Holomorphic Gravity. Christiaan L. M. Mantz

Transcription

1 Holomorphic Gravity Christiaan L. M. Mantz

2 The Waterfall by M.C.Escher is displayed on the cover. The water symbolizes anti-symmetric space-time-momentum-energy curvature, better known as torsion.

3 Holomorphic Gravity On the Reciprocity of Momentum and Space Christiaan L. M. Mantz November 28, 2007

4 Master s Thesis in Theoretical Physics Christiaan Laurens Michael Mantz Spinoza Institute Institute for Theoretical Physics Utrecht University Supervisor: Dr T. Prokopec

5 Abstract In an attempt to generalize general relativity, two new theories, Hermitian gravity and holomorphic gravity, are being proposed. Space-time is generalized to space-time-momentum-energy and hence the general principle of covariance is extended. These theories contain features as a Hermitian metric on complex manifolds, maximal accelerations, dynamical torsion and possibilities for removing singularities. We have indications that the theories of Hermitian and Holomorphic gravity yield general relativity at large scales and a theory equivalent to general relativity at very small scales, where the momenta and energies are very large.

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7 Contents 1 Preliminary Motivations 1 2 General Relativity Canonical Metric Formulation Vielbein Formulation: Noncoordinate Bases Second Order Formalism Cosmology Reciprocity of Momentum and Space Motivations Reciprocal Relativity Hermitian Geometry Complex Manifolds Definitions Complexifications Almost Complex Structure Hermitian Manifolds Complex Gravity Complex Metrics Equations of Motion in Complex Gravity Einstein s Theory of Generalized Gravity and Complex Geometry The Limit to General Relativity Hermitian Gravity The Hermitian Metric Flat Space The Hermitian Equations of Motion The Limit to General Relativity Holomorphic Gravity The Holomorphic Metric Flat Space Second Order Formalism The Limit to General Relativity The Schwarzschild Solution Cosmology The Full Complex Metric The Full Complex Line Element Flat Space The Limit to General Relativity

8 vi CONTENTS 9 Conclusions 71 A Transformation Sheet 73 B Waves 75 Acknowledgements 76 Bibliography 79

9 Chapter 1 Preliminary Motivations In order to specify an event in our physical world one needs a number of dimensions to do so. One would like to say for instance, where and when something happened. Dimensions can be viewed at as a set of universal numbers in order to locate events. In ancient Greece, people already realized that one needs three dimensions in order to specify an event in space and one dimension in order to specify an event in time. Space and time were thus thought of as being invariant subspaces; they were considered to be absolute in the sense that every observer would agree on space and time intervals between any two events. It was Albert Einstein in 1905 [1] who realized that the two spaces, time and space, were actually approximately invariant in the limiting case of observers travelling at low speeds relative to each other; only at relative speeds, small compared to the speed of light, they would approximately agree on the observed space and time intervals. But at increasing relative speeds, speeds not small compared to the speed of light, the difference in space and time intervals, measured in their frame of reference, would differ more and more. Einstein realized that there was a new invariant observable quantity, namely the space-time interval; all observes travelling with any relative speed would agree on any space-time interval between any event. Hence in his theory of special relativity, space and time were relative to moving observers and space-time had become absolute. But how do we know that space-time is just not simply another limit of a bigger space? It has become common in modern physics to consider more dimensions than the four known space-time dimensions. This is done in an attempt to unify the four fundamental forces in nature. The three forces unified in the standard model already suggest additional dimensions in order to specify an event; there are observable parameters, which depend on the energy at which they are measured, called running parameters. The electric charge for example depends on the energy at which it is measured. So one could argue that in order to specify an event, described by the standard model, one needs besides four space-time dimensions also four additional momentum energy dimensions. Einstein argues in 1907 [2]: So far we have applied the principle of relativity, i.e., the assumption that the physical laws are independent of the state of motion of the reference system, only to non accelerated reference systems. Is it conceivable that the principle of relativity also applies to systems that are accelerated relative to each other?. He then introduces a principle that we know now as the equivalence principle; one cannot tell locally whether an observer is accelerating or is placed in a gravitational field. If we follow Einstein s principle of relativity closely, the laws of physics are independent of the system of reference, one could argue that there must be a similar principle of equivalence between rotating observers and observers placed in a torsion field, a gravitational field which could cause observers to rotate. Different approaches have been suggested. One can consider observers with angular momentum [3], coupling to and generating torsion fields. But one can also consider observers with spin, but then the torsion generated by the spin is not dynamical. Theories of generalized gravity, in which dynamical torsion is present, have been proposed [4], but faces possible stability issues [5]. Hence one can argue that a theory of gravitation is needed in which one can consider rotating frames, implying

10 2 CHAPTER 1. PRELIMINARY MOTIVATIONS that one should regard accelerations, in order to transform from an inertial frame to a rotating one, analogous to velocity boosts, which transform inertial frames into each other. One of most pressing problems of the laws of modern physics is our lack of understanding of how quantum mechanics and the theory of general relativity are related. Quantum mechanics possesses a symmetry between the space and the momentum coordinates; any wave function expressed in space coordinates can be fourier transformed into a wave function that lives in momentum space. This reciprocity between space and momentum is not found in the theory of general relativity. Quantum mechanics also embodies the Heisenberg uncertainty principles, which describe the minimum uncertainty in simultaneous measurements of position and momentum. The theory of general relativity describes momenta as tangent vectors and hence these can precisely measured simultaneously with position. Max Born suggested that this reciprocity of space and momentum and the uncertainty principles should be present in a new theory reducing to the laws of general relativity for distances being very large compared to energy and momentum quanta [6]. He argued also that this theory should reduce, on the other hand, to a theory which looks exactly like the theory of general relativity, but then the coordinates are energy-momentum coordinates instead space-time coordinates and they are large compared to the time and space quanta. Finally, he points out that somewhere in the intermediate region, however, this new theory should reduce to the laws of quantum mechanics. Reciprocal relativity my be realized in black hole space-times close to the singularity. From the indications above we can form a raw sketch of a larger theory generalizing and perhaps unifying the laws of general relativity and quantum mechanics: The standard model suggests that we need additional momentum energy coordinates in order to specify an event. According to the principle of relativity, the laws of physics should not depend on any reference frame and hence new observable invariants should be formulated in terms of the eight space-timemomentum-energy coordinates; space-time become relative to the absolute space-time-momentumenergy space, meaning all relatively non inertial observers moving agree on the value of the observable. The quantum mechanical laws suggest that space and momentum are reciprocal variables. The theory should contain both inertial and rotating frames with accelerations relating the two, which according to the equivalence principle implies a dynamical theory of torsion. The theory behaves in the limits as pointed out by Max Born. The aim of this thesis is to pursue some of these ideas and to derive a new theory along these lines. Let our journey begin!

