AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy | Gravity, black holes and physics |
Chapter 1
GRAVITATION AND ITS PLACE IN PHYSICS
1.1. Historical development of knowledge
about gravity
1.2. Newton's law of gravitation
1.3. Mechanical LeSage hypothesis
of the nature of gravity
1.4. Analogy
between gravity and electrostatics
1.5. Electromagnetic
field. Maxwell's equations.
1.6. Four-dimensional
spacetime and special theory of relativity
1.2. Newton's law of gravitation and its modifications
Until the middle of the
17th century, there were two completely different and seemingly
unrelated doctrines of motion side by side: terrestrial
mechanics
dealing with the motion of ordinary bodies and celestial mechanics studying the motion of planets
and stars.
I. Newton followed Galileo's kinematics and built the dynamics of
motion of bodies summarized in three well-known Newton's laws (see §1.6 "Four- dimensional spacetime and special
theory of relativity ", passage
" Newton's
classical mechanics ") . The resulting classical mechanics was (and is still to this day)
able to explain all the movements of bodies that we encounter in
everyday life.
^{ }Newton's predecessor, J. Kepler,
summarized a large number of astronomical observations to date
and traced the general laws governing the motion of planets in
the solar system :
These empirical Kepler's laws served I. Newton as a starting point for
establishing an even more fundamental law that governs not only the
planets, but all bodies "in heaven and on Earth."
^{ }Above all, Newton realized that
the motion of the planets did not conform to the law of inertia.
The planets move in curved orbits (ellipses) around the Sun, so
they must be subjected to a force directed at the Sun - an
attractive force emanating from the Sun. Kepler's third law
applied to the special case of circular orbit says that the
squares of periods 4 p ^{2}
. r ^{2} / v ^{2} are proportional to the squares
of the radius r . Then centripetal acceleration v^{2}/r must be proportional to 1/r^{2}
. A similar lead can be shown for elliptical movement. Due to
Newton's second law on force causing the planet mass m
such centripetal acceleration must be therefore proportional to the m /
r ^{2} . According to the law of action and
reaction, however, the planet must act with a force of the same
magnitude as the Sun acts on the planet, and this force will be
proportional to the mass M of the Sun. The mutual attractive force
between the planet and the Sun will therefore be proportional to m.M / r ^{2} .
The law of general
gravitation
By analyzing Kepler's laws, Newton found that the motion of
planets in the solar system can be easily explained by the
hypothesis that each two bodies are
attracted
to each other by a force directly proportional to the
mass m_{l} and m_{2 }of each and
indirectly proportional to the square of the distance r among them - Newton's law of
gravitation :
m_{l} . m_{2} F = G . ------------ . r° , r° = r / r , r^{2} |
(1.1) |
where r ° is a unit vector indicating the direction
from the body m _{l} to the body m _{2}
. The coefficient of proportionality G - the gravitational
constant -
must be determined empirically (from observation or experiment),
see below.
^{ }In addition to deriving the form
of the law of attraction between celestial bodies, Newton showed
that this force has the same nature as the gravity
of the earth, forcing all free bodies to fall to the
ground with acceleration. Newton, by comparing the acceleration
of the Moon's motion as it orbits at the appropriate distance
around earth and by the acceleration of free-moving
bodies at the earth's surface, he found that the magnitudes of
these accelerations correspond to the law of inverted squares and
therefore agree with the law (1.1), provided that both the fall
of bodies and the orbit of the Moon around the Earth are
conditioned by the Earth's gravity.
^{ }The finding that the force that
compels the orbit of the planet around the sun or moon around the
planet is the same force that causes the fall of bodies to the
ground and called weight , or gravity , united on a common basis before
completely different phenomena and areas:
mechanics, the force
of gravity and "heavenly "mechanics. Equation (1.1) is
therefore called Newton's law of
general gravity .
Gravity
- an irremovable all-pervading force^{ }^{ }
The electrical action between charged bodies (which can be
attractive or repulsive according to the signs of charges)
depends on the material environment and can in principle be shielded (even by
an alternating electromagnetic field using a Faraday cage
) . Gravity, which
is always attractive, however, is an all-pervading
force that
cannot be removed in any way. Gravity cannot be "blocked the
way" by anything, it cannot be
shielded .
If we inserted some massive material between two gravitational
bodies, we would only achieve an amplification
of the total
gravitational action: the gravitational force of the inserted
material would add up in vector to the original gravity.
Universal action of gravity^{ }
Since the attractive force acting on each
body is proportional to the mass of that body, Newton's law is
also consistent with Galileo's law of free fall, according to
which all bodies fall to the ground with equal
acceleration regardless of their mass and composition . 2.2) .
^{ }According to 2.Newton's law of classical
mechanics - the law of force and acceleration -
the force F needed to set a body in motion is proportional
to its mass m and acceleration a : F = m . a.
If this force is gravity, then for a test body of mass m
moving under the influence of gravity of the central body of mass
M (eg Earth)
at a distance r , we can write Newton's law of gravitation
on the other side of the equation: m . a = G. Mm
/ r ^{2} . On
the left and right side of the equation there is a mass m of
the test body, which can be shortened , so the
acceleration no longer depends on the mass of the observed body -
it depends only on the mass of the central gravitational body and
the mutual distance: a = G. M / r ^{2} . As a result, the gravitational force gives
all bodies the same gravitational acceleration , which explains Galileo 's law
of free fall. Although a heavier body is subjected to a greater
gravitational force, which at the same time must do more
"work" to accelerate it - this heavier body puts more
"resistance" to acceleration.
^{ }Further physical knowledge has shown that
this property of the same - universal - action
of gravity applies not only to ordinary matter ,
but also to their building particles such as
electrons, protons, neutrons, as well as to electromagnetic fields
and waves such as light and its quantum. photons
(undoubtedly also applies to other particles
and antiparticles ) . It was this
fact that proved to be the most important property of gravity and
the key to its new understanding withingeneral theory of
relativity (§2.1-2.4 of Chapter
2, especially §2.2 " Universality - a
basic property and the key to understanding the nature of gravity ") .
Gravitational
bodies^{ }
Fig.1.0. Newton's law of general gravity.
Above: Basic situation of gravitational action
between two (idealized) point bodies with masses m _{1} and m _{2} .
Middle: Newton's law of gravitation also applies to the
gravitational action between spherically symmetric bodies of
finite dimensions.
Bottom: A spherically symmetric hollow body
(spherical layer or shell) has zero gravitational force inside
the cavity, outside the body the gravitational force is given by
the standard Newton's law.
Fig. 1.0 shows three situations in which this
law of gravity applies exactly . The basic
theoretical situation according to the upper part of the figure
consists in the gravitational action between two " point
" bodies , in practice bodies whose
dimensions are negligibly small in comparison with their mutual
distance. However, Newton's law of gravitation also applies
exactly to the gravitational action between bodies of finite
dimensions with a spherically symmetric
distribution of mass density (middle part of the
figure). And not only in the outer space around the body, but
also inside the spherically symmetric body - the
gravitational force F (r) at a distance r from the center
is given by the amount of mass m (r) = 4 p _{0 }n ^{r} r ( r ). r ^{2} dr contained
in an imaginary sphere of radius r : F (r) = Gm (r) / r ^{2} ; matter in the outer
shells is not applied, its gravity acting from different
directions is canceled (this situation is
also discussed in §2.4, passage " Time dilation inside gravitational bodies ") . In the lower part of
Fig. 1.0 there is a curious theoretical case of a spherically
symmetric hollow body - a spherical layer or
shell (it does not occur in nature) . The test specimens located inside the cavity
are not subjected to any gravitational force (the
gravitational field inside has zero intensity ,
there is a constant gravitational potential, gravitational forces
from different directions are canceled) ,
but outside the body the gravitational force is exactly given by
Newton's law.
