Mach’s principle in flat and curved spaces.
Application to cosmology (*)
R. L. SIGNORE
BREVATOME - 25, rue de Ponthieu, 75008 Paris, France (ricevuto il 30 Ottobre 1996; approvato il 25 Giugno 1997)
Summary. — Mach’s principle is applied to a curved space. It is shown that the gravitational constant may be considered to depend on space curvature. This dependence introduces a new term in the right-hand side of Einstein’s equations, leading to a new cosmological force. This force is either repulsive or attractive depending on the curvature sign. A flat expanding universe appears to be the only stable dynamical solution of Einstein’s equations. This would explain why our universe is flat. In a flat space, Mach’s principle is used to propose an explanation for the coincidence between some very large numbers.
PACS 04.20 – Classical general relativity.
1. – Introduction
Mach’s principle [1, 2] deals with a fundamental issue of physics: the origin of inertia. According to Mach, inertia is due to the distribution of matter in the universe. Stated another way “matter there governs inertia here” [3]. This principle was used by Einstein in the development of general relativity [4].
A quantitative treatment of this principle has been proposed by D. W. Sciama [5-7]. According to D. W. Sciama, a scalar potential f and a vector potential AKmay be defined depending on the matter contained in the universe. A field is also defined with two parts, a “gravoelectric” part EKand a “gravomagnetic” part HK. The last one is zero since curl AKis zero. The first one is
E K 4 2 grad f 2
g
1 ch
¯A K O¯t . (1)This is called by D. W. Sciama the “Law of Inertial Induction” by analogy with the law of induction in electromagnetism. The potential at a test particle at rest in the universe is equal to f and the vector potential vanishes. When the particle moves
(*) The author of this paper has agreed to not receive the proofs for correction.
relatively to the universe with a velocity 2 nK(t), in the rest frame of the particle the universe moves with velocity nK(t). The vector potential AK no longer vanishes but has the value
A
K
4 (fOc) nK(t) , (2)
and the field takes the form
E
K
4 2(fOc2)(¯ nKO¯t) . (3)
In the rest frame of the particle, the force created by the field EK should be exactly compensated by the inertial force, so that the inertial force has its origin in potential vector AKand hence in all matter of the universe.
In Sciama’s theory, the volume of the universe taken into account for the evaluation of the potentials f and AKis a spherical volume of radius Ct, where t 43t0O2, t0 being the age of the universe. This volume corresponds to the Hubble sphere. The scalar potential f is taken to be the Newtonian potential at the center of this sphere, namely
f 42 2pGrc2t2.
(4)
In a recent paper [8], I proposed to slightly modify the way of calculating such a potential. Instead of limiting the volume of the universe to the Hubble sphere, I suggested to take into account the causal sphere of the test body. The idea is that, at any point of the universe, the influence of the surrounding universe, whatever the nature of the involved phenomenon, is actually limited by the particle horizon of the point, i.e. by the causal sphere. The radius of this causal sphere is 3 ct0 in the matter-dominated era and 2 ct0 in the radiation-dominated era, where t0 is the age of the universe, whereas the radius of Hubble sphere is only half. Since the main contribution of matter to local inertia comes from distant matter (see D. W. Sciama, p. 39 of ref. [2]) the extension in volume from the Hubble sphere to the causal sphere has a significant consequence on the numerical value of the potential.
The second modification I proposed concerns the method for computing potential f. Instead of using a Newtonian definition, I suggested to rely upon relativistic cosmology. This theory discloses the universe behavior through a scale factor R which is determined by two differential equations, the so-called Friedman equations. Gravitation appears to be linked to the deceleration of the universe expansion, the deceleration being due to the space-time curvature. At a distance r the deceleration vector is simply given by
G K 4 (R n n OR) rK. (5)
A quantity W may be defined as the integral of this field along the causal sphere radius from the center to the particle horizon:
W 4
(Rn n
OR) r dr . (6)
This quantity is a gravitational potential. As it takes into account all the matter within the causal sphere it may be called the gravitational potential of the universe.
Taking the value of R
n n
OR from Friedman equations, it immediately appears that the potential W takes a spectacular and simple value:
W 42 c2.