11 Chapter 2 General Relativity The theory of general relativity is governed by nonlinear second order differential equations constraining the metric tensor. These differential equations are the Einstein equations. The formalism in which one writes a theory in terms of a second order differential equation is called second order formalism. In this case the connection coefficients are the Christoffel symbols and they are completely determined by the metric tensor. It is however not clear a priori that, if one starts with the Einstein-Hilbert action, one will obtain Einstein s equations. One can formally derive second order equations of a theory by taking the related variables as independent and deriving the first order differential equations constraining these variables. When a theory is formulated in terms of first order differential equations, the formalism in which this is done is called first order formalism. In this approach to gravity there are two sets of independent dynamical variables and the two first order differential equations on the two dynamical variables are the equations of motion of the theory. We will treat two possibilities for choosing these dynamical variables; first we will go through the canonical metric formulation, using the metric tensor and a connection as dynamical variables and then we will derive the equation s of motion for the vielbein and spin connection. The relations between the two formulations are also shown. 2.1 Canonical Metric Formulation In this section we will treat the metric and the connection as independent dynamical variables. The two first order differential equations constraining these two variables can be obtained by varying the Einstein-Hilbert action S = d d x gr, (2.1) where R = g µν R µν, with respect to the metric and the connection. The tensorial object, R µν, is known as the Ricci tensor and is completely determined by the connection coefficient. The geometrical meaning of this object lies in the parallel transport of a vector around a closed loop. Parallel transporting a vector around an infinitesimal closed loop is equivalent to operating with a commutator on this vector. So in order to obtain the relation between the Ricci tensor and the connection coefficient we calculate the commutator of two covariant derivatives [ λ, ν ]V σ = λ ν V σ Γ α λν α V σ + Γ σ λα ν V α (λ ν), where we have just written out some of the covariant derivatives in terms of partial derivatives and connection coefficients. Continuing this process we arrive at λ ν V σ + λ Γ σ λαv α Γ α λν α V σ + Γ σ λα ν V α + Γ σ λαγ α νβv β Γ σ να λ V α (λ ν).

12 4 CHAPTER 2. GENERAL RELATIVITY Relabelling some dummy indices and eliminating some terms that cancel when anti symmetrized we obtain We can write this equation as ( λ Γ σ νµ ν Γ σ λµ + Γ σ λαγ α νµ Γ σ ναγ α λµ)v µ 2Γ α [λν] αv σ. where the Riemann tensor is defined as [ λ, ν ]V σ = R σ µλνv µ T α λν α V σ, and the torsion tensor is defined as follows R σ µλν λ Γ σ νµ ν Γ σ λµ + Γ σ λαγ α νµ Γ σ ναγ α λµ T α λν 2Γ α [λν]. (2.2) The geometrical interpretation of the Riemann tensor is that it measures the difference between the two transported vectors; the two vectors end up at the same point but there is an angle between the two of them. The way in which the vectors differ, after parallel transporting them, is a property of the manifold. The torsion tensor doesn t measure the difference between two vectors that are at the same base point as the manifold, but instead it measures the difference between these two base points after parallel transporting the two vectors. The failure of closing loop is also a property of the manifold. We can think of the Riemann tensor as a multi linear map of three vector fields to another vector field and notate it in this way as follows R(X, Y )Z X Y Y X [X,Y ] Z. Similarly we can think of the torsion tensor as a bilinear map mapping to vectors fields two another vector field T (X, Y ) X Y [X, Y ]. We are now ready to vary the Einstein-Hilbert action with respect to the metric and the connection in order to derive the first order constraints on the metric and the connection. This is done as follows δs = d d xδ( gg µν )R µν + d d x gg µν δr µν = d d x g(r µν 1 2 Rg µν)δg µν + d d x gg µν δr µν, where we have made use of the following identity for the variation of the determinant of g where δ g = 1 2 g δg = 1 2 ggµν δg µν, (2.3) δg = g(g µν δg µν ) = g(g µν δg µν ). In the last step we have expressed the variation of the metric in terms of the variation of the inverse metric by taking the variation of the following identity g µλ g λν = δ µ ν giving us this identity δg µν = g µρ (g νσ δg ρσ ).