^{ }In all other configurations of asymmetric
inhomogeneous mass density distribution in bodies of finite
dimensions, the law of inverted squares applies only approximately
, these bodies cannot be completely replaced by their center of
gravity (due to the quadratic nonlinearity
of the law of gravitation) . In the general
case of a source of volume V , in which the mass
distribution is given by a function of the local density r ( r
), the gravitational force acting on the test particle mgiven
by the vector volume integral F = -Gm ò _{V} r ( r´ ). I r - r´ I ^{-2} ( r - r ´) ^{0} dV ´
of the
vector sum of contributions from individual material elements of
different density r ( r ) in the source.
Deviations from homogeneity and spherical symmetry^{ }
For large material bodies in space (with a diameter greater than
about 1000 km), strong gravity automatically ensures an
approximately spherical shape with an almost spherically
symmetrical density distribution. However, if such a body rotates
, it will not be spherical, but only axial symmetry
(cf. §3.6 " Kerr
and Kerr-Newman geometry
" and §4.4 " Rotating and electrically
charged Kerr-Newman black holes
").
^{ }However, within terrestrial-type planets,
there are certain local inhomogeneities in the
mass density distribution that cause minor anomalies and
fluctuations in the surrounding gravitational field. This is
manifested by minor irregularities and deviations from the Kepler
orbits(see below " Motion
in Newton's gravitational field ")orbiting satellites. These irregularities are more
pronounced in the low orbits of the satellites, decreasing at
greater distances until they practically disappear. Gravimetric
measurements on the Earth's surface and analysis of deviations in
the orbits of satellites can be used in geology to detect the internal
inhomogeneities of the structure of our planet - and
thus to identify possible mineral deposits.
Gravitational
constant^{ }
For the universal constant G , acting as a coefficient of proportionality
in the law of gravity (1.1) - Newton's gravitational
constant -
the value was experimentally determined
G » 6.67384. 10 ^{-11} kg ^{-1} m ^{3} s ^{-2} . | (1.2) |
A very small value of
the gravitational constant G indicates that the gravitational
action is relatively weak and can only be more pronounced
if at least one of the interacting bodies has a considerably
large mass. Therefore, the mutual gravitational attraction of
ordinary macroscopic bodies is not applied in practice, we do not
observe it in everyday life and it can be demonstrated and
measured only by highly sensitive methods (such
as Cavendish and Eötvös measurements) . Gravity is the domain of very material bodies in the universe.
^{ }The weakness of the gravitational
interaction, among other things, makes it very difficult
to accurately measure the gravitational constant. The
first measurement of the gravitational constant was made as early
as 1798 by H. Cavendish, who used two lead balls weighing 730 g,
mounted on a horizontal arm suspended on a thin fiber. This
mechanical system functioned as a torsion balance ,
which was fitted with a mirror from which a beam of light was
reflected indicating the deflection of the arm. Cavendish
approached the bullets from both sides with a pair of larger lead
balls (160kg) and measured the torsional weight arm deflection
using a mirror. Cavendish's experiment was originally designed to
determine the mass and density of the Earth; later analyzes of
its results determined the value of the gravitational constant G » 6,74.10
^{-11} kg ^{-1} m ^{3} s ^{-2} ,which
differs by only 1% from the current value.
^{ }Despite the fact that the gravitational
constant G is one of the most important
and most basic natural constants, in comparison with other
physical constants we know its value with only a relatively small
accuracy of about 10 ^{-4} (to 4 decimal places). The reason for this
"lousy" accuracy is the above-mentioned weakness
of the gravitational interaction. Perhaps future planned
highly sensitive quantum-radiation measurements using
excited atoms in various quantum states, subjected to the
gravitational action of test masses, can provide some hope for refining
the value of the gravitational constant .
Gradients of gravitational forces - tidal
forces^{ }
The gravitational field, excited
around bodies according to the law of gravity, is inhomogeneous
, in the vicinity of a gravitational body it is significantly
stronger than at greater distances. Thus, when a finite-sized
body is affected by the gravity of another body, the
gravitational force on the near and far sides may be slightly different
- a gravitational force gradient is created ,
given by the vector difference in gravitational force
(acceleration) between the two locations. If this body is not
absolutely rigid (in §1.6 we will see that
the theory of relativity "forbids" absolute rigidity) , it leads to deformation body shape
(eg, originally a spherical shape extends into an ellipsoid).
Gravitational gradients causing this effect are called tidal
forces ( tide = rapids,
slapping, swelling of water ) , or ebb
forces . Thus, tidal forces are created due to the non-zero
dimensions of mutually attractive objects: when these
two bodies are attracted, a greater attractive force will act on
each of them on the side closest to the other body than on the
opposite (averted) side.
^{ }The name comes from the fact that these
forces are well known from our ordinary earthly life - they cause
sea tide and outflow. This is the gravitational
action of the Moon (to a lesser extent the Sun). During the daily
rotation of the Earth around its axis, its individual regions
move to the Earth-Moon line and tidal forces cause deformations
in this direction. The Earth's crust deforms relatively little (but see below) , but large masses
of ocean water can respond to the Moon's attractive gravitational
forces almost instantly - they can move around the Earth's
surface and ocean levels periodically deform ,
increase and decrease several times a day even a few meters. The
Moon attracts water more strongly in the oceans that are closest
to it - a small "bulge" forms on the surface of the
ocean, which manifests itself as a tide at the coast.
The magnitude of tidal forces
can be determined from the law of gravity (1.1) as^{ }vector
difference of gravitational force between two given places
around a gravitational body *). If we have a (spherical)
gravitational body of mass M , in whose gravitational
field there is another (test) spherical body of diameter r
and mass m at a distance R , then a tidal force F_{grad} will act on the
edges of this body in the direction of the joint
(with the basic body M ) :
^{«} F _{grad} = F (R + r) - F (Rr) = G. [mM /
(R + r) ^{2}
] - G.
[mM / (Rr) ^{2} ] » - 2 GmMr / R ^{3} .
This tidal force acts outwardly and stretched
body in the direction towards the connector gravitated body M
. In an analogous manner, the tidal force acting on a body of
diameter r in the direction perpendicular
to the junction of the two bodies can be determined as a vector
difference:
_{á }^{â} F _{grad} = G. [mM / R ^{2} ] .sin (2r / R) » GmMr / R ^{3} .
In the direction perpendicular, the tidal force is about half and
points inwards - in this direction it compresses
the body. Thus, tidal forces " stretch
" the bodies in the direction of their joint and at the same
time " compress " them in the
perpendicular direction.
*) The resulting simplified expressions ( » ) are created by linearization
- in algebraic expressions or in Taylor power expansion
only members with first powers r , R are left ,
while members with higher powers are neglected.
Tidal forces are more pronounced in principle in three basic
situations (and their combinations) :
1. Close circulation of
two bodies, such as tight binary stars. Tidal forces periodically
deform the partially elastic material of both bodies, with
viscous friction generating heat therein . As a
result, part of the kinetic energy of the orbital motion is
converted into thermal energy, the orbits approaching,
the circulation time is shortened ...