(7)
In Sciama’s theory, this value is deduced theoretically
(
see eq. (6))
but the potential given by the Newtonian equation (4) is not exactly in accordance with this theoretical value.Returning to Sciama’s equation (3)
E
K
4 2 (fOc2)(¯ nKO¯t) , (3)
it immediately appears that the quantity 2(WOc2) is exactly equal to unity, so that the field EK acting on a test body of gravitational mass mg produces a gravitational force equal to mg(¯ n
K
O¯t). This result may be re-interpreted by saying that there is an inertial mass miequal to mg, which very clearly establishes the origin of inertia.
Sciama’s theory leads to another result: because the gravitational potential is equal to 2c2, the total energy (i.e. inertial plus gravitational energy) of a particle at rest in the universe is zero (ref. [5], p. 38, par. 4i):
mic21 mgW 40 .
(8)
This result is in full agreement with the flatness of space in the Einstein-de Sitter model, where there is an exact compensation of mass energy and potential energy.
2. – The problem with a curved space
The foregoing results are actually limited to a flat space, i.e. an Einstein-de Sitter universe. When the space component is curved, either positively or negatively, the values of scalar and vector potentials are changed. The advantage of considering the potential as the integral of the deceleration field, as in eq. (6), is that the new value of the potential may be computed by still using the results of the relativistic cosmology in a curved space. In my recent paper [8], this computation was done in a positively curved space. In the matter-dominated era the result is
W 42 c2h2
O2( 1 2 cos h) , (9)
and in the radiation-dominated era the result is
W 42 c2h2O2 sin2h ,
(10)
where h is the arc parameter measure of time [9].
These potentials are no longer constant. They are continuously increasing in absolute value, becoming more and more negative.
At early times, when h is small, a limited development may be used for the circular functions cosinus or sinus, so that the potential W takes the approximate values
W 42 c2
( 1 1h2 O12 ) (11)
in the matter-dominated era [10] and
W 4 (2c2O2 )( 1 1 h2O3 )
(12)
in the radiation-dominated era.
The dependence of W on time makes the gravitational force EK given by eq. (3) dependent on time as well. This would apparently lead to an inertial mass changing with time. As to the net energy density, it would no longer be equal to zero but would become more and more negative as time passes. So, energy does not seem to be conserved. In an unlimited universe, the conservation-of-energy principle is rather difficult to define but in a closed universe, such as a positively curved space, the universe has a finite volume ( 2 p2R3) and so a finite energy and this principle should hold. Apparently, it does not.
The goal of the present paper is to attempt to solve these difficulties.
3. – Mach’s principle in a curved space
In a positively curved space, the field acting upon a test body moving relatively to the universe is still given by eq. (3); however, the potential is no longer equal to 2c2. It is given by eqs. (9) and (10). These equations could be written in a general form:
W 42 c2OK ,
(13)
where K depends on the era:
i ) in the matter-dominated era , K 42(12cos h)Oh2; (14)
ii ) in the radiation-dominated era , K 42 sin2hOh2. (148)
Accordingly, the force FK acting upon a test body of gravitational mass mg is F K 4 mgE K , i.e. F K 4 ( 1 OK ) mg(¯ n K O¯t) , (15)
so that the inertial mass miis no longer equal to mg but to mgOK . Then
mgOmi4 K . (16)
The factor K given by eqs. (14) and (148) is smaller than unity, which means that, in a positively curved space, the inertial mass is greater than the gravitational mass. This is not in violation of the equivalence principle since K is still the same for all test bodies whatever their composition, which is the actual meaning of the equivalence principle. In a flat space, the ratio mg/mi is equal to unity because G is given the value 6.67 3 10211kg21m3s22. It is always possible to make the ratio m
g/mi equal to unity by choosing appropriate units and, particularly, by taking an ad hoc value for the gravitational constant G. Multiplying the usual value of the gravitational constant by a factor equal to K would increase (mgOK ) by the same factor and make (mg/mi) equal to unity. Then, it is possible to keep the ratio mg/miequal to unity by changing G into
G 4KG0,