13 2.1. CANONICAL METRIC FORMULATION 5 We still have to work out the variation of the Einstein-Hilbert action with respect to Γ ρ µν. In order to do this we need some more identities, which are derived below. Suppose we have this line element S = d d x gb µ A ν g µν, where A ν ν φ and B µ ν ψ. We now vary this action with respect to ψ and obtain after partial integration δs φ = d d x g( ν + 1 ν gg µν B µ )δφ g = d d x g ν g µν B µ δφ = d d x g( ν + Γ α αν)g µν B µ δφ, where the second step follows from the fact that the integrant transforms as a scalar and hence the two terms between the brackets must transform covariantly. From this observation we can now define Γ ν µν 1 g µ g. (2.4) We had promised that the connection coefficients were completely independent variables, but it clear that this statement is weakened by this identity. The connection coefficient is however still not completely determined by the metric and so we have to proceed our work. With identity for the square root of the determinant of the metric (2.3) we can write the connection coefficient with summed indices (2.4) as Γ ν µν = 1 g µ g = 1 2 gαβ µ g αβ = 1 2 g αβ µ g αβ. (2.5) With these identities at hand we can proceed with the variation of the Einstein-Hilbert action with respect to the connection δs Γ ρ µν = d d x gg µν δrµλν. λ Writing out the Riemann tensor in terms of the connection coefficients yields d d x gg µν [ λ δ(γ λ νµ) ν δ(γ λ λµ) + Γ λ λαδ(γ α νµ) Γ λ ναδ(γ α λµ) + Γ α νµδ(γ λ λα) Γ α λµδ(γ λ να)] = d d x g[ λ g µν δ(γ λ νµ) 1 g λ ( g)g µν δ(γ λ νµ) + ν g µν δ(γ λ λµ) + 1 g ν ( g)g µν δ(γ λ λµ) + g µν Γ λ λαδ(γ α νµ) g µν Γ λ ναδ(γ α λµ) + g µν Γ α νµδ(γ λ λα) g µν Γ α λµδ(γ λ να)], where we have partially integrated and thrown away boundary terms. Collecting the connection coefficients by introducing Kronecker delta s we obtain d d x g[ λ g µν 1 λ ( g)g µν + δ ν g λ α g µα + 1 α ( g)g µα δ ν g λ +g µν Γ α αλ g µα Γ ν αλ + g αβ Γ µ βα δν λ g αν Γ µ λα ]δ(γλ νµ). Using the identities for the connection coefficient with summed indices (2.4, 2.5) gives us d d x g[ λ g µν + δλ ν α g µα + Γ β βα gµα δλ ν g µα Γ ν αλ + g αβ Γ µ βα δν λ g αν Γ µ λα ]δ(γλ νµ) = d d x g[ λ g µν g µα Γ ν αλ g αν Γ µ λα + δν λ( α g µα + g αβ Γ µ βα + Γβ βα gµα )]δ(γ λ νµ). (2.6)

14 6 CHAPTER 2. GENERAL RELATIVITY When writing the partial derivatives and connection coefficients as covariant derivatives the previously stated equations (2.6) look like If we then operate on this equation (2.7) with δ λ ν we obtain λ g µν + δ ν λ α g µα = 0. (2.7) (d 1) ν g µν = 0, meaning ν g µν = 0 for d 1, which then can be plugged into the equations of motion of the connection (2.7), yielding the equations of motion of the connection in it s well known form (2.8b), which are given below. But since the Riemann tensor vanishes for d = 1, these equations of motion (2.7) are always equivalent to the well known form of the equations of motion (2.8b). The equations of motion of the Einstein-Hilbert action in it s known form are R µν 1 2 Rg µν = 0 λ g µν = 0, (2.8a) (2.8b) where the first set of equations are called Einstein s equations and the second set of equations the metric compatibility equations. The metric compatibility equations are, as said before, equivalent to the equations of motion obtained by the variation of the Einstein-Hilbert action with respect to the connection (2.6). Furthermore, the Einstein tensor is defined as and the non-metricity tensor is defined as G µν R µν 1 2 Rg µν (2.9) Q µν λ λ g µν. (2.10) Using these definitions (2.9, 2.10), we can now write the equations of motion of the Einstein-Hilbert action (2.8a, 2.8b) as follows in vacuum Q µν λ = 0 G µν = 0. When matter is present - we will come back to this issue in greater detail later in terms of a scalar field representing matter - these equations of motion become and 1 δl g δg µν = G µν = κt µν 1 δl g δγ λ µν = λ g µν + δ ν λ α g µα = κ M µν λ, where κ and κ are constants to be determined by the taking the Newtonian limit and were the objects T and M represent conserved currents obtained by the variation of the matter part of the action with respect to the metric and the connection respectively. The Einstein tensor turns out to be a divergenceless quantity as the it should be, since energy is conserved µ G µν = 0, µ T µν = 0, (2.11) implying that the covariant derivative of both sides of Einstein s equations vanish. The reader is invited to check that the Einstein tensor is indeed a divergenless quantity.

15 2.2. VIELBEIN FORMULATION: NONCOORDINATE BASES Vielbein Formulation: Noncoordinate Bases In this section we will treat the veilbein and the spin connection as independent variables. These variables are then, as said above, considering the theory via first order formalism, constrained by two first order differential equations, which are the equation of motion of the Hilbert-Palatini action 1 S[e, ω] = where the curvature of ω is given by = M M ɛf e e ɛ abcd ɛ αβγδ ( e a αe b βf cd γδ ), (2.12) F c γδd = γ ω c δ d δ ω c γ d + ω c γ eω e δ d ω c δ eω e γ d and e a α is the vielbein. We will explain the relation between the metric and the connection and the vielbein 2 and the spin connection below. We will also show how the Einstein-Hilber action and the Hilbert-Palatini action are related. The metric in terms of vielbeins is then defined 3 as g µν e a µη ab e b ν, (2.13) where η ab is the Minkowski metric, µ a spacetime index and a a flat Minkowski index. The physical interpretation of the vielbein is that it acts as a local observer; in his Lorentz frame, at spacetime point x γ, the vector v µ (x γ ) is being measured by this observer as v a (x γ ) = e a µv µ (x γ ) [7]. The metric g µν is used to raise and lower spacetime indices, denoted by Greek letters µ, ν and the Minkowski metric is used to raise and lower flat Minkowski spacetime indices, denoted by Latin letters a, b from the beginning of the alphabet. There is an additional symmetry, a local Lorentz symmetry, acting on the flat indices of the vielbein leaving the metric (2.13) invariant. The Lorentz connection for parallel transporting these indices just as the affine connection is used for the parallel transport of the spacetime indices is called the spin connection and denoted by ω a µ b. A tensor with mixed indices transforms both under local lorentz transformations (LLT s) and general coordinate transformations resulting in the mixed tensor transformation law 4 T a µ b ν = Λ a aλ b b x µ x µ x ν x ν T aµ bν. The covariant derivative acting on flat Minkowski spacetime indices gives rise to spin connection coefficients just as it yields ordinary connection coefficients, when acting on spacetime indices, in the following manner µ T a b = µ T a b + ω a µ ct c b ω c µ b T a c. Demanding that a tensor is independent of the the basis it is projected on, we can derive a relation between the two connections. Considering the tensor T, we first project it onto a purely coordinate basis T = µ T ν dx µ ν = ( µ T ν + Γ ν µλt λ )dx µ ν. 1 This is the Hilber-Palatini action in four dimensions. In d dimensions the number of anti symmetrized vielbeins is d 2. 2 The vielbein is also know as the tetrad. 3 We won t make the distinction between eµ a and it s transpose e a µ from now on, since it s not important for our purposes. 4 In this subsection we follow partly the book of S.Carrol [8].