2. A rotating body orbiting near another
gravitational body. This is the case of the rotating Earth,
around which the Moon orbits, mentioned above. In addition to the
deformation of ocean levels (tides), there are also periodic
deformations of the earth's crust and mantle, which are
transformed by viscous friction into heat
heating the planet's interior. It is estimated that approximately
40% of geothermal energy comes from this
gravitational-tidal mechanism inside the Earth (most geothermal energy comes from the radioactive decay
of natural isotopes - see §1.4 " Radionuclides
", passage " Geological significance of
natural radioactivity
" in the book " Nuclear Physics and Physics
ionizing radiation "). This tidal
"warm-up" mechanism also takes place on some moons
around the outer planets of the solar system, which would
otherwise be very cold. An example is Jupiter's moon Io
, in which heating by intense tidal influences leads to strong
volcanic activity.
Interestingly, the tidal
dissipation of rotational energy has apparently slowed and stopped
the earlier rotation of the Moon , which is now facing the
same side of the Earth. Furthermore, since the Moon orbits the
Earth in the same direction as the Earth's rotation about its
axis, tidal forces in the Earth's material cause a gradual transmission.the
rotational momentum of the Earth to the orbital momentum of the
Moon, whose orbit, as a result, moves away from the Earth by
about 40 mm per year. And the Earth's rotation slows down
slightly. In the hypothetical case of a
counter-rotating orbit, the Moon would approach the Earth.
3. Compact gravitationally
collapsed bodies with an extremely strong gravitational
field - neutron stars and especially black holes - cause such
strong gravitational gradients and enormous tidal forces
in their vicinity that even free-falling bodies are intensely
stretched in the radial direction and compressed in the vertical
direction ( there is a kind of
"spaghetti") so that in the end
they end up as a "string" of atoms or elementary
particles (§4.2, passage "An
observer falling into a black hole ").
^{ }In tidal phenomena,
the laws of conservation of energy and
momentum apply . The kinetic energy of the
rotation of bodies and their circulation is partially converted
into thermal energy in the material of bodies. The momentum is
"redistributed" between the rotational motion around
its own axis and the orbital motion of bodies around a common
center of gravity. The intensity and character of
these transformations depend on the individual geometric, kinetic
and material configuration of the bodies involved.
Body
motion in Newton's gravitational field
Within classical physics motion of bodies in Newtonian
gravitational field controls the basic second
Newton's law
of force F and acceleration a : a o d ^{2 }y / dt ^{2} = F , where after the force F is substituted gravitational force by
inverse square law (1.1). Newton's law of gravitation naturally
follows not only the original Kepler's
laws , but also
other observed properties and possibilities of planetary motion,
star motion in binary and multiple systems and within galaxies,
hydrostatic equilibrium in stars (except
for the final stages). and other effects.
Motion of a body in a
centrally symmetric gravitational field . Escape
velocity.
In addition to the idealized and simple case of a homogeneous
gravitational field (such as free fall and
throw of bodies in the Earth's gravitational field) , the simplest and basic task for
moving a body in a gravitational field is when we have one point (or centrally
symmetric) gravitational
body of mass M and investigate the motion of a small test particle in its gravitational field. It is therefore important to investigate the movement of
the test particle mass m in a centrally symmetric field
excited Newtonian gravitujícím body M . On the test
particle^{ }m at a distance r
from the center of the body M , according to (1.1), a gravitational
force F = GmM /
r ^{2} will act , which according to the law of force and acceleration will give it acceleration and
according to the equation m . a
= G. mM / r ^{2} . The mass of the test body m is
shortened here, so that the gravitational
acceleration
will be
a º d ^{2} r / dt ^{2} = G. M / r ^{2} ; | (1.3a) |
has a centripetal direction and does not depend
on the mass m - all bodies fall with the same
acceleration . At the point source r = 0, the gravitational
acceleration of the test particle would be theoretically
infinite.
^{ }If the spherical gravitational body M
is of finite radius R (situation
according to Fig. 1.0 in the middle) , the
simple dependence (1.3a) will apply only outside the body for
r> R, while inside for r <R the dependence will be more
complex, depending on the density r (r ) mass distribution. The
gravitational force F (r) at a distance r from the center
is given by the amount of mass m (r) = 4 p _{0 }ò ^{r} r ( r ). r ^{2} dr contained
in an imaginary sphere of radius r : F (r) = Gm (r) / r ^{2} (the mass in the outer shells does not
apply, its gravity acting from different directions is canceled) . The gravitational acceleration of the test particle m
for a spherical gravitational body M of radius R is
therefore:
a (r) = G. M / r ^{2} for r> R; a (r) = G. [ 4 p _{0 }ò ^{r} r ( r ). r ^{2} dr ] / r ^{2} for r <R. | (1.3b) |
The gravitational acceleration for an extended
body M is finite everywhere, in the center r = 0 it is zero
(the gravitational force is zero there) .
^{ }It is also useful to determine the potential
energy U (r) of the test particle m in the
gravitational field at r . It is the work done against the
gravitational force F (r) = GmM / r ^{2} , necessary to "pull" a particle of mass m
from the point r of the gravitational field to an infinite
distance r ® ¥ , where gravity
no longer acts and the potential energy is there already zero. It
is therefore given by the integral U (r) = Gm _{r }ò ^{¥} (M / r ^{2}) dr = - GmM / r. The gravitational potential j
(r), which is the potential energy of the body m per unit
of its mass U(r) / m , is
j (r) = - G. M / r . | (1.4a) |
This simple relation applies for all distances r
only for the point source M ; in its
place r = 0 the potential acquires an infinite value (similar to the gravitational force F) . If the spherical body M is of finite radius R
(situation according to Fig. 1.0 in the
middle) , a simple dependence will apply
only outside the body for r> R, while inside for r <R the
dependence will be more complex, depending on the density profile
r (r)
of mass distribution:
The amount of mass contained in an imaginary sphere of radius r
is m _{r} = 4 p
_{0 }ò ^{r} r ( r ). r ^{2} dr , so
the gravitational force per unit mass of the test particle will
be F (r) = G. ( _{0} ò ^{r} 4pr (r) r ^{2} dr ) / r ^{2} and the potential will be the integral of this force j (r) = - _{r }ò ^{¥} F (r) dr. If we divide the space around the body M
into the outside r> R and the inside r <R, the
gravitational potential will be formed by two components:^{ }
j (r) = - G. M / r for r> R; j (r) = -G .M / R - G. [ _{0 }ò ^{R} F (r) dr ....... ] for r <R. ... (.... divorce ....) | (1.4b) |
The gravitational potential for an expanded
body M is finite everywhere (if, of
course, the mass density is finite) , at
the center r = 0 it has a maximum value (even
if the gravitational force is zero there) ;
somewhat lower on the surface r = R: j (R) = -GM / R.
^{ }The gravitational potential j plays an important
role in the general theory of relativity , as the
physics of gravity and spacetime (starting
with §2.4 " Physical laws in curved spacetime " and others) .
(Please do not confuse the designation
"phi" with the coordinate "j" in the polar
coordinate system!)
^{ }The simplest motion of a body in
the central field is radial motion
in the direction of
the body M , or in the opposite direction,
"away" from the gravitational body. The result of this movement will depend on the radial velocity
v of movement
of the body in the place of the distance r from the center
of the central body M .
^{ }As it moves outwards from the body M
, the velocity of the body v gradually decreases with
increasing r due to gravitational attraction . According
to the law of conservation occurs converting the kinetic energy
of motion mv ^{2} /2 potential energy G.m.M / R in gravitational field.
If m.v^{2}/2
= G.m.M / r, ie the sum of the kinetic and potential energy of
the particle is zero, the particle continues to move to infinity (where there is zero potential energy) and stops there.