where G0is the value of the gravitational constant in a flat space. With the value of K given by eqs. (14) and (148) one obtains the following values for G, depending on the era:
i ) in the matter-dominated era , G 42G0( 1 2cos h)Oh2; (18)
ii ) in the radiation-dominated era , G 42G0sin2hOh2. (188)
Equation (18) may also be written in a more symmetrical form:
G 4G0
[(
sin (hO2)O(hO2)]
2. (18)
At early times, G takes the form:
i ) in the matter-dominated era [10], G 4 (12h2
O12 ) G0; (19)
ii ) in the radiation-dominated era , G 4 (12h2 O3 ) G0. (198)
Similar results would be obtained for a negatively curved space, but hyperbolic functions (ch x, sh x) should replace circular functions (cos x, sin x). Considering only early times for the sake of simplicity, the gravitational constant will be given by expressions similar to eqs. (19) and (198) but with a plus sign instead of a minus sign. Hence, for any curvature, positive or negative, defined by the usual parameter k (k 411 or k421), the gravitational constant would be:
i ) in the matter-dominated era , G 4G0( 1 2kh2O12 ) ; (20)
ii ) in the radiation-dominated era , G 4G0( 1 2kh2O3 ) . (208)
Reconsidering now the issue of net energy of a test body in a curved space, the left-hand side of eq. (8) may be written as
mi[c21 (mgOmi) W] . (88)
As W is now equal to 2c2/K
(
eq. (13))
and (mg/mi) 4K
(
eq. (16))
, obviously the second term within the bracket is equal to 2c2, which means that the net energy is still zero and remains equal to zero as time passes.Otherwise stated, in a positively curved space, the gravitational part of the energy,
i.e. mgW , is increased by a factor 1 OK, but in the same time the inertial energy, i.e.
mic2, is increased by the same factor due to the increase of the apparent inertial mass
mi, so that the net energy remains zero.
Accordingly, changing the gravitational “constant” keeps the ratio (mg/mi) equal to unity and simultaneously saves the conservation-of-energy principle.
4. – Cosmic dynamics
4.1. Friedman equations. – If the gravitational constant G were made dependent on space curvature, the solution of Einstein’s equations would change. The gravitational constant G appears actually only in the proportionality coefficient between Einstein’s tensor Gaband stress-energy tensor Tab:
Gab4 ( 8 p GOc4) Tab.
For a flat space (k 40), G remains equal to G0 and the solution of Einstein’s equations is the classical one leading to a scale factor R increasing with time:
R 4R0t2 O3Ot02 O3.
(22)
When the space is curved, G takes a different value given by eqs. (20) and (208). For the sake of simplicity, let us consider only the case of the very beginning of the universe when radiation is dominating and when h is small. In this case, eq. (208) applies. The new Einstein’s equations are
Gab4 ( 8 p G0Oc4) Tab2 ( 8 p G0Oc4)(kh2O3 ) Tab.
(23)
The first term in the right-hand side is the usual one, but the second term is new. In the usual comoving frame, the component T00of the stress energy tensor is rc2and the component T11is pR2O( 1 2 ks2), where r is the energy density and p the pressure. The two Friedman equations become
3 R n 2 OR21 3 kc2OR24 8 p G0r 2k8pG0h2rO3 , (24) 22 R n n OR 2 R n 2 OR22 kc2OR24 8 p GpOc22 k8 p G0h2pO3c2, (25) where R n and R n n
denote the first and second derivatives of the scale factor R with respect to time.
A straightforward combination of eqs. (24) and (25) leads to the value of R
n n
OR:
R
n n
OR 4 2 ( 4 pO3 ) G0(r 13pOc2) 1k(4pO3) G0(h2O3 )(r 1 3 pOc2) . (26)
In a positively curved space, eq. (26) becomes
R
n n
OR 4 2 ( 4 pO3 ) G0(r 13pOc2) 1 (4pO3) G0(h2O3 )(r 1 3 pOc2) . (27)
4.2. A new cosmological force. – The first term of the right-hand side of eq. (26) is the usual one. It is the only one if space is flat (k 40). The minus sign means that, in a flat space, R
n n
is negative so that expansion is decelerating. This can be viewed as the effect of an attractive force corresponding to the Newtonian attractive gravitational force.
The second term of the right-hand side of eq. (26) is a new one which is positive when k 41
(
positively curved space, as in eq. (27))
and negative when k 421 (negatively curved space). This term corresponds to what could be called a “cosmological function”. This function introduces a new force which could be called a force “C”, to underline that it has a Cosmological origin and is linked to Curvature.In the radiation-dominated era the equation of state gives p 4rc2
O3 so that force C takes the form
C 4k(8pO9) G0h2r .
