(1/2)
This redshift is commonly called 'gravitational redshift'.
+-----------------------------------+
| GRAVITATIONAL REDSHIFT |
| |
| lambda[m/s] / lambda[m/m] = |
| - 1 / (1 - r| / r[m/s]))^(1/2) |
+----------------------------------+
The effect is very small in the only credible example Einstein had
at hand, the Sun. r| for the Sun is 2.9Km but the light is emitted
from the photosphere whose r[m/s] is 692,000Km. Hence
lambda[m/s]/lambda[m/m] = 1 / (1 - r| / r[m/s]))^(1/2)
lambda[m/s]/lambda[m/m] = 1 / (1 - (2.9) / (692,000))^(1/2)
= 1 / (1 - (4.19075E-6))^(1/2)
= 1 / (0.999995809)^(1/2)
= 1 / (0.999997905)
= 1.000002095
The wavelengths of the light, in a spectral line, differ by only 2
parts in 1 million! While spectrometers were able to sense such small
shifts, the shift on the Sun was smothered by the vastly larger
Doppler motions of the photosphere, electromagnetic line broadening,
rotational broadening, and many other well known and large factors.
Some residual shift was found and plausibly it could be a relativity
effect. But it was not at all convincing for most astronomers.
In the 1930s with the study of white dwarf stars, a much larger
redshift was found. White dwarfs are solar mass stars that have
contracted to about the size of the Earth. r[m/s] is about 1/100 that
for the Sun.
lambda[m/s]/lambda[m/m] = 1 / (1 - (2.9) / (6,920))^(1/2)
= 1 / (1 - (4.19075E-4))^(1/2)
= 1 / (0.9995809)^(1/2)
= 1 / (0.9997905)
= 1.0002096
The redshift is now about 2 parts in 10 thousand. In addition to the
more enhanced redshift, a white dwarf tends to have a much calmer
photosphere than the Sun. The surface gravity being much stronger, the
gases are held more tightly and move around far less. The two allow
the effect to be far easier to detect and confirm. This finding was
more convincing but still not conclusive.
Mercury's Orbit Precession
------------------------
We can not go thru the actual calculation for this test; it is
very tricky. But we can see that there can be a precession in a
reduced scenario. Allow Mercury to orbit the Sun in a circular orbit
of radius r[m/s] = 57.9E6Km. Here the time unit is taken as the
orbital period, which, after all, is nothing but a certain number of
seconds. So
delT[m/s] / delT[m/m] = 1 / (1 - r| / r[m/s]))^(1/2)
= 1 / (1 - (2.9) / (57.9E6))^(1/2)
= 1 / (1 - (5.0089E-8))^(1/2)
= 1 / (0.9999999499)^(1/2)
= 1 / (0.999999975)
= 1.000000025
Which is to say the orbital period we experience at Earth for Mercury
is 2 parts in 100 million larger than that felt by Mercury itself.
When we sense that Mercury completed one revolution around the Sun, in
our 88 days, Mercury in itself already completed that circuit and
nudged a little into the next one. If we mark a point on the orbit to
clock off laps, we see that when Mercury crossed that mark as felt on
Earth, Mercury itself saw its own self cross the mark a short time
ago. Let that mark be, as is common for orbital mechanics, the
perihelion point.
Then each lap of Mercury thru its perihelion as Earth senses it,
Mercury completed that lap and left the perihelion behind. So to us
the perihelion nudged forward in the orbit a bit, not much, each lap.
The fractional orbital period excess is one minus the total ratio we
found above, or 2.504E-8. In angular measure this amounts to
AngExc = (2.504E-8 lap) * (1.296E6 arcsec/lap)
= 3.245E-1 arcsec
Mercury's period is 88d and there are 36525d in one Julian century.
So Mercury runs (36525d)/(88d) = 415 laps/century. Each accumulates an
additional 3.245E-1 arcsec, so
AcumAngExc = (3.245E-1 arcsec/lap) * (415 lap/century)
= 13.469 arcsec/century.
