sign in sign up
<<
Home , Gravitation (book), Gravity

arXiv:1904.05706v3 [gr-qc] 29 Jan 2020 ∗ Keywords orsodn author Corresponding suewa ed n hc disa cinprinciple. an admits which and theo fields Einstein’s weak of assume expansion of non-relativisti large covariant This structure. causal dilati non-relativistic gravitational p includes passed that gravity are Newtonian , gravitational and light relativity of general deflection of tests classical three the that strate ttmnsaotrltvsi ffcsaeotnsbl.I t In subtle. often are effects relativistic about Statements 1 ntttfrTertsh hsk ignsiceTechnis Eidgenössische Physik, Theoretische für Institut nvriyo dnug,PtrGtreTi od Edinburg road, Tait Guthrie Peter Edinburgh, of University 2 rvt,rltvt,nnrltvt,ato,1 action, non-relativity, relativity, Gravity, : 3 colo ahmtc n awl nttt o Mathematic for Institute Maxwell and Mathematics of School rvt ewe etnadEinstein and between Gravity odt,KHRylIsiueo ehooyadSokomUn Stockholm and Technology of Institute Royal KTH Nordita, sa rte o h rvt eerhFudto 2019 Foundation Research Gravity the for written Essay ensHansen Dennis easnpy.tzc,jhroge.cu,obers@nbi. [email protected], [email protected], olgtlsakn2,S-0 1Sokom Sweden. Stockholm, 91 SE-106 23, Roslagstullsbacken ofagPuiSrse2,89 üih Switzerland. Zürich, 8093 27, Wolfgang-Pauli-Strasse lgase 7 K20 oehgnØ Denmark. Ø, Copenhagen DK-2100 17, Blegdamsvej 4 h il orIsiue oehgnUniversity, Copenhagen Institute, Bohr Niels The wrsfrEsy nGravitation. on Essays for Awards 1 el Hartong Jelle , aur 0 2020 30, January Abstract 1 ∗ 2 n il .Obers A. Niels and , rvt hoyaie rma from arises gravity c aeyprhlo precession, perihelion namely , /c neet hl eann a retaining while effects on yo rvt htde not does that gravity of ry retyb netninof extension an by erfectly h ohcueZürich, Hochschule che 2 i sa ewl demon- will we essay his expansion. H F,UK. 3FD, EH9 h ku.dk lSciences, al 3,4 iversity, In 1916 formulated the three classical tests of his , namely [1]:

1. the precession of the perihelion of ’s

2. the deflection of light by the , and

3. the effect.

Subsequently all of these have been tested experimentally with wonderful agreement. Originally these tests were devised as it was known that Newton’s law of universal gravitation gave inaccurate predictions. All three tests are described in the general relativity framework by studying the Schwarzschild solution and its . We will argue that these effects do not require Einstein’s notion of . The three classical tests are actually solely strong gravitational field effects. This follows from an extension of Newtonian gravitation that includes time dilation [2]. This novel theory is obtained from a covariant off shell 1/c2 expansion of general relativity, leading naturally to a gravity theory with manifest non-relativistic symmetries [3, 4]. It is much richer than Newtonian gravity and can account for many results we know from general relativity. This opens the way for a more nuanced discussion about the status of gravity and its quantum version in the non-relativistic domain. The Einstein says that locally the laws of reduce to those of special relativity. On the other hand, insisting on local Galilean relativity will lead to Newton–Cartan geometry, pioneered by Cartan [5, 6] and further developed in e.g. [7, 8].

The Galilean algebra consists of time and translations H, Pa, Galilean boosts Ga and Jab = −Jba, where a,b = 1,...,d are spatial indices. The relevant local symmetry algebra is actually its central extension, known as the Bargmann algebra, where the central N (appearing in [Pa, Gb] = Nδab) can be interpreted as . One can obtain Newton–Cartan geometry by gauging the Bargmann algebra [9, 10], which in turn can be obtained via the Inönü–Wigner contraction of the direct sum of the Poincaré algebra and a U(1) generator. Since not all relativistic systems possess a U(1) symmetry this has the serious drawback that it is not a generic limit that always exists. Another way to obtain the non-relativistic limit of relativistic physics is to re-instate factors of c and to perform a 1/c2 expansion up to the desired order. This approach can always be done and is thus generic, but as we will see it is not necessarily equivalent to the above mentioned Bargmann limit. The Poincaré algebra has generators TA =