(28)
At early times, the arc parameter h is proportional to t1 /2(see ref. [9]). Accordingly,
h2is proportional to the ordinary time t. But the density r is proportional to t22. Then, force C changes as 1 /t . It is stronger and stronger as one gets closer and closer to the universe origin.
4.3. The effect of force C on space curvature. – Force C is either repulsive or attractive depending on the sign of space curvature. It is repulsive when the space is positively curved (k 411) and repulsive when the space is negatively curved (k421). Whatever its characteristic, the force C always cancels space curvature. Its effect makes space flat. Force C can be compared to a negative feedback mechanism reducing an error signal. Because force C varies as 1 Ot, it is very strong just after the Big Bang so that the universe is immediately and forcefully flattened. Any discrepancy with respect to flatness is erased by force C.
It has been pointed out that the apparent flatness of our universe is very unlikely because the existence of a flat space about 15 billion years after the Big Bang would require an extraordinary coincidence at the very beginning between the energy density and the critical value. So far, no convincing explanation has been suggested to account for such a coincidence. The foregoing considerations present a simple explanation. According to the force-C hypothesis, it would be no coincidence at all that the universe is flat because a flat expanding space is the only stable dynamic solution of Einstein’s equations. Any other curved solution would give rise to a force erasing the curvature by pushing or pulling space towards a flat geometry.
4.4. Relation with the classical cosmological constant. – Einstein’s equations (21) may be rewritten with a cosmological constant L:
Gab2 Lgab4 ( 8 p GOc4) Tab.
(29)
The equation giving the ratio R
n n OR becomes R n n OR 4 2 ( 4 pO3 ) G(r 1 3 pOc2) 1c2 LO3 . (30)
Comparing eq. (30) with eq. (27) shows that the new term in the right-hand side of eq. (27), for a positively curved space (k 41), corresponds to a cosmological constant given by
( 4 hO3) G0(h2O3 )(r 1 3 pOc2) 4c2LO3 . (31)
In the radiation-dominated era where p 4rc2/3 the corresponding value of L is L 44pG0r( 2 h2O3 c2)
(32)
instead of Einstein’s classical value
L 44pG0r . (33)
Despite the similarity of expressions (32) and (33) there are two main differences between them. Firstly, the cosmological term of eq. (32) is time dependent. Secondly, the effect of the cosmological term of eq. (32) is to force the positively curved space to expand towards a flat expanding space, whereas Einstein’s cosmological constant of eq. (33) is intended to exactly counterbalance the crushing of space to get a static equilibrium. Eddington [11] showed that this equilibrium is actually unstable. If a slight disturbance weakens the gravitational attraction, the repulsion due to the cosmological constant produces an expansion, that increases the average distance between bodies so that the attraction becomes weaker and weaker and the expansion speed increases. Similarly, if a slight disturbance strengthens the attraction, an
Fig. 1. – The cosmological force is repulsive for a positively curved space (k 41) and attractive for a negatively curved space (k 421). For k40 (flat space) the cosmological force is zero. The flat expanding space is the only stable dynamical solution.
accelerated contraction is produced. This is not the case in the dynamic equilibrium due to force C where stability is obtained (see fig. 1).
4.5. Cosmology with varying G. – Let M be the mass of the Hubble sphere, which is generally taken into account to apply Mach’s principle, and L be the Hubble length. The potential at the sphere center is then approximately 2GM/L. As M/L is not constant, if a constant effect on inertia is to be obtained (or, equivalently, a constant equality between gravitational energy and inertial energy), G cannot be constant in time. There is then a need for a theory wherein G is allowed to vary. Brans and Dicke [12] proposed a scalar-tensor theory which allows G to vary with expansion.
Taking the causal sphere instead of the Hubble sphere and computing the potential by the method explained in the foregoing
(
eq. (6))
allow one to avoid this difficulty since the potential thus obtained is no longer dependent on time but is constant (2c2). The gravitational constant no longer needs to vary with time.5. – Mach’s principle and very large numbers
5.1. The two groups of very large numbers. – The universe provides us with natural units for measuring length, mass, time, etc. For example, a natural unit of length is the size of a nucleon. This unit is known as the fermi. When the large-scale universe is measured with such natural units, it has approximately a size of 1040fermi.
