Is Gravity Actually the Curvature of Spacetime?
aa r X i v : . [ g r- q c ] M a y Is Gravity Actually the Curvature of Spacetime?
Sebastian Bahamonde , ∗ and Mir Faizal , † Laboratory of Theoretical Physics, Institute of Physics,University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia. Department of Mathematics, University College London,Gower Street, London, WC1E 6BT, United Kingdom. Department of Physics and Astronomy,University of Lethbridge, 4401 University Drive,Lethbridge, Alberta T1K 3M4, Canada and Irving K. Barber School of Arts and Sciences,University of British Columbia - Okanagan, 3333 University Way,Kelowna, British Columbia V1V 1V7, Canada
Abstract
The Einstein equations, apart from being the classical field equations of General Relativity, arealso the classical field equations of two other theories of gravity. As the experimental tests ofGeneral Relativity are done using the Einstein equations, we do not really know, if gravity is thecurvature of a torsionless spacetime, or torsion of a curvatureless spacetime, or if it occurs due tothe non-metricity of a curvatureless and torsionless spacetime. However, as the classical actions ofall these theories differ from each other by boundary terms, and the Casimir effect is a boundaryeffect, we propose that a novel gravitational Casimir effect between superconductors can be usedto test which of these theories actually describe gravity.Corresponding author: Sebastian Bahamonde Submission date: 31 March 2019
Essay written for the Gravity Research Foundation 2019 Awards for Essays onGravitation ∗ email: [email protected], [email protected] † email: [email protected] eneral Relativity (GR) is one of the most well tested theories in Nature, but in all thosetests, what is actually tested are the predictions made by the Einstein equations [1]. Itis possible to construct two other geometrical theories describing gravity, which are funda-mentally different from GR, but whose classical field equations are the Einstein equations.To understand these theories, we first note that the spacetime has to be described by adifferential manifold in any geometrical theory of gravity. Now a general affine connectionΓ αµν , on such a manifold, can be decomposed into three pieces [2, 3]Γ αµν = (cid:8) αµν (cid:9) + K αµν + L αµν . (1)The first term (cid:8) αµν (cid:9) is the standard Levi-Civita connection , which is obtained from the metric.The second term K αµν is the contortion tensor , which is obtained from the torsion tensor T αµν as K αµν ≡ (1 / T αµν + T ( µαν ) . The last term L αµν is the disformation tensor , whichis constructed from the non-metricity tensor Q αµν ≡ ∇ α g µν as L αµν ≡ (1 / Q αµν − Q ( µαν ) .GR is described by a torsionless spacetime ( T αµν = 0), which satisfies the metric com-patibility condition ∇ α g µν = 0 ( Q αµν = 0). So, as K αµν = L αµν = 0 in GR, the affineconnection of Eq. (1) can be written in terms of the Levi-Civita connection as Γ αµν = (cid:8) αµν (cid:9) .The curvature tensor constructed from this Levi-Civita connection ¯ R αβµν is used to obtainthe Einstein-Hilbert action S G = 116 πG Z √− g ¯ R , (2)where g is the determinant of the metric g µν and ¯ R ≡ g βν ¯ R αβαν is the scalar curvatureobtained from ¯ R αβµν . Einstein equations are the classical field equations obtained from thisaction. Teleparallel Gravity (TG) is another geometrical theory of gravity, whose classical fieldequations are the Einstein equations. In this theory, the general connection of Eq. (1) isequated to the
Weitzenb¨ock connection , and so the curvature of spacetime vanishes. Thus,TG is constructed using such a curvatureless spacetime, which satisfies the metric compat-ibility condition Q αµν = 0 ( L αµν = 0) [4–9]. This theory is constructed from the torsiontensor in the tetrad formalism, and its action is given by S T = − πG Z eT , (3)where e = √− g is the determinant of the tetrad and T is the scalar torsion (which isconstructed from contractions of the torsion tensor).2t may be noted that in curvatureless spacetime of TG, the curvature tensor ( R αβµν )obtained from the general affine connection of Eq. (1) vanishes, but the curvature tensorconstructed using the Levi-Civita connection ( ¯ R αβµν ) does not vanish. In TG, the scalarcurvature R obtained from R αβµν is related to the scalar curvature ¯ R obtained from ¯ R αβµν ,as R = ¯ R + T − (2 /e ) ∂ µ ( eT λλµ ) = 0, so we can write¯ R = − T + B T , (4)where B T = (2 /e ) ∂ µ ( eT λλµ ) is a boundary term. Thus, the action for GR given by Eq. (2)and the action for TG given by Eq. (3), differ from each other by the boundary term B T .It is also possible to formulate a Theory of Non-Metricity (TNM) to describe gravity[10–14]. This theory is also called Coincident General Relativity or Symmetric TeleparallelGravity, as it has certain features which resemble both GR and TG, but we shall call it asTNM, as the theory is based on the concept of non-metricity. In this theory, both the torsiontensor and R λµνβ vanish, and gravity is produced because of the non-vanishing non-metricitytensor, ∇ α g µν = Q αµν = 0 ( L αµν = 0). The action for this theory is constructed using thenon-metricity scalar Q (which is obtained from the non-metricity tensor