arXiv:1106.2037v1 [gr-qc] 10 Jun 2011 Contents space curved of theory field vierbein Einstein’s II eaiitccia atri uvdspace curved in matter chiral Relativistic VIII. I.Enti’ action Einstein’s VII. I.Connections III. V Curvature IV. I rvttoa action Gravitational VI. X Conclusion IX. I ahmtclframework Mathematical II. .Mteaia constructs Mathematical V. .Acknowledgements X. .Introduction I. .Rcitensor Ricci D. .Fl rvttoa cin15 action gravitational Full D. .Fedeuto ntewa edlmt16 limit field weak the in equation Field D. .Smlrt oYn-il ag hoy3 theory gauge Yang-Mills to Similarity A. .Reancrauefo h ffiecneto 8 connection affine the from curvature Riemann A. connection Affine A. basis Local A. .Cneuneo erdpsuae11 postulate tetrad of Consequence A. .Fe edgaiainlato 12 action gravitational field Free A. .Lgaga est om115 1 form density Lagrangian A. .Ivrac nfltsae17 space flat in Invariance A. .Tta postulate Tetrad C. .Ivratvlm lmn 12 element volume Invariant C. .Ato o rvttoa ore14 source gravitational a for Action C. .Frtodrflcuto ntemti esr16 tensor metric the in fluctuation First-order C. .Cvratdrvtv fasio ed18 field spinor a of derivative Covariant C. .Si connection Spin B. field Vierbein B. .Reancrauefo h pncneto 10 connection spin the from curvature Riemann B. .An oncini em ftemti esr12 tensor metric the of terms in connection Affine B. .Vrainwt epc otevebi ed14 field vierbein the to respect with Variation B. .Lgaga est om216 2 form density Lagrangian B. .Ivrac ncre pc 17 space curved in Invariance B. eiaino h h iseneuto n hnteDrceq incl Dirac are the English then into and translated the equation manuscripts Using 1928 Einstein original the space. is curved the theory in of field derivation theory vierbein curved field the in quantum convent Thus, transported relativistic the repre field derivative. of vierbein quantum covariant place the 4-spinor the of the a applications take to important Ein to correction most fields theory. the Yang-Mills vierbein of a the one strictly of were potential terms gravity vector is in if the derivative) be covariant like g would (the exactly for it derivative not theory the but the field to field as gauge correction taken gauge a field a is vierbein of theory the role on vierbein vierbe based Einstein’s the is field. It following Einstein. reviewed by is 1928 theory Relativity General 2008) 20, 01 January MA (Dated: Base, Force Air Hanscom Laboratory, Research Force Air Yepez Jeffrey ewrs iren eea eaiiy rvttoa ga gravitational relativity, general vierbein, Keywords: 11 12 12 15 17 20 20 2 4 8 7 7 6 6 5 4 g hoy ia qaini uvdspace curved in equation Dirac theory, uge o rprinlt h irenfil as field vierbein the to proportional not uded. nfil hoyapoc rpsdin proposed approach theory field in h otntrlwyt ersn a represent to way natural most the aiy h irenfil lyn the playing field vierbein the ravity; etto sfrtedrvto fthe of derivation the for is sentation ti icvrdtesi connection spin the discovered stein Appendices irenfil hoy rsne sa is presented theory, field vierbein edde nYn-il theory–the Yang-Mills in does field pc,yedn h orc omof form correct the yielding space, aini uvdsae Einstein’s space. curved in uation sur ot ftemti tensor metric the of root” “square I.Ivrat n covariants and Invariants III. oa ffiecneto.T date, To connection. affine ional I itn aalls n oainlinvariance rotational and parallelism Distant II. I h edeuto ntefis approximation first the in equation field The II. I. .Teudryn edequation field underlying The I. 731 isens12 aucito itn parallelism distant on manuscript 1928 Einstein’s References n isens12 aucito nfiainof electromagnetism unification and on gravity manuscript 1928 Einstein’s Bi n metric and -Bein 21 21 20 1 1 1 4 2 1 I INTRODUCTION

I. INTRODUCTION the publisher’s typist, it can cause confusion.1 As far as I know at the time of this writing in 2008, the two The purpose of this manuscript is to provide a self- 1928 manuscripts have never been translated into En- contained review of the procedure for deriving the Ein- glish. English versions of these manuscripts are provided stein equations for gravity and the Dirac equation in as part of this review–see the Appendix–and are included curved space using the vierbein field theory. This gauge to contribute to its completeness. field theory approach to General Relativity (GR) was In the beginning of the year 1928, Dirac introduced discovered by Einstein in 1928 in his pursuit of a unified his famous square root of the Klein-Gordon equation, es- field theory of gravity and electricity. He originally pub- tablishing the starting point for the story of relativistic lished this approach in two successive letters appearing quantum field theory, in his paper on the quantum theory one week apart (Einstein, 1928a,b). The first manuscript, of the electron (Dirac, 1928). This groundbreaking pa- a seminal contribution to mathematical physics, adds per by Dirac may have inspired Einstein, who completed the concept of distant parallelism to Riemann’s theory his manuscripts a half year later in the summer of 1928. of curved manifolds that is based on comparison of dis- With deep insight, Einstein introduced the vierbein field, tant vector magnitudes, which before Einstein did not which constitutes the square root of the metric tensor incorporate comparison of distant directions. field.2 Einstein and Dirac’s square root theories math- Historically there appears to have been a lack of in- ematically fit well together; they even become joined at terest in Einstein’s research following his discovery of the hip when one considers the dynamical behavior of general relativity, principally from the late 1920’s on- chiral matter in curved space. ward. Initial enthusiasm for Einstein’s unification ap- Einstein’s second manuscript represents a simple and proach turned into a general rejection. In Born’s July intuitive attempt to unify gravity and electromagnetism. 