Two groups of very large dimensionless numbers having a value of approximately 1040have been defined [13]:
i) A so-called N1group, dealing with subatomic quantities. If a nucleon of mass
Mn were a black hole, it would have a Schwarzschild radius of 2 GMn/c2, or approximately GMn/c2. This quantity is called the gravitational length of the nucleon and is noted ag. Actually, the radius of the nucleon is much larger and is noted a. The
ratio a/agis a very large number:
N1faOagfaO(GMnOc2) 40.231040.
(34)
Other similar very large numbers may be found using, e.g., the nucleon Compton length, instead of the nucleon radius (the ratio is then equal to 1.5 31038).
ii) A so-called N2group, dealing with cosmic quantities. For example, the Hubble length L, when expressed in the nucleon unit a, is
LOa4531040.
(35)
Other similar very large numbers may be found by using as a unit the electron Compton length or the nucleon Compton length (the numbers are, respectively, 4 31038
and 7 31041).
The similarity of the values of the N1and N2numbers (about 1040) is rather striking. P. A. M. Dirac [14] hypothesized the existence of a physical relation between these two groups explaining the coincidence and why it remains permanently. But, at present, as far as I know, no physical theory has managed to connect the two groups of numbers.
5.2. Connection with Mach’s principle in flat space. – These considerations about very large numbers may be related to Mach’s principle in the following way.
If the coincidence is to be permanent, a first difficulty arises because the first group
N1, e.g. a/ag, does not depend on time, whereas the second group N2, e.g. L/a , is time dependent. To avoid this difficulty, P. A. M. Dirac [14] suggested that G should decrease with time. As the Hubble length increases with time as t, the gravitational constant G should decrease as 1 /t . Then, both N1and N2numbers are increasing with time and may remain approximately equal.
Taking the Hubble length L as a cosmic parameter is rather awkward since this quantity changes with time. Furthermore, this length has only a theoretical interest (it is the location where the recession speed is equal to the speed of light) but has no real physical significance. The foregoing considerations about Mach’s principle show that there is a cosmic quantity which remains constant as time passes despite the expansion: it is the Machian effect of the causal sphere on any test particle and this effect is measured by the constant potential 2c2. Furthermore, this quantity has an essential physical meaning, since it is at the origin of inertia.
To obtain a dimensionless number one has to measure this potential by taking another potential linked with the nucleon as a unit. An obvious unit would be the potential at the center of a nucleon. This potential Wn is approximately 2MnG/a . With
Mn4 1.67 3 10227kg and a 410215m the value of this potential is
WnA 210222m2s22. (36)
With this value as a natural potential unit, the universe potential Wu reflecting the Machian effect on inertia is
WuOWn4 c2O102224 1039.
(37)
This value is indeed within the range of the N2numbers.
i) As N2is time independent, the similarity of its value with that of an N1number belonging to the first group is permanent and there is no need to let G change with time.
ii) A simple explanation for the mysterious connection between N1and N2can be suggested. Let N1 be the ratio between the actual size of a nucleon and its Schwarzschild radius Gmn/c2:
N14 aO(GMnOc2) , (38)
and let N2 be the ratio between the gravitational potential (2c2) at the center of the causal sphere and the gravitational potential (2GM/a) at the center of a nucleon:
N24 c2O(GMnOa) . (39)
Even though these two numbers are dealing with completely different domains (N1 with the subatomic world and N2with the cosmos) it appears very simply that they are necessarily equal since
N1faO(GMnOc2) 4c2O(GMnOa) f N2.
(40)
Accordingly, the similarity between the values of N1and N2numbers is not a mere coincidence but is a necessity strongly connected to Mach’s principle.
5.3. The unity of the universe. – Equality (40) between an N1 number and an N2 number illustrates the great harmony of the universe and its unity. The unity of the universe may be emphasized in another way, in connection with the value 2c2 illustrating the Machian effect of the universe. Let us consider a Planck particle with a mass equal to the Planck mass m* given by
m * 4 (ˇcOG)1 O2,
(41)
and a size equal to the Planck length a* given by
a * 4 (GˇOc3)1 O2.
