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Appendix A
MACH'S PRINCIPLE AND
GENERAL THEORY OF RELATIVITY
A.1. Mach's
principle and the origin of inertia
A.2. Brans-Dicke's theory of
gravitation
A.3. Mach's principle and experiment; inertia anisotropy
A.4. The status of Mach's principle in the
general theory of relativity
A.1. Mach's principle and the origin of inertia
There are basically two opposing views on the relationship between local and global laws of physics - the relationship between the laws of "laboratory physics" and the properties and structure of the universe as a whole. The first approach is that we try to derive the properties of the universe on the basis of knowledge of local physical laws - their extrapolation and synthesis. This is how we proceeded throughout this book.
The second approach, on the other hand, is based on the idea that local physical laws have their origin in the global structure of the entire universe, or that they are at least affected by the overall distribution of matter in the universe. This second approach is related to the so-called Mach principle, whose relationship to the general theory of relativity (and thus to current ideas about gravity and the structure of space-time) will be briefly discussed here.
In his critique of Newton's conception of absolute space and time, E.Mach [175] expressed some ideas which, together with Riemann's ideas on non-Euclidean geometry and its possible relationship with physics, played an important heuristic role for Einstein in creating his general theory of relativity. A.Einstein summarized and formulated more precisely Mach's ideas - this created Mach's principle, which, however, as it turned out later, is not included in the general theory of relativity.
If we follow the laws of motion in a non-inertial frame of reference (such as a rotating system - Newton's well-known example of a rotating bucket of water), inertial forces such as centrifugal and Coriolis forces will manifest here. According to Newton, these forces arise during accelerated motion relative to absolute space. Mach argued conversely that the inertial forces generated by the accelerated motion relative to the distant stars, or rather unequal movement relative to the sum mass of the universe. Mach, which followed the views of Leibniz and especially Berkeley, therefore, not considered the inertia as an independent property of the body itself, but as property conditional by all others bodies distributed in the space.
The most common formulations of Mach's principle are as follows :
During the development of the physics of gravity and cosmology, a very wide range of different (often contradictory) views and opinions emerged on the question of the validity of Mach's principle and its relation to the general theory of relativity. Even from the current point of view, it is possible to have a fundamentally double view of the origin of inertial forces manifested in non-inertial frame of reference :
There is a simple objection to the logic of approach b). Let us observe a given frame of reference with respect to some very distant small body which is at rest (even if this body actually moved quite unevenly, it will seem to us that it is at rest, because its movements will be practically unobservable due to the great distance). If we have a non-inertial frame of reference in which inertial forces act, it would certainly not be reasonable from the fact that the non-inertial system also manifests itself in its uneven motion relative to said body, to conclude that the observed inertial forces have their origin in this distant body!
Einstein mentioned three phenomena in GTR that could allegedly correspond to Mach's principle :
a) An accelerating force will act on a body if nearby bodies accelerate, the direction of this force being the same as the direction of acceleration of the surrounding bodies.
b) The rotating hollow body will create an array of Coriolis forces inside, causing the moving bodies to deflect in the direction of rotation, as well as an array of radial centrifugal forces.
c) The inertia of a body increases when heavy bodies are concentrated in its vicinity.
We first notice the
effect c), which, unlike the first two,
is a direct and immediate consequence of Mach's principle and can
also be formulated as follows: the inertial mass of a body
depends on the distribution of matter in its vicinity; if there
were no other masses in the universe, the solitary body would have no any
inertia (the inertial mass would be zero). This effect is
certainly not included in the general theory of relativity *):
according to the principle of equivalence, each body in a locally
inertial system has the same inertial mass as in the special
theory of relativity. A given force (eg the force of a tensioned
spring) always imparts the same acceleration to a given body in a
locally inertial frame of reference, regardless of the presence
of near or far masses.
*) The fact that Einstein
initially considered the phenomenon c) as a result of GTR, was caused by a misinterpretation of
the relevant equations, which reflected the effect of the
particular coordinate system used.
Let's ask a question: Will the manifestation of inertia forces in a hypothetical completely empty universe in which there will be no other materials except non-inertial studied (eg. rotating) reference frames? It is clear that this is not possible to verify in practice, the real universe is filled with matter. However, Einstein's gravitational equations (without the cosmological term) allow for some exact simple solutions in empty space :
1. Minkowski mass-free spacetime, in which each test particle has a non-zero inertial mass corresponding to the law of inertia in the special theory of relativity.
2. Schwarzschild solution (see §3.4), in which there is no global inertial system, but at each point it is possible to introduce a locally inertial reference system in which the inertial mass of the test particles does not depend at all on the body exciting the Schwarzschild gravitational field.