This is quite one third of the value calculated by the strict
method, which is 43 arcsec/century. It is meant here only to show that
there can be an alteration in the motion of Mercury that is due to
spacetime warping in the vicinity of the Sun.
Bending of Starlight
------------------
The last of the three classical tests in astronomy for relativity
was the deviation of a star's position on the sky when that star is
seen next to the Sun. This comes about from the passage of the light
thru a stronger spacetime warp near the Sun somewhat like passing thru
a medium of stronger refractive index. The light is 'refracted' from
its straight path to appear as if it came from a little ways farther
from the Sun on the sky. With no derivation here we note that Einstein
predicted a displacement of 1.75 arcsec outward from the Sun for a
star at the Sun's limb. As it is utterly impossible to see stars near
the Sun this test was at first considered fatuous.
Eddington proposed to try the test during a solar eclipse. With
the Moon covering the Sun and the sky being a mid twilight darkness,
stars could be photographed near the Sun. For the eclipse of May 1919
his team did get pictures with several stars in the field of the Sun.
He compared these with pictures taken when the Sun was remote from
this area of the sky, when that part, in Taurus, was in the night sky.
He did get displacements of 1 to 2 arcsec outward from the Sun.
This was a very persuasive evidence for the relativity theory. On the
other hand observational difficulties cast some doubt on the results.
Did the camera change optical properties by day versus at night? Did
the gelatin on the plate expand in the day? Did the atmosphere refract
the stars thru daytime turbulence?
Astronomers Neglect Relativity
----------------------------
Because the effects of relativity were so tiny and did not upset
any of the work of astronomers in the 1920s thru the 1950s, most
simply skipped over relativity. They learned it thru their schooling
but, so what?
In the 1920s thru the 1950s astronomy simply did not work with
phaenomena or objects where the relativity effects became important.
Even tho they agreed that the Mercury orbit precession was real, it
was just as well to treat it as a perturbation added onto the strict
Newton orbit behavior.
In the 1960s the whole picture of astronomy changed overnight. The
emerging radio astronomy and improving optical instruments and the
placement of astronomy satellites in Earth orbit revealed new objects
and phaenomena. Many of them were of such violent or extreme behavior
that no tweaking of Newton physics could explain them. They did yield
successfully to Einstein physics. Today no astronomer can ignore
Einstein physics; it is the primary method of dealing with the cosmos.
The home astronomer so far lacks almost entirely any schooling in
relativity and is often made to feel it is too far beyond his
abilities. Certainly relativity can be utterly arcane and complex,
well outside the ordinary maths and science of the home astronomer.
But much can be learned and understood with skills not above
elementary calculus.
Features of Schwarzschild Spacetime
--------------------------------
A closer look at the Schwarzschild spacetime equation reveals to
critical features.
delT[m/s] = delT[m/m] / (1 - (r | /r[m/s]))^(1/2)
When r[m/s] = r|, the (1-(r|/r[m/s])) term becomes zero and the whole
right side become infinite. That is, as we edge closer to the mass,
approaching a distance r| from it, the time dilation experienced in
the safe frame grows without limit. The clock on the movable platform
is felt to run slower and slower until, at r| away, it stops
completely.
When r[m/s] < r|, the parenthesis term becomes negative and the
squareroot is not defined. We plain have no physics to describe what
happens to the s observer once he's closer to the mass than r|. Being
that the Schwarzschild surface is merely a mathematical one and not a
solid barrier, the s observer can transgress thru it without
impedance.
Near the Schwarzschild Surface
----------------------------
When the spaceship, the m frame, is near the Schwarzschild surface
its time is severely dilated as felt by the Earth. In addition, the
light from it is severely redshifted. We at Earth sense the spaceship
to run slower and slower as it approaches r| and the wavelength of its
light, radio, television, &c waves to grow without limit.
In the ultimate step from just outside the surface to on it, the
last wavecrest is emitted toward Earth but the very next one takes
infinite time to come out. The wavelength is now infinite. Similarly,
the last clock tick is emitted but the very next one is infinitely
long in coming.