^{ }Thus, the so-called escape
velocity v_{2} is very important , which is the smallest radial
velocity needed for the body m to leave the gravitational
field of the central body M from a distance r
forever and move to infinity :
v _{2} (r) = Ö ( 2 G M / r). | (1.5) |
The escape velocity does not depend on the mass m
or the composition
of the escaping body (universality of
gravity) , it
depends only on the mass M of the gravitational body and the
radius r from which the escaping body starts (for a body starting from the Earth's surface the escape
velocity is about 11.2 km / s - the so-called second
cosmic velocity , which is why we mark it v_{2} , the escape velocity
from the "surface" of the Sun is significantly higher,
about 617 km / s) . In §4.2 (section " Complete gravitational collapse.
Black hole. ") We will see that in the gravitational collapse of
massive stars, the escape velocity can reach the speed of light -
thus creatingblack hole .
^{ }Another important "cosmic"
velocity in the central gravitational field is the velocity
of the circular orbit v_{1} , which a body of mass m must reach in the
azimuthal direction in order to balance the centrifugal force F _{rot} = mv^{2} /r with the
attractive force F _{g} = GmM / r^{2 }of the gravitational
of a body M so that the body can orbit in a
circular path of radius r :
v _{1} (r) = Ö ( G M / r). | (1.6) |
Again, it does not depend on the mass m
or the composition
of the orbiting body (universality of
gravity) , it
depends only on the mass M of the gravitational body and the
radius r along which the body orbits (for satellites orbiting the Earth in low orbit near the
planet 7.9 km / s - the so - called first cosmic
velocity , that's why we mark it v_{1} ) .
By comparing equations (1.5) and (1.6) we see that the escape
velocity v_{2}
is Ö2 (= 1.414) - multiple of the
circular orbital velocity v_{1} .
Note:^{ }
The additional cosmic velocities mentioned in cosmonautics (3-6)
are given by the specific situation of the Earth and its orbit
around the Sun. They are therefore not of universal significance
and will not be considered in our general treatise on
astrophysics and gravity.
^{ }We will analyze more complex types of
movements below in the reduced motion of two bodies :
Gravitational
motion of two bodies
In the case of Newton's gravitational field, it is shown (will be shown below) that the motion of two
bodies can
be converted to the problem of the motion of one test body . So let's have two bodies with
masses m _{1} and m _{2} , which act on each other by
gravity according to Newton's law (and are otherwise free). The
equations of motion of these bodies will then be
d^{2}r_{1} m_{1}m_{2} r_{1}r_{2} d^{2}r_{2} m_{1}m_{2} r_{1}r_{2} m_{1} ------ = -G --------- ----- , m_{2} ------- = -G --------- ----- , dt^{2} r^{2} r dt^{2} r^{2} r |
where r _{1} , r _{2} are the position vectors of the bodies m _{1} and m _{2} with respect to the given reference point O (origin of the reference system), r _{12} = r _{2} - r _{1} is the position vector of the body m _{2} with respect to the body m _{1} (ie r _{21} = - r _{12} ) ar = | r _{12} | = | r _{21} | is the distance between the two bodies. Subtracting both equations gives an equation describing the relative motion of the body m _{2}relative to the body m _{1} :
d
^{2 }r
m
_{2}
( m _{1} + m _{2} )
r m _{2} ------ = -G ---------------- -----, dt ^{2} r ^{2} r |
(1.10) |
where r = r _{12} is the position vector of the body m _{2} with respect to the body m _{1} .
It is advantageous to place the origin of the O coordinates in the common center of gravity of both bodies. Then m _{1} applies . r _{1} + m _{2} . r _{2} = 0, so that the position vectors of the individual particles are to their distance vector r associated relationships
r _{1} = [m _{2} / (m _{1} + m _{2} ] r , r _{2} = [-m _{1} / (m _{1} + m _{2} ] r . | (1.11a) |
If we introduce a quantity
m _{1} . m _{2 }_{m = ----------------- }m _{1} + m _{2} | (1.11b) |
called reduced mass , the equation of motion (1.10) can be written in the form
m. d ^{2 }r / dt ^{2} = F (r), F (r) = -G (m _{1} m _{2} / r ^{2} ). r / r. | (1.12) |
This relationship has the form of equation of motion of a single particle of mass m moving in an external field F (r) which is formed by the m _{1} and m _{2} and is symmetrical relative to the origin of the coordinate r = 0. The determination of the motion of two interacting bodies is thus reduced to the problem of the movement of a single imaginary body m in a centrally symmetric field around a fixed center of gravity. If this problem has a solution r = r (t), it is easy to determine the individual trajectories r _{1} = r _{1} (t) and r _{2} = r _{2} (t) on the basis of relations (1.11a) of the original bodies m _{1} and m _{2} .
If we multiplies equation (1.12) the scalar vector v = r^{. }, we can write them in the form after editing
d / dt [ ^{1} / _{2} M ^{2} + U (r)] = 0; U (r) = -Gm _{1} m _{2} / r, |
here ^{1}/_{2} mv^{2} is the kinetic energy and U (r) is the potential energy of the body m , related to the field strength in the relation
F = - ¶ U (r) / ¶ r = - dU / dr. r / r. |
It follows that the total energy E of a particle is a constant independent of time:
^{1}/_{2} mv ^{2} + U (r) = E = const. , | (1.13) |
which expresses the law of conservation of energy during the motion of a body of mass m in Newton's central field.
Trajectories of bodies
If we introduce the polar coordinates r, j in
the path plane (motion in the central field is planar) , it is
possible to divide the equation of motion (1.12) into radial and
tangent components :
d ^{2} r / dt ^{2} - r. (d j / dt) ^{2}
= - (1 / m). dU / dr = - (G / m). m _{1} m _{2} / r ^{2} ,
(1 / r). ^{d}/_{dt} (r ^{2} .d j / dt) = 0. |
(1.14) |
Solution of the second equation
r ^{2} .d j / dt = L = const. | (1.15) |
expresses the law of conservation of the momentum when moving in the central field and at the same time Kepler's law of surfaces (area velocity (1/2) r ^{2} j^{.} is constant over time), where the quantity L is the momentum per unit mass of the particle ("specific momentum") L = J / m.
Equation (1.14) can then be written in the form with respect to (1.13)
m. d ^{2} r / dt ^{2} = mL ^{2} / r ^{3} + U (r). | (1.16a) |
However, it is easier to follow the law of conservation of energy (1.13 ) rewritten in polar coordinates
(1.16b) |
which is an equation describing the radial component of motion. From the law of conservation of momentum (1.15) rewritten in the form d j = (J / mr ^{2} ) dt and from equation (1.16b) the equation between r and j
(1.17) |
expressing the shape of the trajectory.
Equation (1.16a), resp. (1.16b) shows that the radial part of the motion corresponds to a one-dimensional motion in a central field with an " effective potential "
composed of both the gravitational potential energy U (r) and the centrifugal potential energy J ^{2} / 2mr ^{2} . Values of r at which is V_{ef }(r) = E, and thus according to (1.16b) r ^{. }= 0 , correspond to the turning points of the path, at which the distance function from the center r (t) transitions from increase (distance) to decrease (approach) or vice versa. These turning points determine the range of distances from the center at which the particle m can move. A graphical representation of the course of the effective potential for the motion of a body in the Newton gravitational field of a central body is shown in Fig. 1.1a. If there is only one turning point r = r _{min} , it is the movement of unrestricted (infinite), begining and terminating at infinity. If there are two turning points r = r _{min} (r ^{..} <0) ar = r _{max} (r ^{..} > 0), then the whole trajectory lies inside the annulus r _{min} <r < r _{max} - this is a limited movement (finite) , in this case after the ellipse . The idea of the effective potential is a very useful tool in the study of the motion of bodies in centrally (or axially) symmetrical fields, as we will see in §3.4,4.3 and §4.4 in the analysis of the motion of bodies in the gravitational fields of black holes.