Such simple and proven solutions (along with some others, such as Gödel's solution [104]) can be described as "anti-Mach". They appropriately document the fact that Mach's principle in its original formulation is incompatible with the general theory of relativity because it contradicts the principle of equivalence.
Another objection to Mach's principle is that it is closely linked to direct immediate action at a distance. It is not clear how objects (perhaps galaxies) located at distances of, say, ~ 10^{8} light-years from us can react immediately and have a feedback effect on the accelerating test body located here at a given point in time. These inertial forces should be retarded just like any other field. The answer to this objection can be that the field by which distant galaxies act at a given place (where we accelerate a test body) is already there, because it reached here before the speed of light and captures the state, in which these distant galaxies existed many millions or billions of years ago. This field can therefore be manifested by an inertial force simultaneously with a change in the speed of the monitored body. If we assume that the field is responsible for the inertia of the gravitational field (as it is in the spirit of GTR - see §2.1 -2.3) is thus advocated "Mach's principle" equivalent claims "all matter in the universe gives the gravitational field, creating the geometry of spacetime and this spacetime geometry causes the inertial properties of the test body". However, spacetime is locally Euclidean (principle of equivalence) and the local inertial properties of bodies do not depend at all on matter that excites the global gravitational field. The content of Mach's principle expressed in the original formulation has in fact disappeared - if we look for the origin of inertia in gravity, we are left with again the general theory of relativity...
Phenomena a) and b), often mentioned in support of
Mach's principle, actually occur according to GTR **).
However, they have virtually little to do with Mach's principle;
are actually effects entrainment
local inertial system, when a moving mass will change the
geometry of space in its surroundings so that the local inertial
system are different from those here would normally
(meaning entrainment inertial system angular momentum of a
rotating black hole was discussed in §4.4). These effects are
substantially the same nature as the ordinary gravitational
attraction, when in the
vinicity of a
material body (perhaps a planet or a star) the local inertial
systems fall with acceleration, while without the presence of a
gravitational body they would merge with the global inertial
systems at infinity. However, in these mentioned locally inertial
systems, according to the principle of equivalence, the test bodies will have exactly the same inertial properties, such
as the STR, regardless of the largest accumulation of material objects
in their surroundigs.
**) Thirring and Lense [248]
investigated the case where there is a massive hollow sphere in
an infinite empty asymptotically flat space. If a test piece
rotates around its center inside it, centrifugal and Coriolis
forces will be manifested on it. Conversely, if the body remains
at rest and the hollow sphere rotates, the body will again be
subjected to centrifugal and Coriolis forces, but significantly
smaller than in the previous case (they will be proportional to
the ratio of the gravitational radius of the hollow sphere to its
actual radius), although relatively it is the same kind of
movement.
In czech literature [132], [133], [265] there is sometimes an interesting modification of Mach's approach to field theory, the so-called megaphysics of Z. Horák ; a monograph has even been published [265] in which this approach is elaborated and presented as physically sound (see, however, critical note * below) . The idea that this theory using Mach principle in its strongest version (eg. in the empty space becomes the speed of light is equal to zero!), in accordance with the general theory of relativity can not agree , as is clear from the above arguments. Compliance with GTR can be proved only for a significantly weaker variant of the Mach principle.
The whole core of the misunderstanding in the supposed agreement between Mach's principle and the general theory of relativity probably lies in the overly straightforward identification of gravity and inertia. In §2.1 we have shown that gravity and inertia are "two sides of the same coin" and that they have a common physical nature, but it cannot be inferred from this that they are completely identical. In the general theory of relativity, it is not true that when gravity ceases to act, inertia ceases to exist. Metric tensor g_{ik} (which is always non-zero) can in this case be placed on the diagonal shape, which "dictates" to bodies the coommon inertia, known from the special theory of relativity. The only thing that is affected by the distribution and motion of matter are the geometric properties of spacetime, and thus determination of (local) inertial frames of reference of individual bodies.
When it became clear
that Mach's principle in its original strong version was not
contained in Einstein's general theory of relativity, opinions of physicists
split into roughly three groups. Some simply rejected Mach's
principle as incorrect. Others chose a more cautious
position and trying to find suitable weaker
formulation
of Mach principle (keep the "healthy core")
compatible with GTR. Orthodox proponents of Mach's
principle then sought to change the
general theory of relativity "to the image" of Mach's
principle *). Of these efforts to illustrate, below we present
only the best known of them - the Brans-Dicke
theory of gravitation.