16 8 CHAPTER 2. GENERAL RELATIVITY Now we project the same tensor onto a mixed basis and converting it to the same object projected onto a purely coordinate basis T = µ T a dx µ ê (a) = ( µ T a + ω a µ bt b )dx µ ê (a) = ( µ (e a νt ν ) + ω a µ be b νt ν )dx µ e ν a ν = (e λ ae a ν µ T ν + e λ at ν µ e a ν + e λ ae b νω a µ bt ν )dx µ λ = ( µ T ν + (e ν a µ e a λ + e ν ae b λω a µ b)t λ )dx µ ν, where we relabelled indices in the last step. Now we can easily compare the tensorial object T in both projections and obtain the following relation between the two corresponding connections Γ ν µλ = e ν a µ e a λ + e ν ae b λω a µ b = e ν ad µ e a λ. (2.14) The spin connection in terms of the connection projected coordinate basis is then ω a µ b = e λ b µ e a λ + e λ b e a νγ ν µλ = e λ b µ e a λ, where D α e b β = αe b β +ω b α ce c β is the gauge covariant derivative and where µ e a λ = µe a λ Γɛ αɛ is an ordinary covariant derivative acting only on curved space-time indices. The covariant derivative of the vielbein vanishes µ e a ν = µ e a ν e a λγ λ µν + e c νω a µ c = 0, where we have used the expression for the connection coefficient in terms of the spin connection (2.14). The spin connection then, serving as a connection in order to construct a covariant derivative, clearly does not transform as a tensor under local Lorentz transformations. It transforms inhomogeneously, in the following manner ω a µ b = Λa aλ b b ω a µ b Λ c b µλ a c. Requiring metric compatibility on the two flat space indices µ η ab = µ η ab ωµ c aη cb ωµ c bη ac = ω µba ω µab we obtain an asymmetry between the indices a and b of the spin connection ω µba = ω µab. (2.15) Before we can begin with obtaining the equations of motions of the Hilbert-Palatini action we will need a number of definitions. A determinant of a matrix, in this case the inverse of the vielbein, is then defined as ɛ abcd ɛ αβγδ e α a e β b eγ c e δ d e 1, (2.16) where e 1 is just shorthand notation for e 1 and where the completely antisymmetric Levi-Civita symbol is defined as +1 for µ 1 µ 2 µ n being an even permutation of 01 (n 1) ɛ µ1µ 2 µ n = 1 for µ 1 µ 2 µ n being an odd permutation of 01 (n 1) 0 otherwise The Levi-Civita symbol does not transform as a tensor. When multiplying the definition of the deteminant of the vielbein (2.16) with ɛ α β γ δ we obtain ɛ abcd e [α a e β b eγ c e δ ] d = e 1 ɛ α β γ δ, (2.17)

17 2.2. VIELBEIN FORMULATION: NONCOORDINATE BASES 9 where we have used the following identity ɛ α β γ δ ɛ αβγδ = ( 1) l 4!δ α [α δ β β δγ γ δ δ ] δ, where l is the number of negative values of the metric (we are neglecting this overall sign whenever it is irrelevant for our purposes). The left hand side of the obtained equation (2.17) does transform as a tensor and it is called the Levi-Civita tensor and denoted by ɛ α β γ δ. Now we can write this equation (2.17) as ɛ αβγδ = e 1 ɛ αβγδ, (2.18) where we have ommitted the primes on the indices. The same procedure for the vielbein, instead of the inverse vielbein, leads to the definition for the Levi-Civita tensor with it s indices lowered ɛ αβγδ = e ɛ αβγδ. We also could have just lowered the indices with the metric, since e = g. transforms as under the mixed transformation law. The vielbein a = Λ a x µ a e µ x µ eµ a taking the determinant of both sides we obtain e(x x µ ) = x µ e(x), where the determinant of a proper Lorentz transformation is one. Objects that transform under the transformation law times a determinant in front, as the determinant of the vielbein does above, are called tensor densities and the power to which the determinant is raised is the weight of the tensor density. It is clear that the Levi-Civita symbol transforms as a tensor density by the definition of the Levi-Civita tensor (2.18); the right hand transforms as a tensor since the left hand side does and e is a tensor density, therefore the Levi-Civita symbol is also a tensor density, but with opposite weight. This leads to the following transformation law for the Levi-Civita symbol ɛ µ 1 µ = x µ 2...µ n x µ ɛ x µ1 µ 1µ 2 µ n x µ2 x µ 1 x µ 2 We state the following identity for contracting two Levi-Civita tensors xµn. (2.19) x µ n ɛ µ1µ2 µ kα 1α 2 α n k ɛ µ1µ 2 µ k β 1β 2 β n k = ( 1) l k!(n k)!δ [α1 β 1 δ α2 β 2 δ α n k] β n k. (2.20) Finally we define the volume element as dx n = dx 0 dx 1 dx n 1 (2.21) = 1 n! ɛ µ 1µ 2 µ n dx µ1 dx µ2 dx µ1. Now we are ready to show how the Hilbert-Palatini action and the Hilbert-Einstein action are related. The Hilbert-Einstein action is again given by S[g, Γ] = d 4 x gr, which can be written as d 4 x er ν1ν2 ν 3ν 4 ɛ σ1σ2ν3ν4 ɛ ν1ν 2σ 1σ 2.