(42)
One can define a classical potential inside such a particle by
W * 42m * G
a * .
(43)
Inserting the values of m* and a* into eq. (43) gives
W * 42 c2.
Otherwise stated, the classical gravitational potential self-energy of a particle having the Planck mass is equal to its rest mass energy [15], and the sum of these energies is zero. This is exactly the situation occurring in a causal sphere of 45 billion light years radius.
5.4. The fundamental role of the potential. – At first sight, it might appear rather arbitrary to focus on potentials when defining large numbers such as N1 and N2, instead of using length as is done traditionally. Actually it is not. When describing nature in the low-field approximation, the governing law is Poisson’s equation
essentially based upon the potential
˜2W 44pGr .
(44)
When dealing with relativistic conditions, the governing metric is the Schwarzschild metric with two potentials g00and g11given by
( g00Oc2) 4g114 ( 1 2 2 GMOc2r) , (45)
wherein the quantity GM/r is actually a potential. Even more, the term GM/c2r is a ratio between two potentials, the potential (2GM/r) with respect to the massive body and the potential (2c2) with respect to the universe wherein this body is necessarily embedded so as to have an inertial mass. Accordingly, the potentials ratio is the essence of the gravitational world.
6. – Conclusion
Measuring the Machian effect of the universe on inertia by taking the integral of the expansion deceleration along the causal sphere radius has the main advantage of giving a simple method for extending Mach’s principle to a curved space. This leads to a modification of the value of the gravitational constant G, depending on space curvature. This modification introduces a new cosmological force that has the effect of erasing any space curvature. This would explain why an expanding universe is flat. Furthermore, this concept leads to a constant value for the potential at the center of the causal sphere. This value makes it possible to define a new very large cosmic number and to demonstrate that this number is necessarily equal to another very large number linked to subatomic quantities. Mach’s principle then gives an insight into the great unity of the universe.
R E F E R E N C E S
[1] Mach’s ideas were first published in 1872 in German. An English translation is available in The Science of Mechanics (Open Court, La Salle) 1960.
[2] For a survey of Mach’s principle, with a reminder of Newton’s and Berkeley’s arguments, see HARRISON E. R., Cosmology, The Science of the Universe (Cambridge University Press) 1989, pp. 176-184 and also SCIAMA D. W., The Physical Foundations of General Relativity (Heinemann, London) 1969, Chapt. 2 and 3.
[3] MISNER C. W., THORNE K. S. and WHEELER J. A., Gravitation (W. H. Freeman and Company, New York) 1973, pp. 734-735.
[4] EINSTEINA., The Meaning of Relativity, 5th edition (Princeton University Press, Princeton) 1988.
[5] SCIAMAD. W., Mon. Not. R. Astron. Soc., 113 (1953) 34.
[6] The work of D. W. Sciama and his collaborators on Mach’s principle is reviewed by TODK. P. in the article Mach’s principle and isotropic singularities, in The Renaissance of General Relativity and Cosmology, A Survey to Celebrate the 65th Birthday of Dennis Sciama, edited by G. ELLIS, A. LANZA and J. MILLER (Cambridge University Press) 1993, pp. 234-247.
[7] For an application of Sciama’s “Law of Inertial Induction”, see BERRYM. V., Principles of Cosmology and Gravitation (Adam Hilger) 1989, pp. 37-39.
[8] SIGNORER. L., Nuovo Cimento B, 111 (1996) 1087.
[9] For the Friedman cosmology in a closed universe and the definition of arc parameter see MISNERC. W. et al., ref. [3], pp. 734-735.
[10] Strictly speaking, at very early times, the universe was dominated by radiation. But one may consider a hypothetical universe always dominated by matter. Anyway, at the very beginning of the matter-dominated era, after decoupling, let us say after a few million years, the age of the universe was still small compared to the billion years of now.
[11] EDDINGTON A. Sir, The Expanding Universe (Cambridge University Press, Cambridge) 1988, p. 50.
[12] BRANSC. and DICKER. H., Phys. Rev., 124 (1961) 925. [13] See HARRISONE. R., ref. [2], pp. 329-345.
[14] DIRACP. A. M., Nature, 139 (1937) 323.
[15] KENYON I. R., General Relativity (Oxford University Press) 1991, Exercise Q12.1, pp. 211 and 223.