*) In the book [265]
"Physical field from the point of view of relativity",
the usual logical procedure of derivation of STR is reversed,
whose basic postulates try to derive within Newtonian mechanics
from certain "megaphysical" considerations about the
gravitational potential c*
of the universe: from conservation
law. the mechanical energy of a body in the gravitational field
of the universe first derives relativistic dynamics, namely the
relation m = m(v) = m_{o}.Ö(1 -
v^{2}/c_{*}^{2}), and on this basis the relativistic
kinematics - Lorentz transformations - is only derived. This
procedure, apart from the fact that it is illogical and obscures
the very physical essence of the special theory of relativity,
can certainly not be considered as proof and from the conformity of Mach's principle
with the theory of relativity, much less for the proof of STR.
Obtaining the correct laws of STR should be considered rather a
coincidence, because the derivation is based on a physically not
entirely correct procedure when confronting the change of
momentum of particle with the
law of conservation of energy (eg the energy of a particle in the
field of force causing particle acceleration ). Moreover, the
very concept in which a certain value C* of the
constant gravitational potential is ascribed physical meaning, in principle not be consistent
with that used by classical mechanics which potential is
determined up to an arbitrary constant, so that the cases of c* =
(arbitrary constant) and C* = 0 are physically completely
equivalent.
A.2. Brans-Dicke's theory of gravitation
According to a strong version of Mach's principle, the inertial masses of individual elementary particles are not natural constants, but are determined by the interaction of these particles with the total mass of the universe, ie the interaction with a certain "cosmological" field related to the density of matter in space. According to the principle of equivalence (inertial mass and passive and active gravity), the mass m of each particle can be determined by the gravitational acceleration G.m/r^{2} given to the next test particle at a distance r. Cosmological Mach field influencing body weight should therefore influence the gravitational constant G.
In Brans-Dicke theory of gravity [32], [69], which is a modification of Einstein GTR in a curved space-time e, except for the usual and well known particles and fields postulates the existence of so-called j -field (zero rest mass - long-range field), which here has some significantly different properties and manifestations than other fields. Overall integral action in this theory consists of three members :
(A.1) |
where G_{o} = const. is the gravitational constant (the reason for the index "_{o}" follows from the next), L_{m} is an ordinary Lagrangian of ordinary particles and nongravity fields, L_{j} =w.j_{,i}j^{ ,i}/j is the Lagrangian of the hypothetical j -field j = j(x^{i}) = j(x, y, z, t). The j -field, however, is part of the first term in (A.1) describing gravity, where the scalar curvature of space-time R is multiplied by the function j. And this is the unusual role of the j-field, as will be seen below.
The differential equations of the field are obtained, as usual, by placing the corresponding variation of the integral of the action (A.1) equal to zero. The variation of the metric g^{ik} leads to gravitational equations similar to Einstein's equations:
R^{ik} - ^{1}/_{2} g^{ik} R = (8pG_{o}/c^{4}) (T_{m}^{ik} + T_{j}^{ik}) . | (A.2) |
The variation according to j gives the equation
(2 w / j) . oj - ( w / j ^{2} ). ( j _{, i }j ^{, i} ) + R = 0 ,
however, where the scalar curvature R is determined according to the first equation (A.2) by the total energy-momentum tensor T^{ik} = T_{m}^{ik }+T_{j}^{ik}, so that after settling an equation is obtained describing how matter as a source excites j -field :
(A.3) |
The energy-momentum tensor T_{j}^{ik }of the field j appears in equation (A.2) together with the energy-momentum tensor of nongravity matter as a source of the gravitational field, which is not strange. However, the "effective gravitational constant" G = G_{o}/j is no longer a universal constant here, because j is generally a function of place and time. This function j is given by equation (A.3), in which the mass of all bodies and non-gravitational fields acts as a source.
The j -field could therefore be the field that mediates Mach's influence of physical laws by the distribution of matter in space, because the variability of the gravitational constant G effectively affects the masses of all particles. According to equation (A.3), the field j depends on the distribution of mass in the whole space. If there is no mass, j is equal to zero, in the integral of action (A.1) the term j.R describing gravity disappears and no gravity exists. From equation (A.3) of the generation of the j -field by the distribution of the mass, it is further seen that the sources excite the j -field inversely proportional to the distance r. If we consider a universe with an approximately homogeneous and isotropic distribution of matter, the mass contained in a spherical shell of radius r and thickness Dr increases with a radius as r^{2}. Therefore, in this (seemingly realistic) case, the field j is determined primarily by distant sources (distribution of matter in distant parts of the universe) and will depend very little on the presence of nearby matter (such as planets, stars or even galaxies) under normal circumstances. This specific property of the Mach force is offered by various megaphysical speculations. Only if there is such a dense mass~energy accumulation of in the vicinity of the considered place that the gravitational potential is close to one (~c^{2} in common units), the j -field will also be significantly affected. This situation would occur in the vicinity of the neutron stars and especially black holes, whose physics in Brans-Dick theory would be significantly different from the physics of black holes in GTR (described in Chapter 4). In Brans-Dicke's theory, for example, there could be longitudinal waves of the j-field; such "longitudinal gravitational waves" could also radiate in a spherically symmetric gravitational collapse.