The shape of the path is obtained by solving equation (1.17). For a Newtonian gravitational field with U (r) = -Gm _{1} m _{2} / r (as well as for any central field with a potential proportional to 1 / r or 1 / r ^{2} ) the integration can be performed analytically :
where the integration constant can be canceled by a suitable choice of the beginning of the reading of the angle j ( j = 0 in perihelion). The trajectory equation can then be rewritten in the form
This is the equation of a conic section with a focus at the origin of the O coordinates , ie in the common center of gravity of both bodies, p is a parameter and e is the eccentricity of the orbit determining what kind of conic :
E <0,
Þ e <1 - ellipse E = 0, Þ e = 1 - parabola E> 0, Þ e> 1 - hyperbola. |
Fig.1.1. Motion of
bodies under the influence of Newton's gravitational
field. a) The course of the "effective potential" V _{ef} (r) controlling the radial component of the motion in the central Newtonian gravitational field. Intersections with the energy line E = const. they determine the turning points at which the radial component of the motion changes direction. Lines E = const. <0 (if they are permissible, ie E> min (V _{ef} )) correspond to finite motion either elliptical (radial component oscillates between r = r _{A} and r = r _{B} ) or circular (constantly r = r _{C} ) motion. If E> = 0, the motion is infinite - it starts and ends at infinity; point D corresponds to the closest approach of the body to the center. b) Finite movement (E <0) of the two bodies interacting gravitationally m _{1} am _{2} happens ellipses having the same eccentricity with the focus in a common centroid C . The problem of two bodies is equivalent to the problem of motion of a body with a reduced mass m in the gravitational field centrally symmetrically to the common center of gravity C . c) Analogously infinite motion (E> 0) occurs after hyperbolách around the center C . d) Cross section of some equipotential surfaces of a system of two bodies M _{1}and M _{2} orbiting a common center of gravity. The bold line indicates the first common equipotential of both bodies - the Roche limit. Furthermore, the intersections of equipotential surfaces are marked - Lagrange libration points L _{1} , L _{2} , ..., L _{5} . |
Since the position vectors r_{1} and r_{2 }of both bodies m _{1} and m _{2} are proportional to the vector r , each of them also describes a conic with a focus in a common center of gravity. As can be seen from (1.11), the ratio r _{1} / r _{2} is the same for any point of the path, so that the bodies move relative to the center of gravity along paths that have the same shape (orbiting, for example, ellipses generally of different sizes but the same eccentricity ) - see Fig.1.1b, c.
The most important case is the gravitationally coupled motion along an ellipse , whose longer half-axis a and shorter half-axis b are given by
It is obvious that the longer semiaxis not depend on the momentum, but only for the energy E . The turning points r _{min} = a (1-e) and r _{max} = a (1 + e), ie, the "perihelion" and the "aphelium" of orbit, are also the roots of the equation V _{ef} (r) = E. The time of one orbit along an elliptical orbit , i.e. the period T , can be easily determined from (1.15) and (1.17) by integration according to time from t = 0 to T and according to j from j = 0 to 2 p . After the adjustment, we get the relationship
(1.18a) |
which is the exact wording of Kepler's third law. If m _{1} >> m _{2} , as is the case, for example, in the solar system, then 3. Kepler's law has the usual form
(1.18b) |
where M = m _{1} denotes the mass of the central body (eg the Sun). The ratio of the squares of the orbital periods and the third powers of the large half-axes is thus approximately the same for all planets, but the relation (1.18a) holds exactly. In the special case of circular circulation (e = 0) it is finally possible that 3.Kepler's law (1.18b) can be expressed in the form
G. M = w ^{2} . r ^{3} . | (1.18c) |
If Newton's law (1.1) applies , then a closed trajectory (in a frame of reference firmly connected to the center of gravity) is based on finite motion . To trajectory finite his motion was closed, the angle must be Dj by which the position vector r is rotated for the time in between two has spots turnover r _{min} and r _{max} , rational multiple of 2p , i.e. Dj = 2 p. M / n, where m and n are integers. Then, for n periods of the radial component of the motion, the body performs m circulates and returns to the starting position. However, in the case of a deviation from the law of inverted squares in Newton's law, this condition is no longer met and the "elliptical" trajectory is not closed. If the deviation is not too large, such a path can be imagined again as an ellipse, which, however, is no longer fixed, but the whole rotates slowly (performs a precessional movement) around the center of gravity. Such precession causes perihelion and aphelium to be in a slightly different place in each cycle. The elliptical orbits of the planets around the Sun actually perform the above-mentioned precessional motion, while the deviation from the law of inverted squares is due to the fact that it is not exactly the central field (gravitational influence of other planets, the Sun and planets are not point). The general theory of relativity shows that Newton's law is not accurate even for the centrally symmetric case for strong gravitational fields; emerging anomalous precession and slide of perihelion was actually demonstrated in Mercury (see §4.3).
The problem of motion of
several bodies
In fact, there are a large number of individual bodies and
formations of different masses in outer space, which are affected
by gravity. Therefore, the motion of planets, moons, and stars in
binary or multiple systems actually differs from the
above-derived simple laws of motion of two bodies in a common
central gravitational field. The study of the motion of more
gravitationally influencing bodies is called the n-body
problem . This problem is very difficult even for the
case of only 3 bodies, it is generally not analytically solvable.
Only in some special cases is it analytically solvable. As early
as 1772, J.L.Lagrange showed that for each system of two orbiting
bodies, 5 significant points can be found in the coordinate
system rotating together with the connection of both bodies, the
so-called pound points . If we place a third body in one
of them, at suitable speeds the movement of all three bodies will
take place again in conic sections.
The problem of movement of the three bodies is well solved if the
weight of the third body is negligibly small with respect to the
two basic bodies, circulating undisturbed around the common
center of gravity. Then, in fact, the general problem of motion
of three bodies breaks down into the motion of two bodies
discussed above and into a separate problem of motion of one
"test" body in the resulting gravitational and
centrifugal field of two basic bodies - see the following passage
:
Binary
system: equipotential surfaces, Roche limit, libration points
So far we have dealt with the mutual motion of two bodies under
the influence of their own gravitational field. The most
important such case is the so-called binary system
- a system of two gravitationally coupled bodies orbiting a
common center of gravity. An example is the binary
systems that often occur in space. The binary system of
bodies M _{1}
and M _{2} is
schematically shown in Fig. 1.1d. To analyze the motion of a
small particle (such as gas atoms in the space of a binary
system) in the gravitational and centrifugal field of a binary
system, it is useful to determine the shape of surfaces that are
places of a certain gravitational potential - equipotential
surfaces. For a system of two bodies of masses M _{1} and M _{2} rotating around a
common center of gravity at an angular velocity w according to Fig.
1.1d, we choose the coordinate system x, y, z rotating together
with the bodies so that the x- axis is identical with the
line M _{1}
and M _{2}
and the origin is in the center of gravity of both bodies. The
gravitational potential j for any point P (x, y, z) will then be
j (x, y, z) = - G M _{1} / r _{1} - G M _{2} / R _{2} - r _{a }^{2} w ^{2} /2,
where r _{1} and r _{2} are the distances of the point P (x, y, z) from the
centers of the bodies M _{1} and M _{2} , r _{by the} distance from the center of gravity of both bodies. The
gradient of the first two terms indicates the gravitational
acceleration that both masses M _{1} and M _{2} cause at the point P (x, y, z) , the third term
expresses the centrifugal acceleration caused by the rotation of
the system.