18 10 CHAPTER 2. GENERAL RELATIVITY One can show indeed that these two expressions are the same by writing out the latter, using identity that contracts two Levi-Civita tensors (2.20). Not paying attention to numerical factors we arrive at d 4 x R ν1ν2 ν 3ν 4 ɛ σ1σ2ν3ν4 ɛ ν1ν 2σ 1σ 2, where we have absorbed the determinant into the Levi-Civita tensor. This then becomes d 4 x e c σ 1 e d σ 2 F ab ν 3ν 4 ɛ σ1σ2ν3ν4 ɛ abcd = S[e, ω], where we have used the definitions of the determinant of the vielbein (2.16, 2.17). This expression then equals the Hilbert-Palatini action (2.12). We are now ready to vary the Hilbert-Palatini action to obtain the equations of motion. Varying the Hilbert-Palatini action (2.12) with respect to the vielbein we obtain δ e S[e, ω] = d 4 x ɛ abcd ɛ αβγδ δ(e a α) ( 2e b βfγδ cd ). Varying the same action with respect to the spin connection we obtain δ ω S[e, ω] = d 4 x ɛ abcd η fd ɛ αβγδ e a αe b β ( γ δω c δ f δ δω c γ f + δω c γ eω e δ f + ω c γ eδω e δ f δω c δ eω e γ f ω c δ eδω e γ f ), which can now be written in terms of gauge covariant derivatives, since the variation to the spin connection is a tensor, yielding d 4 x ɛ abcd η fd ɛ αβγδ e a αe b [ β Dγ δωδ c f D δ δωγ c ] f = d 4 x ɛ abcd η fd ɛ αβγδ [ ( 2D γ e a α e b ) ] β δω c δ f, where we have relabelled indices and used the anti-symmetry of the lower indices of the spin connection (2.15). The equations of motion of the Hilbert-Palatini action (2.12) then become ɛ abcd ɛ αβγδ e b βf cd γδ = 0 (2.22a) ɛ abcd ɛ αβγδ ( D α e b βe c γ) = 0, (2.22b) where we have used definition of connection with summed indices (2.4). These equations can be written in differential geometry notation e F = 0 D e e = 0, (2.23) where we suppressed indices. The canonical metric formulation and the vielbein formulation are not equivalent. The first order differential equations constraining the metric and the vielbein are different. The equation of motion constraining the vielbein will yield us vanishing torsion instead of the metric compatibility equations. The first equation of motion is equivalent to the equation of motion of the Einstein- Hilbert action with respect to the metric (2.8a), since the Einstein tensor is equivalent to the trace of the double dual of the Riemann tensor ɛ µ1µ2µ3µ4 ɛ ν1ν2ν3ν4 R µ3µ 4ν 1ν 2 g µ2ν 3 = g µ1[ν1 g µ2 ν2 g µ3 ν3 g µ4 ν4] R µ3µ 4ν 1ν 2 g µ2ν 3, (2.24)

19 2.2. VIELBEIN FORMULATION: NONCOORDINATE BASES 11 where the Levi-Civita symbols are multiplied with determinants of the vielbein in order to get Levi-Civita tensors so that the indices of the Levi-Civita tensors can be raised and lowered. Raising the Levi-Civita tensor leads immediately to the right hand side, if one considers the definition for contracting two Levi-Civita tensors (2.20) (the Kronecker delta s are multipied by metric tensors). Multiplying by minus one-forth we obtain where we have used the identity 1 4 gµ1[ν1 g µ3 ν2 g µ4 ν4] R µ3µ 4ν 1ν 2 = R µ1ν4 1 2 gµ1ν4 R, g µ1[ν1 g µ2 ν2 g µ k νk g µn νn] g µk ν k = (d n + 1)g µ1[ν1 g µ2 ν2 g µn 1 νn 1] and written out the anti-symmetrized product of metric tensors, yielding six terms; two of those terms give Ricci scalar terms and the other four give Ricci tensor terms such that we precisely get the Einstein tensor. The first set of equations of motion of the Hilbert-Palatini action (2.22a) can be formed into the Einstein equations by writing the set as the trace of the double dual of the Riemann tensor in the following manner ɛ abcd ɛ γδαβ e bβ F cdγδ = ɛ abcd ɛ γδαβ e µ c e ν de bβ R µνγδ = ɛ abcd ɛ γδαβ e µ c e ν de ρ b R µνγδg βρ. Multiplying the set of equation with e σ a in order to get rid of the last Lorentz index a ɛ abcd ɛ γδαβ e µ c e ν de ρ b eσ ar µνγδ g βρ = ɛ σρµν ɛ γδαβ R µνγδ g βρ. This is again the trace of the double dual of the Riemann tensor which is equivalent to the Einstein tensor as shown above in the manipulations (2.24). This concludes the exercise of showing the equivalence of the first set of equations of motion of the Einstein-Hilbert action with respect to the metric with the first set the equations of motion of the Hilbert-Palatini action with respect to the veilbein. The second set of equations of motion of the Hilbert-Palatini action turns out to be equivalent to the vanishing of the connection with lower indices anti symmetrized; the connection turns out to be symmetric. This can be shown through the following manipulations ɛ abcd ɛ αβγδ ( D α e b βe c ) γ = e [α a e β b eγ c e δ] ( d 2e b β D α e c ) γ, where we have used the product rule, relabeled indices and expressed the antisymmetric epsilon tensors in terms of vielbeins. This then becomes e [α a e γ c e δ] ( d 2Dα e c γ) = e [α a e γ c e δ] ( ) d 2T c αγ, (2.25) the last step follows from the definitions of the torsion tensor (2.2) and the connection in terms of vielbeins (2.14). Multiplying with minus three-halves and writing out the anti-symmetrized product of vielbeins yields the equation of motion of the spin connection in the known form α e α a Tbλ λ e α b Taλ λ + Tab α = 2e α [a T b]λ λ T ab α Hab α = 0, (2.26) where Hab is an object of twenty-four degrees of freedom since the lower two indices are antisymmetrized. Multiplying this object with e a α gives us (d 2)T λ bλ = 0, but since for d = 2 both actions are zero, this equation is just equivalent to saying that the contraction of the torsion tensor is zero. Plugging this condition into the equation of motion of the spin connection in the known form (2.26) we obtain the condition that torsion must vanish, that is α Tab = 0. (2.27)

