Using ideas from continuum mechanics we construct a theory of gravity. We
show that this theory is equivalent to Einstein's theory of general relativity;
it is also a much faster way of reaching general relativity than the
conventional route. Our approach is simple and natural: we form a very general
model and then apply two physical assumptions supported by experimental
evidence. This easily reduces our construction to a model equivalent to general
relativity. Finally, we suggest a simple way of modifying our theory to
investigate non-standard space-time symmetries.Comment: 7 pages, this essay received a honorable mention in the 2014 essay
  competition of the Gravity Research Foundatio

Boehmer, Christian G.

Downes, Robert J.

English

arXiv

 Using ideas from continuum mechanics we construct a theory of gravity. We show that this theory is equivalent to Einstein's theory of general relativity; it is also a much faster way of reaching general relativity than the conventional route. Our approach is simple and natural: we form a very general model and then apply two physical assumptions supported by experimental evidence. This easily reduces our construction to a model equivalent to general relativity. Finally, we suggest a simple way of modifying our theory to investigate nonstandard spacetime symmetries. </jats:p

Christian G. Böhmer

Robert J. Downes

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International Journal of Modern Physics D

From continuum mechanics to general relativity

Christian G. Böhmer

CiteSeerX

Using ideas from continuum mechanics we construct a theory of gravity. We show that this theory is equivalent to Einstein's theory of general relativity; it is also a much faster way of reaching general relativity than the conventional route. Our approach is simple and natural: we form a very general model and then apply two physical assumptions supported by experimental evidence. This easily reduces our construction to a model equivalent to general relativity. Finally, we suggest a simple way of modifying our theory to investigate nonstandard spacetime symmetries