The s observer stops, as felt by Earth, at the Schwarzschild
surface and freezes there for all eternity thereafter.
By this feature of the Schwarzschild surface, the first name for a
star that becomes a blackhole was 'frozen star'. But this could also
mean a star which in some sense has a icy cold hard surface. Wheeler
in 1967 at a conference on such stars coined the word 'black hole'.
This was also spelled 'black-hole' but is now ever more prevalently
one word 'blackhole'.
The term when translated into other languages litterally comes out
pretty naughty. In some cases, the English word is inserted within
quotes to avoid using the native term.
Length and Mass near a Blackhole
------------------------------
The gravity field near a blackhole exaggerates the general
relativity effects on length and mass, in addition to time. In special
relativity length and mass are also dilated by the same factor beta:
delR[m/s] / delR[m/m] = 1 / (1 - (v/c)^2)^(1/2)
delM[m/s] / delM[m/m] = 1 / (1 - (v/c)^2)^(1/2)
The kilogram of mass and the meter of length at the spaceship are
sensed to be bigger (longer, greater) at the stational platform
compared to their sizes within the very spaceship. Note well that,
like with the time dilation, works on Einstein physics will invert the
meaning of delR and delM. Be careful to read the text and not simply
copy off the equations.
So we may do the same substitution of (r|/r[m/s]) for (v/c)^2 and
get the dilations of length and time under the influence of gravity:
delR[m/s] / delR[m/m] = 1 / (1 - (r| / r[m/s[))^(1/2)
delM[m/s] / delM[m/m] = 1 / (1 - (r| / r[m/s]))^(1/2)
Thus within gravity both the mass and length unit of the movable
frame are felt to be larger by the safe frame than those within the
movable frame itself.
Geometry near the Schwarzschild Surface
-------------------------------------
Because length is dilated near a blackhole, the geometry based on
length measures is distorted. Consider the whole notion of 'distance
from the blackhole' and 'radius of the Schwarzschild surface'. What
meterstick are we using? We were very careful to note that the 'r' in
the dilation formulae was written 'r[m/s]'. The formulae are cast in a
way that allows us to use the flat undistorted meterstick. It's as if
we overlaid the spacetime near the blackhole with a transparent Euclid
coordinate grid and looked thru it at the blackhole.
So far so good. But the fellow on the movable platform feels the
situation very much different. In particular, due to the length
dilation, the unit of length is smaller to him than what the safe
frame senses. The disparity of his and the safe observer's assessment
of length rapidly grows as the movable observer approaches the
Schwarzschild surface. So in a given distance as seen thru the Euclid
grid by the safe observer, there are more meters for the movable
observer.
The situation can be seen better with a specific example. In
Newton physics we can add mass to a body to increase its radius. The
increment of radius per unit increment of mass (assuming constant
density thruout) is found by differentiating the volume-mass relation
M = 4 * pi * rho0 * R^3 / 3
R^3 = 3 * M / (4 * pi * rho0)
R = (3 * M / (4 * pi * rho0))^(1/3)
= (3 / (4 *p i *rho0))^(1/3) * M^(1/3)
1der(R, M) = (3 / (4 * pi * rho0))^(1/3) * (1 / 3) * M^(-2/3)
The mass increases weakly with addition of matter to the object.
In fact, each new increment of matter, as a layer on the preceding
mass, adds less and less radius!
Now for the blackhole we have an altogether different behavior of
mass versus radius. The mass-radius relation is
R = r|
= 2 * gamma * M / c^2
= (2 * gamma / c^2) * M
1der(R, M) = 2 * gamma / c^2
The radius increases linearly with added mass. From this simple
comparison we see that there is some very weird geometry prevailing
inside a blackhole that is utterly different from what we would expect
in Newton physics. As a matter of fact, the notion of 'geometry' in
any ordinary sense disappears. There is no parameter in this mass-
radius relation that involves geometry, like the number pi. There are
only two external nongeometric constants of physics, gamma and c.