{H, Pa,Ba, Jab}, where Ba are the Lorentz boosts. We first re-instate all factors of c explicitly in the commutation relations and the generators with the polynomial 2 (n) n ring in σ ≡ 1/c . We then obtain the basis of generators TA ≡ TA ⊗ σ , where n ≥ 0 is

2 the level. This is a graded algebra with nonzero commutation relations of the form

(m) (n) (m+n) (m) (n) (m+n+1) (m) (n) (m+n+1) H ,Ba = Pa , Pa ,Bb = δabH , Ba ,Bb = −Jab , h i h i h i (1) (m) where we left out the commutators with Jab . We can quotient this algebra by setting to zero all generators with level n>L for some L [11]. At level n = 0 the algebra is isomor- (1) (1) (1) (1) phic to the Galilean algebra. Including the level n = 1 generators {H , Pa ,Ba , Jab } (1) (1) (1) one finds a commutator [Pa, Gb] = H δab, except that now [H , Ga] = Pa . Hence H(1) is not central like in the Bargmann algebra, and thus the latter is not a subalgebra. This has severe implications: the result of gauging the Bargmann algebra will in general not coincide with the 1/c2 expansion of general relativity [2]! To see how strong field effects are encoded into the non-relativistic limit of general relativity, we define the following covariant 1/c2 expansion (where c is the slope of the 1 light cone in tangent space) of the Lorentzian metric gµν [3, 12]:

1 g = −c2τ τ + h − Φ + · · · , (2) µν µ ν µν c2 µν 1 1 1 gµν = hµν − vˆµvˆν + hµρhνσΦ + 2ˆvµvˆν Φ+˜ · · · , (3) c2 c2 ρσ c4

µ ν where Φ˜ = −vˆ vˆ h¯µν /2. The omitted terms will not contribute to our calculations. Covariance requires diffeomorphisms that are analytic in 1/c2. Expanding Einstein’s equations in 1/c2 tells us that at leading order τ ∧ dτ = 0, so that the clock 1-form µ τ = τµdx describes a foliation of . When dτ 6= 0 two observers with worldlines

γ1, γ2 need not experience the same lapse of time, γ1 τ 6= γ2 τ. In Galilean relativity µν space is measured by the inverse spatial metric h withR signatureR (0, 1,..., 1) satisfying µν µ µ µν h τν = 0. The tensorsv ˆ ≡ v − h mν and h¯µν ≡ hµν − mµτν − mν τµ are projective inverses and furthermore involve the field mµ, which is important as m0 is the Newtonian potential. They satisfy the completeness relations

µ µλ¯ µ µ − vˆ τν + h hλν = δν andv ˆ τµ = −1 . (4)

µν µρ νσ The geometry is described by τµ and h . The subleading fields mµ and h h Φρσ are gauge fields on the geometry. A natural choice for the affine connection is [10]

1 Γ¯λ = −vˆλ∂ τ + hλσ ∂ h + ∂ h − ∂ h . (5) µν µ ν 2 µ νσ ν µσ σ µν 1 − 2 0 0 a b  0 This follows from writing gµν = c EµEµ + δabEµEν and Taylor expanding the vielbeins Eµ and a −2 Eµ in σ = c .

3 νρ This connection is Newton–Cartan metric compatible, i.e. ∇¯ µτν = ∇¯ µh = 0 where ∇¯ is the covariant derivative associated to (5). Notice that the connection has torsion ¯λ λ Γ[µν] = −vˆ ∂[µτν] when dτ 6= 0. Our hand is thus forced: when repackaging gravity in the geometric language of Newton–Cartan geometry, torsion arises naturally. The most general solution to τ ∧ dτ =0 is τ = NdT , where T is a time function and N is the non- relativistic lapse function. As in general relativity we can always use diffeomorphisms to choose T = t as the time coordinate. However, in Newton-Cartan geometry there is not enough freedom to set N to unity: the lapse function is physical. Setting dτ = 0 corresponds to a weak gravitational field. We are now in a position to expand the Einstein–Hilbert action in powers of 1/c2. Since we expand the metric up to subleading orders we will do the same for the action. 4 2 Schematically the Einstein–Hilbert Lagrangian expands as LEH = c LLO + c LNLO + · · ·