^{ }Selected equipotential levels are shown in
cross section perpendicular to the axis of rotation in Fig. 1.1d.
In the vicinity of each of the bodies, the equipotential surfaces
have a slightly deformed spherical shape and are enclosed around
each of them separately. At greater distances, the deformation
increases until these surfaces touch both bodies, and at even
greater distances, both bodies already have common
equipotential surfaces.
^{ }The equipotential surfaces, which touch at
one point, in the inner libration point L_{1} , form the so-called
critical Roche limit - it is the first common
equipotential surface of both bodies. Within this limit, each
particle moves under the predominant gravitational influence of
one or the other body. At the libration point L _{1} , the test particle
can pass from the gravitational sphere of influence of one body
to the area of gravity of the other body.
An interesting and astrophysically important
phenomenon occurs when, for example, the body M _{1}of the gaseous state
fills (or exceeds) the entire space defined by the Roche
boundary. In such a case, the gravitational action of the second
body "draws" or "sucks out" the gas from the
upper layers M _{1} , which overflows around the inner
libration point L _{1} onto the second body M _{2} . This phenomenon often occurs in close binary
stars and can lead to dramatic astrophysical processes,
as will be shown in Chapter 4 on the evolution of stars and black
holes - §4.1, 4.2, 4.8, illustrated in Figure 4.26. In the orbit
of solids, the Roche limit can manifest itself when a smaller
body (eg the moon) approaches its mass around a more massive body
(eg a planet) so much that the libration point L _{1} finds itself inside
this smaller body. In this case, opposing gravitational force
gradients (additionally having a time-varying tidal force
character) can cause this lighter body to rupture
(however, for smaller compact bodies, the strength of the body
material can prevent this).
^{ }In space (in the gravitational and
centrifugal field) around a system of two bodies rotating around
a common center of gravity, there are a total of 5 significant
points L _{1}
, L _{2} ,
..., L _{5} -
the so-called Lagrange libration points (Fig.
1.1d). The libration points are the points where the attractive
and centrifugal forces acting on the test particle are balanced.
The body located at these points in them can remain at rest
against the junction of the two bodies M _{1} and M _{2} . The most important libration point is the already
mentioned internal libration point L _{1} , which is located on the connection between the
bodies. The outer libration points L2 and L3 lie outside the
system on a line passing through both bodies. The exact positions
of all these 3 points depend on the specific weights M _{1} and M _{2} , their distance and
speed of rotation. The libration points L _{4} and L _{5} lie symmetrically outside the line and form isosceles
triangles with the centers of the bodies M _{1} and M _{2} .
We will not deal with a detailed analysis of the motion of bodies under the influence of Newton's gravitational force here, it is a matter of "celestial" mechanics (for a more general case of motion in the gravitational field of a black hole, however, the relevant analysis is performed in §3.4,4.3 and §4.4) .
Astronomical
significance of Newton's law of gravitation
Newton's law of gravitation proved to be very successful in
elucidating all the motions of planets, moons, comets and other
bodies in the solar system. A great triumph was the use of
Newton's law of gravitation to accurately analyze some of the
anomalies in the motion of planets from Kepler's laws, which at
first seemed against the law of gravity. In 1840, astronomers
discovered that the last known planet in the solar system at the
time, Uranus, deviated slightly from the calculated orbit as it
moved. Therefore, there were temporary doubts about the validity
of Newton's law at such great distances from the Sun. However,
further calculations have shown that the anomalous behavior of
Uranus can be fully explained by the gravitational pull of an
even more distant, as yet undiscovered planet, which slightly
diverts the motion of Uranus from its ideal orbit; the position
of a hypothetical planet in the sky was also determined.
The
Atom and the Planetary System: Similarities and Differences^{ }
After discovering that the atom is a system of positively charged
nuclei and negatively charged electrons bound by an electric
force, the already well-researched solar system, bound by
gravitational force, became the inspiration for clarifying the
structure of this system. There is an obvious analogy
at three points:
¨ The electric and gravitational forces
decrease with the square of the distance;
¨ The attractive gravitational force and
the attractive electric force (between charges of opposite sign)
can be permanently compensated in a vacuum by centrifugal force
during orbital motion;
¨ The same Kepler's laws apply to motion
in central gravitational and electric fields.^{ }
^{ }Based on these analogies,
Rutheford's planetary model of the atom was created (see eg
"Nuclear physics and physics of ionizing radiation",
§1.1 "Atoms and atomic nuclei", section " Atom structure "). However, there are also fundamental
differences between the planetary system and the atom : _{ }
¨ The
difference in the properties and strength of electric and
gravitational forces. While the orbits of the planets are stable
for a long time *), the orbiting motion of an electron in an atom
according to Maxwell's electrodynamics would result in intense
radiation of electromagnetic waves, which quickly carry away the
kinetic energy of the orbit.
*) According to the general theory of
relativity, gravitational waves are emitted even during the orbit
of the planets, but their energy is completely negligible and
does not affect the orbits for many millions of years.
¨Huge
difference in size and weight. The planetary system (masses » 10 ^{30} kg, diameter » 10 ^{8} km) can be fully
described by Newton's classical mechanics, while the atom
(diameter » 10 ^{-8} cm) is typically a quantum system.
^{ }These differences forced Bohr's
quantum modification of the planetary model of the atom.
Nevertheless, the planetary idea of ??the atom is used in
illustrative qualitative considerations.
"Cosmographic
Mystery" ?^{ }
In addition to Kepler's basic laws, astronomers have
tried to explain the specific distances of planets
(radii of orbit) and other bodies in the solar system. Already
Kepler is at work "Mysterium Cosmographicum"
tried to explain the distances of planets from the Sun using
"Platon's polyhedra" circumscribed by the spheres of
individual planets. More recently, inspired by Bohr's
model of the atom (and Balmer's series of spectral lines of the
hydrogen atom), Titus, Bode, and Mohorovic (and other authors)
have sought to find a "quantum law" for the distances
(radii of orbit) of planets in the solar system. We
now know that these laws are only apparent - the
structure of the solar system is, in addition to the laws of
gravity and mechanics, the product of complex and often random
processes of its formation (including collisions and
various resonant phenomena) and has nothing to do with quantum
laws. None "fundamental law" for distances or radii of
planetary orbit it does not exist, efforts to
find it result in mere "numerology". "Cosmographic
mystery" is just fiction ...
Regular
and chaotic motion in gravitationally coupled systems
The above analysis of motion in Newton's gravitational field was
performed for simplified cases that could be converted to motion
in the central field. A high degree of symmetry
leads to an integrable problem with a regular
solution , the laws of conservation of motion integrals
will apply. Two gravitationally bound bodies will orbit
around the common center of gravity along stable elliptical
orbits (here we neglect the emission of gravitational waves, in
the case of the solar system also, for example, the pressure of
radiation from the Sun, etc.). In more complex cases of three or
more bodies, the orbits will interact with gravitational
perturbations- the symmetry is broken. Calculations and
computer simulations show that in such complex systems a small
change d_{o
}of the initial conditions causes the
initially close trajectories to diverge exponentially from time t
: d = d _{o} .e^{- }^{l }^{.}^{t} . After a long enough
time, the system eventually becomes chaotic .