Thermodynamics and Blackholes
---------------------------
Recall that in the sections on thermodynamics we examined entropy
and the second law of thermodynamics (SLOT). In a real thermoprocess
the entropy always increases. In an ideal situation of a reversible
process, which never attains in nature, entropy can at best remain the
same. There is no theoretical basis for this observed behavior of
entropy. Despite countless efforts to find one, so far SLOT remains
only the deduction of observation.
Now with blackholes there may possibly be a way to put ordinary
thermodynamics on a solid theoretical foundation. Not for sure but now
there is some hope of doing so. The reason comes from the way
blackholes behave.
When mass is added to a blackhole its radius, the Schwarzschild
radius, increases. If mass could be removed from a blackhole its
radius would decrease. But there is no possible way to get any matter
out of a blackhole once it's added into it. Mass entering a blackhole
is permanently lost to the universe. Hence, the radius of a blackhole
in communication with the universe, by being immersed in interstellar
gas and dust, can only increase its radius. At the very best, in total
isolation, its radius would remain the same.
When two blackholes combine, by collision or slow coalescence,
their masses add and the radius of the new single blackhole is larger.
Given enough time, all blackhole radii will increase in as much as all
blackholes will eventually add matter to them by one way or an other.
The parallel to entropy is striking. It is believed from brute
force experience that in the universe entropy is continually
increasing and any process we do merely augments entropy. Many
physicists are keen on the idea that the relentless increase of
blackhole radius is in some as yet unknown manner linked to the
relentless increase of entropy. In the latter case the behavior of
radius is based on natural outcomes of perfectly developed laws of
physics, those of Einstein,
Can it be that there is, after all, some more fundamental physics
underlying thermodynamics comparable to Einstein physics for
blackholes? Does Einstein physics some how comprehend thermodynamics
and all we know right now is the tiny smidgeon that deals with
blackholes? Are thermodynamics and relativity subsets of some other
allnew regime of physics? At the end of the 20th century no one knows.
The Singularity
-------------
The object that formed into a blackhole is collapsed into a pure
geometrical point at the center of the Schwarzschild surface. This was
not well accepted in the early days of blackhole studies. The concept
of a singularity in physics is not a happy one. Yet, it gradually was
realized that there can be no core or nucleus of finite volume inside
the blackhole and that all the mass making up the blackhole must be
within that one mathematical point in the center.
Arguments against the actuality of a singularity included two of
note to home astronomers. The volume of a gas is a linear function of
its temperature. In fact, this is one way to define the temperature
scale. The gas is confined in a cylinder topped by a weighted piston.
The height of the piston is proportional to the volume because the
lateral expansion of the gas is constrained by the cylinder.
As the gas is cooled, its volume decreases. In the limit, at
absolute zero, the volume is itself zero. So we now have a finite mass
compacted into a zero volume. This is one example of a singularity.
In fact, the world is built to prevent this singularity from ever
occurring. The gas freezes to a liquid, which in turn in almost of
constant volume regardless of further cooling. At absolute zero we
have a nonzero volume with nonzero mass.
In an other example, a telescope, by geometric optics, produces a
point image of incoming starlight. As the beam converges (for a simple
refractor) its power density increases, in watt/meter2. The influx
passes thru smaller and smaller areas downray from the front lens.
At the focal point the area is zero, but with nonzero power. The
singularity of infinite power density results.
In fact, the image, even by the most perfectly crafted optics, is
not a geometric point. It's a tiny but nonzero dot. This dot, of
oneish arcsecond angular diameter in homesize telescopes, is what
prevents unlimited resolution of features on, say, a planets. The dots
overlap from each ray of incoming light. The real result is a diffuse
image of tempered resolution.
The real-world image forming process forces the ideal point focus
into a finite size dot. The singularity is prevents.
Could not some process, as yet unknown, prevent the collapse of a
star (galaxy, &c) into such a point and create a singularity?