The cosmological implications of Brans-Dicke's theory also show some differences from standard cosmology in GTR [69]; above all, a faster cosmological expansion is manifested, which would of course have an effect on the initial nucleosynthesis (§5.4), especially on the helium content. It is also possible to consider cosmological solutions with a free j- field, which are no longer so directly connected with Mach's principle.
Brans-Dicke's theory contains a free dimensionless parameter w, where the value w = ¥ corresponds to the standard general theory of relativity. Experimentally verifying the Brans-Dicke theory, ie determining the parameter w, can be in principle in three ways :
So far, carried out observations of this kind lead to the value of the parameter w to imitat |w| ³ 6. Overall, it can be said that the results of observations are not favorable for the Brans-Dicke theory and rather prefer the general theory of relativity.
A.3. Mach's principle and experiment; anisotropy of inertia
Only an experiment can decide between Mach's principle (in its strong version) and the general theory of relativity. The validity of Mach's principle would lead to an anisotropy of inertia if the distribution of the surrounding masses in space would not be isotropic, eg if the test specimen were located near another very large mass that would not be completely negligible with respect to the mass of all solids in space (if the universe is closed and the "final"; for the infinite universe, Mach's principle loses uts meaning). The inertial mass of the body would then depend on the direction of its acceleration (and the direction of acceleration would generally not agree with the direction of the force, conservative forces would cease to be conservative , etc.).
We really seem to be in a similar asymmetric situation; our Earth lies on the edge of a Galaxy with a mass of ~10^{11 }M_{¤}, so this eccentric position of the Earth should, according to Mach's principle, be manifested by the inertia of each body slightly dependent on the angle between the direction of motion of the body and the line with the center of the Galaxy. If we were to accelerate a body in the direction of the center of the Galaxy and then in the opposite direction, we would measure a slightly different inertial mass (resistance of the body to acceleration) in both cases.
An experiment to verify the isotropy of inertia was performed in 1960 by Hughes et al. [139]. In this experiment, the resonant absorption of photons by Li^{7} nuclei in a strong magnetic field was monitored. If the laws of nuclear physics invariant with respect to rotation, i.e. inertia is isotropic, in the magnetic field ally four split energy sublevels of the ground state nucleus Li^{7} (spin 3/2) will be equidistant and the absorption spectrum will show a single sharp peak. Anisotropy of inertia, on the other hand, would lead to a violation of the equidistance of split energy levels, and thus to the formation of three close absorption lines. Exact spectrometric measurements (performed every few hours to effect rotation of the earth substantially changed the angle between the magnetic field and the direction of the center of the Galaxy) demonstrated no measurable expanding of apsorption peaks (larger than the line width), which shows that the anisotropy of inertia weight cannot be greater than about Dm/m £ 10^{-21}.
A.4. The status of Mach's principle in the general theory of relativity
In conclusion, Mach's
principle is fulfilled in the general theory of relativity only
in the sense that by the statement "The distribution of
matter in the whole universe determines the inertial properties
of bodies" we will understand "The
distribution of matter in space determines the local inertial
system of each body" *). Once such a locally inertial
system is found, the inertial properties of bodies or other local
physical laws no longer depend on the presence and distribution
of near or distant bodies.
*)^{
}An interesting approach
to Mach's principle was developed by Wheeler [276], who does not
try to modify the general theory of relativity, but used Mach's
principle (but in a much weaker wording than the original
formulation) to obtain boundary conditions for Einstein's
equations as a certain cosmological selection principle. In
Wheeler concept is no longer talks about the intantaneous distribution of matter in the universe
should determine the inertial properties of bodies in locally inertial systems, but the
geometry of spacetime is determined by the distribution of
mass-energy in the initial hypersurface spatial-type. Mach's principle in this application
then leads to the requirement that the universe be closed;
geometry of spacetime in the past sthose, present and future, and thus the
"inertial properties" of all particles, are then
determined by entering the three-dimensional geometry of the
closed space at two close moments in time and by entering the
density of the distribution and the flow of energy and momentum.
After a critical
analysis of Mach's principle and its relation to the general
theory of relativity, we see that the original vague formulation
"Inertial properties of each body are given by its
interaction with all matter in space", allowing
misinterpretation, should be replaced by "Properties of
space and time are by the distribution of matter ~ energy in the
universe", embodying the essence of the general
theory of relativity. Only this is the "healthy core" of Mach's principle,
compatible with the general theory of relativity..!..
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Vojtech Ullmann