The degree of linear stability or instability -
"chaotic" of such a system can be characterized by the
so-called Lyapunov time T _{L} = 1 / l , for which the system deviates 2.7 times (this factor
increases each initial deviation); parameter l = 1 / T_{L} is sometimes called Lyapunov's
exponent . For the inner planets of the solar system
(excluding Mercury), the Lyapunov time is estimated to be T _{L} » 5.10 ^{6} years. The high value of this time explains the
extraordinary accuracy of astronomical predictions of planetary
motions over time horizons of hundreds and thousands of years.
However, at intervals of hundreds of millions to billions of
years, the chaotic nature of the planet's orbits would already be
decisive; some of the planets could even leave the bound system
of the solar system. From a general point of view, the behavior
of chaotic systems is outlined in §3.3, section " Determinism - chance - chaos?
".
Physical
significance of Newton 's law of gravitation
Before Newton's law, physics encountered the force acting between
bodies only during their mechanical contact -
impact or friction.
With his law of gravity, Newton first introduced into physics the
concept of direct action of bodies
at a distance ("actio in dis tans")
in empty space *). However, both Newton himself and his followers
were not satisfied with this
idea and tried to find an "environment" transmitting
gravitational force effects and thus explain the nature of
gravity (§1.3). Later developments in physics have shown that
the idea of ??direct action at a distance through empty space is
correct, no environment is needed, but it cannot be an immediate
action (as Newton's law assumes), but always properlyretarded (see §2.1).
*) The development of the concept of the physical
field has shown that even though two bodies do not
physically touch, they "touch" or even intersect
their fields. And that causes their force interaction. In addition to its immediate
practical contribution, Newton's law of gravitation is also of
great unitarianism . The law of gravity also
describes the fall of a stone to the earth, the motion of a
planet around the Sun, or perhaps the motion of a star in a
galaxy. This, for the first time, bridged the gap that (in the
understanding of humans) previously existed between the Earth and
the universe. The same physical
laws have
been shown to apply in the solar system, and probably throughout
the universe
^{ }. Newton synthesis A Kepler's kinematics
of planetary motions with her and Galileo dynamics of movement
earthly bodies is so in the history of the first m case
process, which later in the development of science many times
repeated, and that continues today: the
unification of formerly independent branches of physics
showing that the laws of nature are consistent and mutually
articulated - see Appendix B "Unitary
field theory and quantum gravity".
^{ }Despite all the successes of Newton's
theory, however, one anomaly remained unexplained under Newton's
law of gravitation. It was a peculiarity of the orbit of the
planet closest to the Sun, Mercury. The orbit of this planet,
whose orbit is quite eccentric (and therefore perih é is well
established)lium), is markedly different from Kepler's laws. If
it were a motion of a single body in Newton's centrally symmetric
gravitational field, Mercury would have to orbit along an ideal
constant ellipse with the Sun in focus. The observed rate of
precession and perihelia amounts of even 5600 "for 100 years
although crucial part (approximately 5026") having a
kinematic origin - is caused by the movement of the reference
system. The remaining 575 "/ 100 years actual precession
movement perihelia showing the elliptical orbit of Mercury slowly
rotates. Almost all the feed can be explained disruptive
influence of other planets, p øedevším Venus. After
subtracting the gravitational influence of planets known from
observed of Mercury, however, it does not get an ideal ellipse,
but some very small anomalous
displacement
remains perihelion about 43 "/ 100 years. This anomalous
shift of perihelion remained unexplained in Newton 's theory (efforts to explain it, for example, by the influence of
another unknown planet between Mercury and the Sun were
unsuccessful).
Distribution of kinetic and potential
energy. Viral theorem.
If a system of bodies or particles moves in a force field , the kinetic
energy E_{kin }of motion
and the potential energy E_{pot} in the force field acquire certain values
; the internal energy U is
given by the sum of the kinetic and potential energy of the
system. If this
force field is a gravitational
attraction
, where the interacting force between the
individual particles is inversely proportional to the square of
their distance F ~ r ^{–2} , then between the mean value of the total kinetic
energy <E _{kin} > And
the mean value of the potential energy <E _{pot} > the relation :
2 <E _{kin} > + <E _{pot} > = 0, or
<U> = <E _{kin} > + <E _{pot} > = ^{1}/_{2} <E _{pot} > = - <E _{kin} > .
Thus, the sum of the potential
energy and twice the kinetic energy of the stationary system of
bodies is equal to zero . Or the total energy of the
gravitationally coupled system (at
equilibrium) is equal to half the
mean value of the potential energy of the system (or negatively taken total mean kinetic energy). These important laws, which can be derived in the
framework of classical mechanics [165] is called theorem
virial or virial theorem (lat. vires, virium = power, energy ) . This knowledge plays an important role in astrophysics
- the formation and evolution of stars (§4.1,
passage " Star formation ") , the behavior of
galaxies and galaxy clusters (§5.6, the
passage " Hidden-dark matter in galaxies
and galaxy clusters ") .
Modifications of Newton's Law of Gravity
Minor difficulties in celestial mechanics (whether
real or apparent) raised
various doubts about the accuracy of the law of inverted squares
in Newton's law of gravitation (1.1). Therefore, during the 18th
and 19th centuries, attempts were made to "refine" and
modify Newton's law of gravitation by introducing various small corrections in the law of inverted squares, e.g.
m.
M a
m. M F = - G -------- (1 + ----) r ^{o} , n = 1 or 2; F = - G -------- r ^{o} , r ^{2} r ^{n} r ^{2+ }^{b} (Clairaut's law) (Hall's law) |
where a and b are small constants (corrections) correspondingly modifying the original law of inverted squares to correspond to the observed anomalies. Another modification of the law of gravitation has its origin in Seeliger's well-known "gravitational cosmological paradox" arising from an attempt to use Newton's law of gravitation in an infinite Euclidean space (universe) filled with matter with a constant non-zero density. The law of gravity in the form (1.1) in such a case gives an infinite value of the gravitational potential and infinite gravitational force (a satisfactory solution is obtained only if the mass distribution density in all directions from a given point decreases faster than r ^{-2} ). In order to make the law of gravity compatible with the idea of an infinite space homogeneously filled with cosmic matter, a modification of Newton's law was proposed using an additional exponential factor :
m.
M F = - G ---------. e ^{- }^{e }^{. r} . r ^{o} , r ^{2} |
(1.19) |
where e is a small positive constant. This
modification can be related to the hypothesis of " absorption " of gravity by the environment lying between
gravitational bodies. The law of gravitation considering the
absorption of gravity would indeed have the form F = -G (mM / r ^{2} ). e ^{- }^{mr
}^{r} , where r is
the density of the environment between the bodies M
and m (homogeneous is assumed for simplicity)
and mis the absorption coefficient. Attempts to
demonstrate gravity absorption have not yielded convincing
results. In addition, the absorption of gravity would lead to a
violation of the proportionality between inertial and
gravitational mass, which would lead to an inadmissible violation
of the 3rd Kepler's law. Furthermore, the effect of gravity
absorption on the Earth's surface would cause appropriate
variations in gravity and acceleration (with a period of 24
hours), caused by shielding the gravitational action of the Sun
and the Earth's Moon. In experiments with pendulums no similar
effect was observed, as well as properties of the marine tide
caused by the Moon and Sun e show no observable abnormalities
which could be attributed to changes in tidal acceleration
stemming from absorbing gravity.
^{ }All similar attempts to modify
Newton's law were in the nature of formal ad
hoc hypotheses
, were not substantiated by deeper physical reasons, and
ultimately failed to satisfactorily explain one of the observed
anomalies without other side effects and anomalies contradicting
the results of the observations.