Newton Escapes the Singularity
----------------------------
In Newton physics the creation of a singularity can be avoided. An
element of the body has a weight downward to the center by gravity.
But the compressive or bearing strength of the underlying material can
support it from freefall. In deed, this is the whole reason why we can
build structures that do not collapse under their own weight.
For simplicity's sake we assume here a uniform incompressible
substance of density rho0. This is only to ease up on the maths for
the rho0, being a constant, can filter out of the integrations we will
have to perform.
The weight of this element is the mutual gravity force between it
and the mass within the sphere laying interior to it; recall the work
with Newton's shells. That is,
delF = -gamma * (rho0 * delr * A) * (4 * pi * rho0 * r^3 / 3)
/ r^2
= (-4 * gamma * pi * rho0^2 * A * r/3) * dr
= (-4 * gamma * pi * rho0^2 * A / 3) * r * dr
A is the bottom area of the element, r is the radius from the center
to the element, dr is the vertical thickness of the element. The minus
sign means that the force is directed downward, against the sense of r,
which is directed upward. In some works this minus sign is sucked into
gamma, it being declared a negative number.
F = intl((- 4 * gamma * pi * rho0^2 * A / 3) * r, dr)
= (-4 * gamma * pi * rho0^2 * A / 3) * (r^2 / 2) + const
= (-4 * gamma * pi * rho0^2 * A / 6) * (r^2) + const
= (-2 * gamma * pi * rho0^2 * A / 3) * (r^2) + const
The pressure the element exerts on the underlying material is this
force divided by the bottom area of the element.
P = F / A
= (-2 * gamma * pi * rho0^2 * A / 3 * A) * (r^2) + const
= (-2 * gamma * pi * rho0^2 / 3) * (r^2) + const
= (-2 * gamma * pi * rho^2 * r^2) / 3 + const
Evaluating this between r = 0 at the center and r = R at the
surface, we get
P = (-2 * gamma * pi * rho^2 * R^2) / 3
- (2 * gamma * pi * rho^2 * 0^2) / 3
= (-2*gamma*pi*rho^2*R^2)/3
This is the pressure at the center of the body -- of uniform density
rho0! -- due to the weight of the material it's made of. This pressure
has to be supported by internally generated resistance in the
material.
Because in astronomy we commonly suss out the mass of an object
before we figure out its density, we can turn this expression in terms
of mass by recalling that density = (mass/volume). So we have also
P = (-2 * gamma * pi * rho^2 * R^2) / 3
= (-2 * gamma * pi * (3 * M / (4 * pi * R^3))^2 * R^2) / 3
= (-2 * gamma * pi * (9 * M^2 / (16 * pi^2 * R^6) * R^2) / 3
= (-2 * gamma * pi * 9 * M^2 * R^2) / (16 * pi^2 * R^6 * 3)
= (-18 * gamma * pi * M^2) / (48 * pi^2 * R^6)
= (-3 * gamma * M^2) / (8 *pi * R^4)
+-------------------------------------------+
| NEWTON CENTRAL PRESSURE FOR CONSTANT rho0 |
| |
| P = (-2 * gamma * pi * rho^2 * R^2) / 3 |
| = (-3 * gamma * M^2) / (8 * pi * R^4) |
+-------------------------------------------+
Note that for any nonzero density rho0 (or mass m) and nonzero
radius R the pressure P is always a finite value. Thus, at least in
principle, some material can be found, invented, concocted that will
resist this pressure. And in this way the singularity can be avoided.
We do not say there always is such a material, but only that the hope
exists in the nature of things that such a resistant substance may
exist, yet to be known to us.
Einstein Mandates the Singularity
-------------------------------
When general relativity is applies to the calculation of the
central pressure the maths gets a lot tougher. We have to present just
the ultimate result here, which was first worked out by Tolman,
Oppenheimer, and Volkoff in 1939.
P = (rho0 * c^2) * (1 - (1 - r| / R)^(1/2) / (3
* ((1 - r| / R)^(1/2)) - 1)
= (3 * M * c^2 / (4 * pi * R^3))
* (1 - (1 - r| / R)^(1/2) / (3 * ((1 - r| / R)^(1/2)) - 1)
where we toss the equation into terms of mass.