[13]. It can be shown that the equations of of the leading order Lagrangian LLO are identical to the of the subleading fields appearing in the next- to-leading order Lagrangian LNLO. We can thus throw away LLO and with LNLO. This defines the theory of non-relativistic gravity2 and the action is [2]:

1 S = − dd+1xe vˆµvˆν R¯ − Φ˜hµν R¯ − Φ hµρhνσ R¯ − a a − ∇¯ a 16πG µν µν µν ρσ ρ σ ρ σ Z h 1  + Φ hµν hρσR¯ − 2hρσ a a + ∇¯ a + S , (6) 2 µν ρσ ρ σ ρ σ M   i 1/2 ρ ¯ where e = (−det(−τµτν + hµν )) , aµ ≡ 2ˆv ∂[ρτµ], Rµν the Ricci tensor associated to ¯λ Γµν and SM contains possible couplings. In the absence of matter sources for τµ and demanding that τµ is nowhere vanishing the equations of motion dτ = 0. In this case the field Φµν decouples from the equations of motion for mµ and hµν . An example of a simple matter Lagrangian that does not source τµ is LM = −(d − 2)eρ/2 where ρ is a mass density. The equations of motion lead to d − 2 R¯ =8πG ρτ τ , (7) µν d − 1 µ ν with dτ =0 and Φµν appearing in a decoupled equation. This is the geometrized diffeo- morphism covariant version of the Poisson equation [8]. The theory however allows for more general geometries in which dτ 6= 0. These are the non-relativistic gravity solutions that encode strong field effects. We will now focus on those, i.e. we will assume that there is a matter distribution that sources dτ 6= 0 (such as a spherical non-rotating fluid ) outside of which we have a vacuum solution describing time dilation effects [14].

2Continuing the 1/c2 expansion to even higher orders will bring in relativistic effects to the theory, but we shall refrain from studying them in this essay.

4 To study the motion of massive on a background with dτ 6= 0 we must perform an expansion of the relativistic proper time worldline action. We need to expand ˙ µ 1 µ ˙ µ µ the embedding scalars as τµX = c τµy˙ + · · · and hµν X = hµν x˙ + · · · . The resulting reparameterization invariant action is

1/2 d 2 S ∝ dλ (τ yµ) − τ yµa x˙ ν − h x˙ µx˙ ν . (8) dλ µ µ ν µν Z "  #

In a gauge in which the term in brackets in the integrand of the action (8) is a constant (−2E > 0) on shell and using τµ = N∂µT , one finds the equations of

d d (τ yµ) − τ yµa x˙ ν = N 2 (yµ∂ T )=1 , (9) dλ µ µ ν dλ µ 1 − x¨λ + Γ¯λ x˙ µx˙ ν = hλσ∂ N 2 , (10) µν 2 σ 1 1 1 E = − + h x˙ µx˙ ν . (11) 2 N 2 2 µν

The last term in (10) is the negative gradient of the potential −N −2/2. The term µ τµy acts as a Lagrange multiplier implementing conservation of energy E in (11). We thus see that the expansion of timelike geodesics leads to bound (E < 0) non-relativistic particles with potential energy −N −2/2. The 1/c2 expansion for null geodesics leads to identical equations except with E = 0. It can be shown that the most general 4-dimensional static spherically symmetric vacuum solution to the equations of motion of (6) is given by

r τ = 1 − s δ0 , (12) µ r µ r r hµν = diag 0, 1 − s , 1/r2, 1/ r2 sin2 θ , (13) r     with mµ =0 and Φµν = 0 and rs an integration constant [14] where the coordinates are xµ = (t,r,θ,φ). We can also obtain the above geometry by performing a 1/c2 expansion of the Schwarzschild solution, treating the rs as a c-independent integration constant [3, 4]. This expansion terminates after one order, so that it is a solution of non-relativistic gravity. The time dilation is described by the lapse function