Boehmer, CG

Downes, RJ

UCL Discovery

arXiv:1405.4728v1  [gr-qc]  19 May 2014From continuum mechanics to general relativityChristian G. Bo¨hmer∗and Robert J. Downes†∗Department of Mathematics,†Centre for Advanced Spatial Analysis,University College London,Gower Street, London, WC1E 6BT, UK19 May 2014This essay received a honorable mention in the 2014 essay competition ofthe Gravity Research Foundation.AbstractUsing ideas from continuum mechanics we construct a theory ofgravity. We show that this theory is equivalent to Einstein’s theoryof general relativity; it is also a much faster way of reaching generalrelativity than the conventional route. Our approach is simple andnatural: we form a very general model and then apply two physicalassumptions supported by experimental evidence. This easily reducesour construction to a model equivalent to general relativity. Finally,we suggest a simple way of modifying our theory to investigate non-standard space-time symmetries.∗c.boehmer@ucl.ac.uk†r.downes@ucl.ac.uk1Einstein himself, of course, arrived at the same Lagrangian butwithout the help of a developed field theory, and I must admitthat I have no idea how he guessed the final result. We have hadtroubles enough arriving at the theory - but I feel as though hehad done it while swimming underwater, blindfolded, and withhis hands tied behind his back!Richard P. Feynman, Feynman lectures on gravitation, 1995.1 IntroductionEinstein’s theory of general relativity is one of the crowning achievements ofmodern science. However, the route from a special-relativistic mind gameto a fully-fledged theory capable of producing physical predictions is longand tortuous. As Feynman notes, Einstein completed this journey withseemingly superhuman intuition: for the rest of us, lengthy mathematicalpreparation is the key to developing the theory.The purpose of our essay is to showcase a new approach to generalrelativity. Using well-known machinery from continuum mechanics [1] weconstruct a general Lagrangian which describes a space-time continuum.We make two physical assumptions which reduce our Lagrangian (relativelypainlessly) to that of general relativity, the Einstein-Hilbert action. We alsosuggest a simple way of modifying our theory to investigate different modelsof gravitation with alternative physical assumptions.Readers with a background in continuum mechanics should be familiarwith our construction while this essay serves as an introduction to generalrelativity for members of this community.2 Mathematical preliminariesThere is strong experimental evidence that space and time should not betreated as independent concepts, but that we should think of a 4-dimensionalspace-time continuum when describing nature. Such a continuum is mod-eled mathematically as a 4-dimensional manifold equipped with Lorentzianmetricds2 = ηijδiαδjβdxαdxβ := (dt)2 − (dx1)2 − (dx2)2 − (dx3)2 (2.1)which is used to determine distances between points. We adopt Einstein’ssummation convention whereby summation is carried out over repeated in-dices. Here xκ, κ = 1, 2, 3 are the standard Cartesian coordinates and x0 = t2is time. This is the usual setting for special relativity, a theory containingno gravity. The Kronecker δ’s (where i, j = 0, 1, 2, 3) indicate that all localcoordinate frames are aligned. This over-complication will become useful tous further on.Light propagating in a vacuum described by space-time (2.1) travels instraight lines. However, experimental evidence suggests that in our universelight rays are deflected by massive bodies. We argue that this observationindicates (2.1) should be modified when gravity is taken into account. Todo this we assume that our local coordinate system changes from point-to-point on the manifold in the presence of gravity. Mathematically, we modifyour metric by attaching to every point a frame or tetrad eiα(xκ, t), whichdescribes this change of coordinatesds2 = ηijeiαejβdxαdxβ := gαβdxαdxβ . (2.2)The sixteen components of the frame can be thought of as a set of orthonor-mal vectors attached to each point of the manifold, where i labels eachvector and α their components. At any fixed point P of our manifold, thefunctions eiα determine how the local coordinates at P are related to thelocal coordinates at any other point Q.The frame will play the role of the dynamical variable. We are thereforemodelling a situation in which physical information related to the gravita-tional field is encoded in the frame. Gravity manifests itself as the differencein alignment of local coordinates at different points; this is the reason weused Kronecker δ’s in (2.1) when dealing with the special-relativistic case.We can also view the frame as a pseudo-orthogonal matrix. This suggeststhat we can introduce the inverse frame ejβ which, together with the frameitself, will satisfy the equalitieseiαeiβ = δβα, eiαejα = δij . (2.3)The position of the indices allows us to differentiate between the frame andits inverse without using different notation.To progress further we need to measure the gravitational field strength.It seems natural to differentiate the first expression from (2.3) with respectto the coordinate xγ , say, which giveseiα∂γeiβ + eiβ∂γeiα = 0. (2.4)This motivates the introduction of the rank-3 tensor, antisymmetric in thefirst and third indicesKαβγ = gγγ′Kαβγ := gγγ′eiα∂βeiγ′ , (2.5)which we take as our measure of field strength and call the contortion tensor.33 Our approachSo far we have introduced the frame eiα and linked this to the metric (2.2).We then identified a possible gravitational field strength tensor (2.5). Thenext logical step is to construct an admissible Lagrangian from which wecan determine a set of field equations via a variational principle. Here ‘ad-missible’ means we take a very general Lagrangian and apply two physicalassumptions to isolate physically meaningful situations. These assumptionsare discussed below.As the reader will no doubt have noticed, we aim to model the gravi-tational field in a manner very different to general relativity. Indeed, thisis a good time to emphasise