20 12 CHAPTER 2. GENERAL RELATIVITY This condition is called the torsion-free condition. The torsion tensor is equivalent to the anti symmetric part of the connection, using again the definition of the connection in terms of the vielbein and the spin connection (2.14). It should be according to the definition of the torsion tensor (2.2). We can also write the equation of motion of the spin connection 2.25 as In the vacuum this equation becomes D β e [α a e β] b + e[α a e β] b T γ βγ = 0. (2.28) ( ) D β e [α a e β] b = 0. When matter is present the equations of motion of the Hilbert-Palatini action become and ɛ abcd ɛ αβγδ e b βfγδ cd = δl δe a = Ta α α ɛ abcd ɛ αβγδ ( D α e b β e c ) δl γ = δωδ ad = M δ [ad], (2.29) where the objects T and M represent conserved currents obtained by the variation of the matter part of the action with respect to the vielbein and the spin connection respectively. In the vacuum Einstein s equations become equivalent to the Ricci tensor being zero. But when matter is present this is not the case. Similarly when matter is present, the equation of motion of the spin connection (2.29) does not reduce in general to the vanishing torsion condition. This theory, as specified by its equations of motion (2.22a, 2.22b), is often called Einstein-Cartan theory. 2.3 Second Order Formalism In this section we will obtain from the two first order equations of motion (2.8a, 2.8b), constraining the metric and the connection, one second order equation in terms of the metric only, by expressing the connection in terms of the metric uniquely. It turns out that this is only possible in the vacuum. The second set of equations of motion of the Einstein-Hilbert action (2.1) are the metric compatibility equations (2.8b), which are also given by λ g µν = 0, (2.30) where the indices of the metric are both low which can be easily derived by operating with a covariant derivative on δ µ ν = g µɛ g ɛν, which is zero. We know that in vacuum the anti-symmetric part of the connection is zero because of the torsion free condition (2.27). Using this condition, we can then use the metric compatibility equations (2.30), in order to solve for the connection in terms of the metric. We can do this by adding three permutations of the metric compatibility equations and solving for the connection, using that it is symmetric in its lower indices. Doing so we obtain Γ ρ µν = 1 2 gρɛ ( µ g ɛν + ν g µɛ ɛ g µν ). (2.31) This connection is sometimes called the Levi-Civita connection, but is also called the Christoffel connection, among other names. Plugging the expression of the Levi-Civita connection (2.31), into the first set of equations of motion of the Einstein-Hilbert action (2.1), we obtain the Einstein s equations (2.8a), which are now second order equations in terms of the metric only. We can write Einstein s equations in their full glory R µν 1 2 g µνr = 8πG c 4 T µν,

21 2.3. SECOND ORDER FORMALISM 13 assuming that the variation of the matter part of the lagrangian is non zero, yielding a non zero momentum energy tensor. As one can see there is no second order differential equation constraining torsion, but only a first order differential equation (2.22b). This implies that Einstein-Cartan theory contains no dynamical torsion degrees of freedom. We must admit that we have reached a dead end in our search for a dynamical theory of torsion Matter According to the theory of general relativity matter sources spacetime curvature. We will consider matter in the form of a scalar field, for educational purposes. We might need some of the formulas stated later in our attempt to generalize the theory of general relativity. The action for a scalar field is given by S φ = d n x [ g 1 ] 2 gµν µ φ ν φ V (φ). (2.32) The equations of motion of this action are φ dv dφ where = g µν µ ν. If we define the energy-momentum tensor to be T µν = 2 g δs M δg µν the energy-momentum tensor in the case of the scalar field φ is given by = 0, (2.33) T µν = µ φ ν φ 1 2 g µνg αβ α φ β φ g µν V (φ) (2.34) and it s trace T = g µν T µν then becomes in an arbitrary number of dimensions D ( ) 2 D T = g µν µ φ ν φ DV (φ). (2.35) 2 Note that the energy conservation equation (2.11), is satisfied for the scalar field using the equation of motion of the scalar field (2.33) µ T µν = When operating on the Einstein equations ( φ dv ) ν φ = 0. dφ R µν 1 2 g µνr = 8πGT µν (2.36) with the metric tensor we obtain the following relationship between the Ricci scalar and the trace of the energy-momentum tensor ( ) 2 R = 8πG T D 2 and Einstein s equations then become [ ( ) ] 1 R µν = 8πG T µν g µν T. (2.37) D 2

22 14 CHAPTER 2. GENERAL RELATIVITY Plugging in the expressions for the energy-momentum tensor and it s trace (2.34, 2.35), we obtain Einstein s equations with the scalar field as source term [ ( ) ] 2 R µν = 8πG µ φ ν φ + g µν V (φ). (2.38) D 2 Alternatively we can express the energy-momentum tensor in terms of the variables ρ and p. We assume the perfect fluid form for the stress energy tensor T µν = (ρ + p)u µ U ν + pg µν, where in the rest frame of the fluid U µ = c( 1, 0, 0, 0). Often the energy-momentum tensor is given, in it s rest frame, in the following diagonal form T µ ν = diag( ρ, p, p, p), (2.39) which is a coordinate invariant expression, where ρ is the density and p the pressure. The trace is then given by 2.4 Cosmology T = ρ + 3p. In this section we derive the Friedmann equations starting from a manifold, which is isotropic and homogenous such that it can model our expanding universe, where the line element is [ ] dr ds 2 = dt 2 + a 2 2 (t) 1 κr 2 + r2 dω 2. (2.40) Setting the spatial curvature constant, κ, to zero we obtain ds 2 = a 2 (τ) [ dτ 2 + d x 2], where dt = adτ. These coordinates are called conformal coordinates. Plugging this metric into the expression of the Christoffel symbol we obtain Γ ρ µν = a ( δ 0 a ν δµ ρ + δµδ 0 ν ρ η ρ0 ) η µν, (2.41) where a = da dτ. Plugging the Christoffel symbol (2.41) into the Riemann tensor we obtain [ a ( ) ] a 2 ( ) (ηµν R µν = a 2δ 0 a νδµ 0 ) a 2 + [2δ 0 a νδµ 0 + 2η µν ] and the Ricci scalar [ R = 6 a a 3 = 6 ä a (ȧ ) ] 2 a when contracting with the inverse metric g µν = 1 a η 2 µν, where ȧ = da dt. Einstein s equations in four dimensions are R µν = 8πG(T µν 1 2 g µνt ). Because of the isotropy and homogeneity symmetries imposed, the Einstein s equations contain only two independent equations due to spatial spherical symmetry, namely one spatial one (ȧ ) 2 ä a + 2 = 4πG(ρ p), (2.42) a