N(r)= 1 − rs/r, which is exactly the same as in general relativity. This proves that our non-relativisticp gravity theory passes the third classical test as it is a pure time-dilation effect. For motion in the equatorial plane (θ = π/2) equations (10) and (11) lead to

dr 2 r4 r r4 = − 1 − s κ + r2 , (14) dφ b2 r a2      

5 where we treat the radial coordinate r(φ) as a function of the angular variable φ. The parameters a and b have of length and depend on the angular and energy of the . This equation is equivalent to the equation for motion in the equatorial plane for both massive (κ = 1) and massless (κ = 0) particles in general relativity [15]. We have thus demonstrated that the first and second classical tests are also passed by the theory described by the action (6)! In general solutions of the equations of motion of (6) are exact/approximate solutions of general relativity if the latter’s 1/c2 expansion terminates/extends beyond subleading order [4]. For example FRW-type and the associated are fully described by non-relativistic gravity coupled to non-relativistic fluids [14]. However the nature of the 1/c2 expansion is such that the equations of motion of (6) do not admit solutions: they are true relativistic phenomena. Finally, the 1/c2 expansion of general relativity is a simplification even if we have many more fields. This is because there is a preferred foliation of time and interactions become instantaneous. For the same reason the quantum version of non-relativistic grav- ity might be more attainable than its notoriously difficult relativistic parent. In view of the Bronstein cube of Gc¯h-physics, where one typically approaches (relativistic) from the quantum field theory or general relativity corner, this could open up a third road by approaching it from the non-relativistic quantum gravity corner.

References

[1] A. Einstein. Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik, 354:769–822, 1916.

[2] Dennis Hansen, Jelle Hartong, and Niels A. Obers. Action Principle for Newtonian Gravity. Phys. Rev. Lett., 122(6):061106, 2019.

[3] Dieter Van den Bleeken. Torsional newton-cartan gravity from the large c expansion of general relativity. Class. Quant. Grav., 34(18):185004, 2017.

[4] Dieter Van den Bleeken. Torsional newton-cartan gravity and strong gravitational fields. In 15th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic (MG15) Rome, Italy, July 1-7, 2018, 2019.

[5] É. Cartan. Sur les variétés à connexion affine et la théorie de la rélativité généralisée (première partie). Ann. Éc. Norm. Super., 40:325, 1923.

[6] É. Cartan. Sur les variétés à connexion affine et la théorie de la rélativité généralisée (première partie)(suite). Ann. Éc. Norm. Super., 41:1, 1924.

6 [7] L. P. Eisenhart. Dynamical and Geodesics. Annals of Mathematics, 30:591, 1928.

[8] A. Trautman. Sur la theorie newtonienne de la gravitation. Compt. Rend. Acad. Sci. Paris, 247:617, 1963.

[9] Roel Andringa, Eric Bergshoeff, Sudhakar Panda, and Mees de Roo. Newtonian Gravity and the Bargmann Algebra. Class.Quant.Grav., 28:105011, 2011.

[10] Jelle Hartong and Niels A. Obers. Hořava-Lifshitz gravity from dynamical Newton- Cartan geometry. JHEP, 07:155, 2015.

[11] Oleg Khasanov and Stanislav Kuperstein. (In)finite extensions of algebras from their Inonu-Wigner contractions. J. Phys., A44:475202, 2011.

[12] G. Dautcourt. On the of General Relativity. Acta Phys. Pol. B, 21:755, 1990.

[13] Dennis Hansen, Jelle Hartong, and Niels A. Obers. Non-relativistic expansion of the Einstein-Hilbert Lagrangian. In 15th Marcel Grossmann Meeting on Recent De- velopments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories (MG15) Rome, Italy, July 1-7, 2018, 2019.

[14] Dennis Hansen, Jelle Hartong, and Niels A. Obers. Non-Relativistic Gravity and its to Matter. 2020.

[15] Sean M. Carroll. Spacetime and geometry: An introduction to general relativity. San Francisco, USA: Addison-Wesley, 2004.

7