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Cosmological constant implementing Mach principle in general relativity

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Abstract

We consider the fact that noticing on the operational meaning of the physical concepts played an impetus role in the appearance of general relativity (GR). Thus, we have paid more attention to the operational definition of the gravitational coupling constant in this theory as a dimensional constant which is gained through an experiment. However, as all available experiments just provide the value of this constant locally, this coupling constant can operationally be meaningful only in a local area. Regarding this point, to obtain an extension of GR for the large scale, we replace it by a conformal invariant model and then, reduce this model to a theory for the cosmological scale via breaking down the conformal symmetry through singling out a specific conformal frame which is characterized by the large scale characteristics of the universe. Finally, we come to the same field equations that historically were proposed by Einstein for the cosmological scale (GR plus the cosmological constant) as the result of his endeavor for making GR consistent with the Mach principle. However, we declare that the obtained field equations in this alternative approach do not carry the problem of the field equations proposed by Einstein for being consistent with Mach’s principle (i.e., the existence of de Sitter solution), and can also be considered compatible with this principle in the Sciama view.

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Notes

  1. More specific, the strong version of the Mach principle (namely, if there is no matter then there is no geometry). However, the resulted theory only satisfies its weak version (namely, the matter distribution determines the geometry).

  2. Throughout this work, the signature is \((-,+,+,+)\), the lower case Greek indices run from zero to three, and we use units in which \(\hbar = 1= c\).

  3. Although, Einstein gave convincing argument against the absolute time by use of the operational method, but Bridgman [7] declares his failure in carrying it over into GR.

  4. That is usually used as a model for a matter field conformally coupled to gravity. This action may also be considered corresponding to the action of Brans–Dicke theory in the special case of \(\omega =-3/2\), where under transformation \(\overline{g}_{\mu \nu } = \varphi ^2 g_{\mu \nu }\), one can get it as \(S[\overline{g}_{\mu \nu }]=\frac{1}{2}\int {d^4}x\sqrt{-\overline{g}}\left( \frac{1}{6}\overline{R}+\frac{1}{2}\lambda \right) \) without the scalar field [22]. Nevertheless, it should be noted that the scalar field in the Brans–Dicke theory has a different meaning than the one in our approach. In this theory, the value of the gravitational constant varies with position as the consequence of Mach’s principle, i.e. the effect of matter distribution in the universe on the value of this constant at every point. However, in action (2), G has been replaced by a scalar field as the consequence of the conformal invariance and the arbitrariness of unit systems at every point, actually, the same as the scalar field proposed by Dirac in the Weyl–Dirac action [23].

  5. In a spacetime with arbitrary dimension n, the conformal invariance can be achieved if it is accompanied by the appropriate rescaling of the scalar field as \(\varphi \rightarrow \Omega ^w(x)\varphi \), where the conformal weight w depends on the dimension of the scalar field of the model and is \(w=1-n/2\), see, e.g., Ref. [24]. However, the resulted two new fields still remain independent.

  6. However, in general, one can assume that the metrics in the gravitational and the matter parts are different (although conformally related), and achieves interesting results, see, e.g., Ref. [19].

  7. Note that, in four dimensions, with the conformal transformations (3) not only the field equation (4) is conformally invariant, but if, in addition to (3), the traceless symmetric energy–momentum tensor of the matter transforms as \({\tilde{T}}_{\mu \nu }=\Omega ^{-2}(x)\, T_{\mu \nu }\), then the field equation (5) and the conservation equation \(\nabla _{\mu }T^{\mu \nu }=0\) are also conformally invariant, i.e. \({\tilde{\nabla }}_{\mu }{\tilde{T}}^{\mu \nu }=0\) [20].

  8. This condition is not a holonomic constraint, and we have introduced an auxiliary field \(\alpha (x) \) to have such a condition, wherein with the obtained constant value of \(\varphi \), the field equations also yield \(\alpha =9\lambda T^{\mu }{}_{\mu }/(2\Lambda ^2)\).

  9. It also infers that the inertial energy of a particle (with mass m) is due to the gravitational potential energy of the matter of the universe upon it, i.e. \(mc^2-GM_0m/R_0\sim 0\), more or less a mathematical formulation of the Machian point of view [28, 29].

  10. The constant value of \(\varphi \) and those approximations also give \(\lambda =5/(9M_0R_0)\) and \(\alpha =-5/2\). In addition, the proportionality factor 5 in Eq. (13) can be improved. That is, if one considers all the employed approximations as \(\Lambda =a\, R_0^{-2}\), \(R=b\, R_0^{-2}\), \(T^{\mu }{}_{\mu }=-c\, M_0 R_0^{-3}\) and \( R_0/M_0=d\, G\), hence, instead of the proportionality factor 5, one will get the equality factor \(d(b+4a)\). Furthermore, one can usually consider \(d\sim 1\sim b\) up to the first order, and also uses the relation \(\Lambda =3\, \Omega _{\Lambda }H^2\) while assumes \(R_0=\beta \, H_0^{-1}\) (i.e., \(R_0=\beta \, d_H\) at the present epoch). Thus, with the Planck data [30] for \(\Omega _{\Lambda }\) at the present epoch, if the resulted factor is supposed to be \(8\pi \), it will yield \(\beta \simeq 2\).

  11. Note that, there is no explanation in this paper about the meaning one can attribute to the scalar field; and however, as mentioned, the approach of this work inspired us in figuring out the way of applying conformal symmetry, and then, breaking it as a method for acquiring a generalization of a theory operationally valid in some scale, for some other scales.

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Acknowledgments

We thank the Research Office of Shahid Beheshti University for the financial support.

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  1. Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran, 19839, Iran

    Nadereh Namavarian & Mehrdad Farhoudi

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  1. Nadereh Namavarian
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  2. Mehrdad Farhoudi
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Correspondence to Mehrdad Farhoudi.

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Namavarian, N., Farhoudi, M. Cosmological constant implementing Mach principle in general relativity. Gen Relativ Gravit 48, 140 (2016). https://doi.org/10.1007/s10714-016-2135-1

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