23 2.4. COSMOLOGY 15 since R ii = 2ȧ 2 + aä and and the time component since since and ( 8πG T ii 1 ) 2 g iit = 4πGa 2 (p ρ) ä a = 4 πg(ρ + 3p), 3 (2.43a) R 00 = 3ä a ( 8πG T 00 1 ) 2 g 00T = 4πG(3p + ρ) We can simplify the spatial Einstein s equation (2.42) by plugging in the time component of Einstein s equations (2.43a) in order to obtain (ȧ ) 2 = 8πG ρ, (2.43b) a 3 where the two just obtained independent components of Einstein s equations for our isotropic and homogeneous universe (2.43a, 2.43b) are called the Friedmann equations. The Friedmann equations are also written in terms of the parameter H H = ȧ a, which is the Hubble parameter. It is easy to see that the Friedmann equations are consistent with the energy conservation equation ρ + 3ȧ (ρ + p) = 0, a which is derived from the energy conservation equation µ T µ 0 + Γµ µλ T λ 0 Γ λ µ0t µ λ = 0, where we have used the definitions of the energy-momentum tensor (2.39) and the connection coefficient (2.41). We can express the variables ρ and p in terms of the scalar field φ, by equating the expressions for the energy-momentum tensor in terms of ρ and p and this same tensor in terms of φ (2.39, 2.34), yielding the following expressions for the density and for the pressure consecutively ρ = 1 2 φ a 2 ( φ) 2 + V (φ) (2.44) p = 1 2 φ 2 1 6a 2 ( φ) 2 V (φ). (2.45) For an isotropic and homogenous universe, φ = φ(t), the equation of motion for the scalar field (2.33), becomes φ + 3H φ + dv dφ = 0 (2.46)

24 16 CHAPTER 2. GENERAL RELATIVITY and the Friedmann equations then become and (ȧ ä a = 8 3 πg( φ 2 V ) (2.47) a ) 2 = 4πG 3 ( φ 2 + 2V ). (2.48) In four dimensions we also obtain the following expressions for the density and the pressure ρ = 1 2 φ 2 + V (φ) p = 1 2 φ 2 V (φ), where we have used expressions for the density (2.44) and the pressure (2.45) and the fact that φ = 0. In the next subsections we study the Friedmann equations for different physical circumstances Kination Consider a massless scalar field minimally coupled to gravitation, implying V = 0, in the homogeneous FLRW background. The equation of motion for the scalar field (2.46), then becomes φ + 3H φ = 0 and the second Friedmann equation (2.48), is then given by The equation of motion for the scalar field can then be written as (ȧ a ) 2 = 4πG 3 ( φ 2 ). (2.49) t (a3 φ) = 0. For a(t 0 ) = 1 this equation is solved by φ φ = 0 a. Plugging this into the second Friedmann equation 3 (2.49), we can now solve this equation for the scale factor a in terms of t, namely a 3 = 12πGφ 2 0 (t t0 ) Slow roll inflation During slow roll inflation the potential energy dominates over the kinetic energy, which means V >> φ 2, and we neglect the φ term in the equation of motion of the scalar field. The equation of motion for the scalar field (2.46), then becomes and the second Friedmann equation (2.48), is then given by 3H φ + dv dφ = 0 (2.50) (ȧ a ) 2 = 8πG V. (2.51) 3

25 2.4. COSMOLOGY 17 We are going to solve these equations for two potentials. The first potential we consider is V = 1 2 m2 φ 2, leading to the two equations 3H φ + m 2 φ = 0 and ( ) ȧ 2 a = 4πG 3 m2 φ 2. Solving for φ(t) we obtain The Hubble parameter is then φ = φ 0 ± m 2 12πG (t t 0). H = 4πG 3 mφ 0 + m2 3 (t t 0). The second potential we consider is V = λφ 4. Solving this potential for φ(t) we obtain φ = φ 0 e 2λ 3πG t, leading to the following solution for the Hubble parameter Matter Era H = H 0 e 8λ 3πG t. The era in the universe which is dominated by matter is mathematically modelled by the Friedmann equation (2.48) and the equation of motion of the scalar field (2.46) for a potential V = m2 φ 2 2. These equations can be simplified by introducing a new variable a 3 2 φ = ϕ yielding the following equation The solution of this equation is Radiation Era ϕ + m 2 ϕ = 0. (2.52) ϕ = c 1 cos(mt) + c 2 sin(mt). The era in the universe which is dominated by radiation is mathematically modelled by the Friedmann equation (2.48) and the equation of motion of the scalar field (2.46), but with a different potential, namely V = λφ 4. These equations be simplified by introducing a new variable a 3 2 φ = ϕ yielding the following equation ϕ + λϕ 3 = 0. The solution to this equation are Jacobi elliptic functions.