^{ }The difficulty in shifting
Mercury's perihelion was not serious enough to threaten Newton's
theory more seriously; some hypotheses, such as that the Sun is
slightly flattened and the gravitational field is therefore not
exactly spherically symmetrical, could explain similar effects.
However, Newton's theory has some more serious conceptual shortcomings , which manifested themselves in
confrontation with a newer knowledge of the laws of nature. From
the point of view of the depth of knowledge, it can be considered
a shortcoming that Newton's law of gravitation does not explain
the exact equality (proportionality) of gravity and inertia. This
equality is purely empirical here and has the character of chance
(for more details see §2.1).
Velocity of gravitation^{ }
The main weakness of Newton's theory of
gravitation, however, is the already mentioned assumption of immediate and immediate gravitational action "
at a distance ". In Newton's law of
gravitation (1.1) time does not appear in any way; according to
him, a change in the position of one body is gravitationally
reflected immediately on other bodies, even
very distant - gravity has infinite speed .
^{ }This assumption has proved to be
incompatible with the knowledge obtained during research into the
phenomena of electromagnetic characterized ch and generalized in Einstein's
special relativity (see §1.6 and 2.1). The need arose to modify Newton's
law by introducing a time factor - retardation reflecting the final velocity of the gravitational interaction *).
Indeed, this approach leads to a satisfactory and consistent
theory of gravity - Einstein's general
theory of relativity (Chapter 2) - which not only incorporated
gravity into the context of modern physics, but even came up with
the idea of the determining role of
gravity for
all the laws of physics, to identify gravity with the properties
of space and time. In addition to its profound conceptual
significance, the general theory of relativity quite naturally
explains the equality of inertia and gravity, the anomalous shift
of Mercury's perihelion, the curvature of light rays in the
gravitational field, and other phenomena and facts beyond
Newton's theory. According to the general
theory of relativity, the commotion in a gravitational field
propagates at the speed of light c , in
the form of gravitational waves ( §2.7 " Gravitational
waves
" ) .
*) The velocity of the gravitational interaction will be dealt
with in §2.5 " Einstein's equation of the
gravitational field " and
mainly in §2.7
" Gravitational waves ",where
in the passage " How fast is gravity? ", general questions of the speed of propagation
of changes in the gravitational field and the possibilities of
its experimental determination will be discussed.
Galactic
Modifications of Newton's Law of Gravitation - MOND
Newton's law of gravitation, in co-production with the
other 3 laws of mechanics, is phenomenally successful
in analyzing all mechanical processes here on Earth, planetary
motions and other bodies in the solar system, star formation
dynamics and evolution (except final stages). ), stellar motions
in binary and multiple systems. It fails only in extreme
situations of strong gravitational fields in
gravitationally collapsed compact objects in the final stages of
massive star evolution (§4.2 " Final
stages of stellar evolution. Gravitational collapse. Black hole
formation. ") And
in analyzing the structure and evolution of the universe on
cosmological scales (§5.4 "Standard
cosmological model. Big Bang. Forming the structure of the
universe. ").
There must be generalized Einstein's law of gravitation
in the general theory of relativity (§2.5 ' Einstein's gravitational field equation ").
^{ }The
surprising incompatibility but appeared at astronomical
measuring the speed of movement - circulation - stars and gas in
the peripheryof galaxies and speed of galaxies
in galactic clusters: these velocities appeared to be
higher than expected based on Newtonian mechanics and
gravity (the velocities of motion and intensity of
the gravitational field are relatively small here - non-relativistic
)^{ }. The visible
matter in galaxies and their clusters is insufficient for the
dynamics of motion in Newton's law analysis. This paradox was
solved in astrophysics by the hypothesis that the dynamics of
galaxies is determined by an additional massive invisible -
hidden dark matter (§5.6,
section " Future
evolution of the universe. Hidden-dark matter. ") .
^{ }However, the
hypothetical dark matter has not yet been explicitly discovered -
to clarify what it is composed of, to detect its particles. An
alternative hypothesis to explain the observed dynamics
of motion in galaxies is a suitable modification of
Newton's theory (and possibly its
relativistic generalization) - the so-called MOND
(Modified Newtonian Dynamics).). The observed astronomical
measurements of the dynamics of the motion of stars in galaxies
and galaxies in galaxy clusters, which are generally attributed
to the gravitational effect of dark matter, try to explain with
the modified form of the law of gravitation .
Such a modification of MOND was proposed in 1983 by M. Milgrom,
in which Newton's law of gravitation F = GM .m / r^{2 }is modified
by the hypothesis that in addition to mass and distance the
gravitational force F_{MOND }also depends on the value of acceleration
:
F
_{MOND}
= G.M.m / [f (a/a_{o}) r ^{2} ] ,
where G is the gravitational constant, M is the
gravitational mass, m is the mass of the test piece, r
is the distance. The modification of the classical Newton's law
is given by the empirical function f
depending on the value a acceleration of
the test body, in relation to a certain empirical constant a _{o} (indicating the scale transition between Newtonian and
MOND dynamics) . The function f is usually chosen
in the form f(a / a _{o} ) = 1 / (1 + a _{o} / a ), or f(a / a _{o} ) = 1 / Ö [1+ (a _{o} / a ) ^{2}]. To
the thus modified law of gravitation explained the observed
velocity curves in galaxies, for the constant a_{o} by the
fitation were determined value a_{o} »
1.2 × 10^{-10} m ^{-2}
.
Note: In MOND, this is a
different (and otherwise motivated) modification
of the law of gravity than the "Modifications of Newton's law of gravitation" mentioned above!
^{ }Experts differ on
these efforts - both the concept of dark matter and MOND are ad
hoc hypotheses . Milgrom's
MOND with its acceleration dependence is an artificial and not
very convincing theory of gravity . Whyshould the
gravitational force depend on the acceleration - and even on a
very small acceleration? It has no physical logic... Most
astrophysicists are more inclined to the concept of dark matter,
in which the dynamics of galaxies can be explained in more physically
justified ways (it is not even excluded that it
may be just a mistake to model the distribution of standard
baryonic matter in galaxies and galaxy clusters..?..) .
Entropic
hypothesis of the origin of gravity - Verlinde
Originally, this is another newer attempt at an alternative
clarification - exclusion - of dark matter. In this hypothesis (dating from 2016), E. Verlinde declared entropy
to be the primary very basic quantity, the growth of
which causes the "entropic force" that
is the essence of gravity. So there is
no gravity! - we observe it only as an apparent
force, a macroscopic manifestation of entropy growth
with random statistical behavior of microscopic particle-quantum
field systems. This entropic force causes the bodies to move and
performs mechanical work. This rather bizarre and definitely very
unlikely hypothesis is described in §B-5, passage
"Entropic
hypothesis of the nature of gravity".
Mechanistic hypothesis of the
origin of gravity - LeSage
is the oldest attempt to simply explain the origin of gravity,
long abandoned - see the following §1.3 "LeSage's hypothesis".
Einstein's law of gravitation -
general theory of relativity
The most perfect and far unsuccessful improvement and
generalization of Newton's law is Einstein's law of
gravitation within the general theory of relativity
- §2.5 "Einstein's
equations of the gravitational field". We will
use it in the vast majority of interpretations of this book.
^{ }In the final
chapters B-1 to B-7 we will also mention some attempts at unitary
and quantum generalizations (eg "Loop
theory of gravity" or "Unification of
fundamental interactions. Supergravity. Superstrings.").
Gravity, black holes and space-time physics : | ||
Gravity in physics | General theory of relativity | Geometry and topology |
Black holes | Relativistic cosmology | Unitary field theory |
Anthropic principle or cosmic God | ||
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