Q αµν ) as S N = − πG Z √− gQ . (5)As TNM is described by a torsionless and curvatureless spacetime, Q can be related to ¯ R (curvature obtained from the Levi-Civita connection) as¯ R = − Q + B N , (6)where B N = (1 / √− g ) ∂ α ( Q αλλ − Q λλα ) is again a boundary term (different from the boundaryterm obtained in TG). So, the action for GR given by Eq. (2) and the action for TNM givenby Eq. (5) differ from each other by the boundary term B N .Even though the actions of GR, TG and TNM differ from each other by boundary terms,they have the same classical field equations (Einstein equations), so they cannot be classicallydistinguish from each other. The only reason for the preferential attention given to GR (overthe other two geometrical theories) is historical and not scientific. However, they can bedifferentiated using quantum effects because these theories are fundamentally different fromeach other and will produce different quantum corrections. We do not have a full theory ofquantum gravity, but it is possible to get an estimate of perturbative quantum gravitational3ffects, using the formalism of effective field theories [15–17]. Thus, the classical actions forGR ( S G ), TG ( S T ) and TNM ( S N ) get corrected by quantum corrections S QG , S QT and S QN ,such that S = S G + S QG , S = S T + S QT , S = S N + S QN . (7)It is not possible to use cosmological and astrophysical observations to differentiate between S G , S T and S N , however, such observations can differentiate between S , S and S . It hasbeen demonstrated that the quantum corrected GR [16] and quantum corrected TG [17]are both consistent with the cosmological data obtained from SNe Ia + BAO + CC + H [18–21], and so at present, quantum corrections cannot rule out either of them. However, itis still possible that future cosmological observations may rule out one of these theories.Even though, at present, we are not able to use quantum corrections to differentiatebetween these theories, it is still possible to use a combination of quantum effects andboundary effects to distinguish them from each other. As the actions of GR, TG, TNMdiffer from each other by boundary terms, and the Casimir effect is a quantum mechanicalboundary effect, a gravitational Casimir effect can be used to distinguish them from eachother. The reflection of gravitational waves in the microwave regime by quantum propertiesof superconductors (Heisenberg-Coulomb effect) [22–25] can produce a novel measurablegravitational Casimir effect [25–29]. In ordinary metal plates, the lattice of ions and electronsmove along the same geodesic, in the presence of gravitational waves. However, when Cooperpairs form below the superconducting transition, they move along a non-geodesic path due totheir quantum non-localizability. It has been demonstrated that this produces a large massconductivity due to an enhanced mass current [25–29]. As the electromagnetic waves arereflected due to the electrical conductivity, this mass conductivity reflections gravitationalwaves [22–25]. Thus, for such systems, a gravitational Casimir effect can be produced [25–29], in analogy with the conventional electromagnetic Casimir effect [30–33].As the actions for GR, TG and TNM are related to each other by boundary terms, we canrelate the gravitational Casimir energy in GR ( h E i G ) [25–29] to the gravitational Casimirenergies in TG ( h E i T ) and TNM ( h E i N ) as h E i T = h E i G + h E i B T , h E i N = h E i G + h E i B N , (8)where h E i B T is the contribution from the boundary term B T , and h E i B N is the contribu-tion from boundary term B N . Since the boundary action for these theories is different, so4 E i B T = h E i B N = 0, thus we obtain h E i G = h E i T = h E i B N . So, these theories will producedifferent gravitational Casimir effects, and such effects can be used to test which of thesetheories is actually the geometrical theory of gravity. It may be noted that the Casimir forcebetween superconductors has been recently experimentally measured [25, 34–38]. Thus, it ispossible to measure the novel gravitational Casimir effect due to the onset of superconduc-tivity between two aluminum nanostrings. With an optomechanical cavity readout, theseexperiments could detect 6 mP a differences in the Casimir force between such nanostrings[25, 34–38]. The magnitude of a gravitational Casimir effect depends on the difference be-tween the change in momentum of the Cooper pair and change in the momentum of theion core [25–29]. Even if a more detailed analysis, reduced the magnitude of this novelgravitational Casimir effect by ten orders of magnitude, it would still remain a measurableeffect, using the currently available technology. It is important to achieve sufficiently ac-curate parallelism between two superconductors at low temperatures to produce this novelgravitational Casimir effect. The technology needed to obtain such an accurate parallelismhas already been used in resonator platforms for superconducting circuits [39, 40]. So, suchan experiment can be performed using the currently available technology, and we can knowwhich theory actually describes gravity in our Universe. Acknowledgments
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