26 18 CHAPTER 2. GENERAL RELATIVITY

27 Chapter 3 Reciprocity of Momentum and Space 3.1 Motivations In order to unify quantum theory with the principle of general covariance, the following suggestion was made by Max Born in 1938 [6]. The principle of general covariance is intimately related to the principle of relativity; both principles imply that the laws of physics are frame independent. Quantum mechanics, however, obeys a certain symmetry between the position and momentum operators, while the theory of general relativity does not. The aim is now to formulate a new theory which incorporates this symmetry and at the same time satisfies the principle of general covariance (we extend the principle of covariance by considering a larger group, of which the lorentz group is a subgroup, as local symmetry group). The theory of general relativity is the limit of this new theory for large distances compared to energy and momentum quanta, while quantum mechanics is the limiting case for distances small enough such that locally the laws of special relativity are sufficient to describe quantum mechanical processes; the effects of space-time curvature are negligible, when considering atomic processes. From the above arguments it is clear that this new theory should respect the principle of general covariance, but, at the same time, must respect this symmetry between position and momentum. Let us first take a closer look at this symmetry The Reciprocity principle In Quantum mechanics the free particle is represented by a plane wave ( ) i A exp p µx µ. The expression for the plane wave is clearly symmetric in x and p, where the coordinates (x 0, x 1, x 2, x 3 ) and (p 0, p 1, p 2, p 3 ) are being used to describe the space-time (ct, x, y, z) and momentum-energy coordinates (E, p x, p y, p z ) respectively. The laws of quantum mechanics extend this symmetry through its transformation laws; one can Fourier transform any wave equation in x space to another one in p space. The representation of the position and momentum variables by the corresponding operators in Hilbert space are as follows: the operator p µ is given by i x whenever the x µ µ operator is diagonal and whenever the p µ operator is diagonal the x µ operator is given by i p. µ Max Born calls this symmetry principle the principle of reciprocity [9]. One can interpret this in the following way: the laws of physics are invariant under the reciprocity transformation x µ p µ p µ x µ. (3.1)

28 20 CHAPTER 3. RECIPROCITY OF MOMENTUM AND SPACE There are a number of indications that could strengthen our belief in the principle of reciprocity 1 ; the canonical equations of classical mechanics x µ = H p µ p µ = H x µ are invariant 2 under the reciprocity transformation (3.1), if one considers only the last three components of the position and momentum four vectors. The commutation relations from quantum mechanics and the components of angular momentum x µ p ν p ν x µ = i δ µ ν x µ p ν p µ x ν = M µν are invariant under the reciprocity transformation (3.1) for all four components of the position and momentum four vectors. It is suggestive that a theory unifying quantum theory and the theory of general relativity should respect the principle of reciprocity General Relativity and the Reciprocity Principle The theory of general relativity describes our universe at large scales (we are not considering cosmological issues as dark matter and dark energy at the moment) and it generalizes classical mechanical ideas as orbits, instead of wave functions, in order to describe particles. The four dimensional line element ds 2 = g µν dx µ dx ν (3.2) is a fundamental notion in the theory of general relativity. It is clear that the theory of general relativity and the way distances are determined (3.2) breaks the reciprocity symmetry (3.1). Demanding that the theory, unifying quantum theory and the theory of general relativity, should respect the principle of reciprocity, we can state a four dimensional momentum-energy line element dσ 2 = γ µν dp µ dp ν, (3.3) which should dominate over the space-time line element (3.2) whenever the momenta are very large compared to this position length scale. According to the classical laws the momentum p µ is given by mx µ, which corresponds to the tangent vector of the path taken. The idea of having a tangent space at each point of the manifold, corresponding to the physical idea of the momentum as tangent vector, is clearly only applicable in the classical roam of physics, when the momenta are small compared to the distances. For the sake of brevity, Max Born called this scale, at which the theory of general relativity is valid, the molar world [6], while he called the small world, which is described by the momentum energy line element (3.3) the nuclear world. The world, which lies in between these worlds on to the energy-momentum and space-time scales, is familiarly called the quantum world. Since theory of general relativity is governed by Einstein s equations R µν 1 2 g µνr Λg µν = κt µν, we can state via the principle of reciprocity the reciprocal Einstein equations P µν 1 2 γµν P Λ γ µν = κ T µν, 1 The name reciprocity stems from lattice theory of crystals, where the motion of the particle in momentum space is described by a reciprocal lattice. 2 The canonical equations of classical mechanics are only invariant under the reciprocity transformation for H(x, p) = H(p, x),

29 3.2. RECIPROCAL RELATIVITY 21 which are supposed to govern the momentum-energy curvature of the nuclear world. We now have a vague sketch in our minds of how the theory should behave in certain limits and how it should obey the principles of reciprocity and general covariance. Our goal is now to write down a theory which obeys all these limits and principles. The principle of general covariance suggests that there should exist a space-time-momentum-energy line element that specifies a corresponding space-time-momentum-energy interval, which is absolute in the sense that all observers would agree on it; interpreting the momentum-energy coordinates as coordinates, specifying non inertial frames, all relatively non inertial moving observers would agree on the measured space-time-momentum-energy interval. Space-time then becomes a relative space with respect to observers moving non inertially with respect to each other and becomes absolute only in the limit of relatively inertial moving observers. We will consider now an example of a space-time-momentum-energy line element and study some of its transformation properties. 3.2 Reciprocal Relativity We are now ready to study the non inertial transformations on the time-space-momentum-energy space (t, q, p, e) -note that we have suppressed two space and two momentum variables - and two metrics, namely the Born-Green metric ds 2 = dt c 2 dq2 + 1 ( b 2 dp 2 1 ) c 2 de2 (3.4) and the symplectic metric ζ = de dt + dp dq, (3.5) which are invariant under these non-inertial transformations. In doing so we are following mainly the article by Stephen Low [10] Non-relativistic Inertial Transformations The non-relativistic limit obtained by letting v c 0 or equivalently by letting the speed of light go to infinity c. In this limit the coordinate transformations are given by the canonical non-relativistic Galilean transformations of Hamiltonian mechanics dt = dt dq = dq + vdt dp = dp de = de + vdp, leaving the symplectic metric (3.5) and the two metrics ds 2 = dt 2 dµ 2 = dp 2 (3.6) invariant; under these non-relativistic inertial transformations, time and momentum are invariant subspaces of the space-time-momentum-energy space. They are absolute in the sense that any non-relativistic inertial moving observers agree on theses observables. For the sake of simplicity we have taken the space and momentum coordinates to be one dimensional. The group leaving these metrics invariant is the contraction of the Lorentz group, that is the Euclidean group lim SO(1, 1) = E(1) T (1). c The group elements of the one-dimensional Euclidean group are the group elements of the Euclidean group and can hence be represented by the matrix φ(v) = lim Γ(v) = v c ) v 1