that we are not considering a geometry withcurvature. If we assume we work in flat space then our metric (2.2) isconstant, and hence raising and lowering indices commutes with partial dif-ferentiation. We need no longer differentiate between co- and contra-variantvectors and can work with Cartesian tensors, which reduces computationalcomplexity significantly. Stipulating that space-time is flat does not removegravity from our theory; it only serves to reduce the metric equation (2.2)to a restrictive constraint on the frame components eiα.4 Cautionary notesWe stress that our treatment of general relativity via continuum mechanicsis often referred to as the ‘teleparallel equivalent of general relativity’, see [2].In this context, the free parameters of a general continuum-based theory arechosen to produce an equivalence with general relativity.Our approach is different. We do not choose specific parameters to forcesuch an equivalence; we retain all degrees of freedom and apply two physicalassumptions which reduce our continuum model to general relativity. Itis the fact that two sensible physical assumptions alone produce generalrelativity that motivates this essay.5 Lagrangian and assumptionsDefine the Lagrangian W with the rank-6 tensor Cαβγλµν asW :=12CαβγλµνKαβγKλµν (5.1)where C takes all permutations over (αβγλµν), which has 4096 possiblecomponents. Using symmetry arguments we will reduce this to something4more manageable. The symmetry properties of K imply C enjoys two sym-metries, namelyCαβγλµν = −Cγβαλµν , Cαβγλµν = −Cαβγνµλ. (5.2)Under (5.2) the number of independent components of C drops from 4096to 576. Irrespective of the symmetry properties of K, we also haveCαβγλµν = Cλµναβγ . (5.3)Including (5.3) reduces the number of independent components from 576 to300. We now state our two physical assumptions.1. Isotropy: the property of a vacuum space-time whereby gravity actsequally in all directions.2. Lorentz invariance: the property of a space-time whereby experimentalresults are independent of the orientation or the relative velocity of thelaboratory through space.Both assumptions are well supported by current experimental evidence andare key to modern field theory.Isotropy is invoked mathematically by stipulating that the tensor C re-mains invariant under local changes of coordinates and, indeed, must becomposed from tensors which obey the same restriction. This means thata rank-6 tensor of our type can only be composed from Kronecker δ’s. Asimple combinatorial exercise demonstrates there are 15 possible ways ofchoosing pairs from a set of 6 possibilities, which indicates thatW =1215∑m=1cmδm(αβγλµν) (5.4)where cm are undetermined parameters and m(αβγλµν) is one of the 15possible combinations of indices. To see the form of the isotropic tensorsδm(αβγλµν), consider the case when the indices retain their original order,indicated by m = 1, say:c1δ1(αβγλµν) = c1δαβδγλδµν . (5.5)We have not yet included the three symmetries (5.2), (5.3). Doing so reducesthe number of independent components to 3! Explicitly we haveW = c1KααβKβγγ + c2KαβγKαγβ + c3KαβγKαβγ (5.6)5where c1, c2 and c3 are the remaining free parameters. We must now includethe second physical assumption, Lorentz invariance. This manifests itself asthe invariance of our Lagrangian under a certain transformation of the frame,namelyeiα 7→ Λijejα (5.7)where Λij(xκ, t) is a member of the Lorentz group satisfyingηijΛimΛjn = ηmn. (5.8)Our metric (2.2) is invariant under such a transformation. The required in-variance of the Lagrangian (5.6) produces a constraint on the undeterminedparameters c1, c2 and c3, specifically thatc1 = −c2, c3 = 0. (5.9)Therefore we have reduced the number of independent components of Cfrom 4096 to 1, and only 1, free parameter!6 The final resultOur one-parameter Lagrangian can now be writtenW = c1 (KααβKβγγ −KαβγKαγβ) . (6.1)This is a significant achievement. We have produced a simple Lagrangianfrom a very general case using two physical assumptions, all without intro-ducing curvature. Furthermore, one can show that our Lagrangian is, up toa surface term, equivalent to the Einstein-Hilbert action of general relativity.At this point we could vary our Lagrangian to determine a set of fieldequations. These will be equivalent to the Einstein field equations, althoughexpressed in terms of the tetrad instead of the metric.As a final thought, it is possible to introduce different physical as-sumptions into our model by replacing isotropy with another symmetry.This asymmetry is at a very fundamental level of the theory and is com-pletely independent of the symmetries of the metric. One can formulate ananisotropic model (Cαβγλµν anisotropic) where the space-time (gαβ) wouldremain isotropic for instance. Our model allows for the straightforward in-troduction of such non-standard symmetries.Our essay demonstrates a very quick method of arriving at general rela-tivity using the machinery of continuum mechanics. It also provides a neatmethod for investigating symmetries of the gravitational field and, hope-fully, the reader has not at any point felt as though they were “swimmingunderwater, blindfolded, and with [their] hands tied behind [their] back[s]”!6AcknowledgementsWe thank Dmitri Vassiliev and Caitlin Dempsey for useful discussions.References[1] A. E. H. Love, A treatise on the mathematical theory of elasticity, Dover,New York (1944); W. Jaunzemis, Continuum Mechanics, The MacMillanCompany, New York (1967).[2] E´. Cartan, “Sur les espaces ge´ne´ralise´s et la the´orie de la relativite´”,C. R. Acad. Sci., 174, 734737 (1922); “Sur les varie´te´s a` connex-ion affine et la the´orie de la relativite´ ge´ne´ralise´e”, Ann. Sci. EcoleNorm. Sup., 40, 325412 (1923); A. Einstein, “Riemann-Geometriemit Aufrechterhaltung des Begriffes des Fernparallelismus”, Sitzungs-ber. Preuss. Akad. Wiss., 1928(XVII), 217221 (1928); “NeueMo¨glichkeit fu¨r eine einheitliche Feldtheorie von Gravitation und Elek-trizita¨t”, Sitzungsber. Preuss. Akad. Wiss., 1928(XVIII), 224227(1928).7

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