+-------------------------------------------------------------+
| EINSTEIN CENTRAL PRESSURE FOR CONSTANT rho0 |
| |
| P = (rho0 * c^2) * (1 - (1 - r| / R)^(1/2) |
| / (3 * ((1 - r| / R)^(1/2)) - 1) |
| |
| = (3 * M * c^2 / (4 * pi *R^3)) * (1 - (1 - r| / R)^(1/2) |
| / (3 * ((1 - r| / R)^(1/2)) - 1) |
+-------------------------------------------------------------+
The first point to discover about the Einstein formula for central
pressure is that it gives a higher pressure for a given combination of
mass and radius. WHich is say, it is harder under general relativity
to sustain a body against its own weight than in Newton physics. This
is illustrated in the table here using the Sun as the example, for
which m = 1.989E30Kg and r| = 2.953E3m. The pressure is in
newton/meter2, also called pascal. For reference, one atmosphere
pressure on Earth is just about 100,000n/m2 = 100,000p.
R, Km P[New], p P[Ein], p
------- --------- ---------
6.96E5 1.342E14 1.342E14
6.00E5 2.430E14 2.431E14
3.00E5 3.889E15 3.889E15
1.00E5 3.150E17 3.151E17
1.00E4 3.150E21 3.152E21
1,000 3.150E25 3.160E29
100 3.150E29 3.247E29
10 3.150E33 4.512E33
6 2.430E34 4.990E34
4 1.230E35 6.089E35
3.6 1.875E35 1.939E36
3.4 2.357E35 7.886E37
3.33 2.562E35 8.145E38
3.323 2.583E35 7.362E39
The relativity effects do not show up until a very compacted state
is reached; we are taking the mss of the Sun and compressing it into
smaller and smaller volume. Near the ultimate state of compression,
explained below, the ratio of Einstein to Newton central pressure
excedes 2,000 to 1!
But there's more.
Unlike in the Newton case, we can not arbitrarily assign values
for R and rho0 and get a finite central pressure P. The denominator
goes to zero -- the pressure goes to infinity! -- when
3 * (1 - r| / R)^(1/2) - 1 = 0
3 * (1 - r| / R)^(1/2) = 1
(1 - r| / R)^(1/2) = 1 / 3
1 - r| / R = 1 / 9
r| / R = 8 / 9
r| = 8 * R / 9
R = 9 * r| / 8
Note well that this is more than the Schwarzschild radius. What
this means is that the matter of a body need fall within 9/8 of its
own r| and then it must collapse into the blackhole. The central
pressure is at this state already infinite before the blackhole is
formed. There is the distinction between conditions to form a
blackhole and those within the blackhole after its creation.
+-------------------------------------+
| LIMITING RADIUS TO FORM BLACKHOLE |
| |
| M = 8/(9*r|) = 4*c^2*R/(9*gamma) |
| |
| R = 9 * r| / 8 |
| = (9 * gamma * M) / (4 * c^2) |
| |
| R^2 = c^2 / (3 * gamma * pi * rho0) |
+-------------------------------------+
Do recall that we derived the Newton, and presented the Einstein,
central pressure equations for a body of uniform density, rho0. They
apply strictly only to such objects. However, it can be demonstrated,
tho not here, that if rho is a inverse function of radius,
continuously decreasing with distance from the center of the body,
there are corresponding formulae that also contain the limit of
infinite central pressure. Thus, because such radial decrement
densities are the usual property of astronomy objects, we believe that
the re is a real absolute limit to the size of bodies in the universe
and that blackholes can actually be created when thee limits are
exceded.
In the case of the Sun, with r| = 2.953Km, the smallest volume we
can compress its matter before it spontaneously falls into a blackhole
is (9/8)*(2.953Km) = 3.322Km. Thus the peculiar cutoff in the table of
central pressures laid out above.