15th, 1925 letter (the most important in the collection) to He originally developed the vierbein field theory ap- Einstein following the appearance of his student Heisen- proach with the goal of unifying gravity and quantum berg’s revolutionary paper on the matrix representation theory, a goal which he never fully attained with this of quantum mechanics, Born writes (Born, 1961): representation. Nevertheless, the vierbein field theory approach represents progress in the right direction. Ein- Einstein’s field theory . . . was intended to stein’s unification of gravity and electromagnetism, us- unify electrodynamics and gravitation . . . . I ing only fields in four-dimensional spacetime, is concep- think that my enthusiam about the success tually much simpler than the well known Kaluza-Klein of Einstein’s idea was quite genuine. In those approach to unification that posits an extra compactified days we all thought that his objective, which spatial dimension. But historically it was the Kaluza- he pursued right to the end of his life, was Klein notion of extra dimensions that gained popularity attainable and also very important. Many of as it was generalized to string theory. In contradistinc- us became more doubtful when other types of tion, Einstein’s approach requires no extra constructs, fields emerged in physics, in addition to these; just the intuitive notion of distant parallelism. In the the first was Yukawa’s meson field, which is Einstein vierbein field formulation of the connection and a direct generalization of the electromagnetic curvature, the basis vectors in the tangent space of a field and describes nuclear forces, and then spacetime manifold are not derived from any coordinate there were the fields which belong to the other system of that manifold. elementary particles. After that we were in- Although Einstein is considered one of the founding clined to regard Einstein’s ceaseless efforts as fathers of quantum mechanics, he is not presently consid- a tragic error. ered one of the founding fathers of relativistic quantum field theory in flat space. This is understandable since in Weyl and Pauli’s rejection of Einstein’s thesis of distant his theoretical attempts to discover a unified field theory parallelism also helped paved the way for the view that he did not predict any of the new leptons or quarks, nor Einstein’s findings had gone awry. Furthermore, as the their weak or strong gauge interactions, in the Standard belief in the fundamental correctness of quantum theory Model of particle physics that emerged some two decades solidified by burgeoning experimental verifications, the theoretical physics community seemed more inclined to latch onto Einstein’s purported repudiation of quantum mechanics: he failed to grasp the most important direc- 1 The opening equation (1a) was originally typeset as tion of twentieth century physics. H = hgµν , Λ α, Λ β , ··· Einstein announced his goal of achieving a unified field µβ ν α theory before he published firm results. It is already hard offered as the Hamiltonian whose variation at the end of the day not to look askance at an audacious unification agenda, yields the Einstein and Maxwell’s equations. I have corrected this in the translated manuscript in the appendix. but it did not help when the published version of the 2 The culmination of Einstein’s new field theory approach ap- manuscript had a fundamental error in its opening equa- peared in Physical Review in 1948 (Einstein, 1948), entitled “A tion; even though this error was perhaps introduced by Generalized Theory of Gravitation.”

2 I INTRODUCTION following his passing. However, Einstein did employ local and Carroll (Carroll, 2004).5 6 However, Weinberg and rotational invariance as the gauge symmetry in the first Carroll review vierbein theory as a sideline to their main 1928 manuscript and discovered what today we call the approach to GR, which is the standard coordinate-based spin connection, the gravitational gauge field associated approach of differential geometry. An excellent treat- with the Lorentz group as the local symmetry group (viz. ment of quantum field theory in curved space is given by local rotations and boosts). Birrell and Davies (Birrell and Davies, 1982), but again This he accomplished about three decades before with a very brief description of the vierbein representa- Yang and Mills discovered nonabelian gauge the- tion of GR. Therefore, it is hoped that a self-contained ory (Yang and Mills, 1954), the antecedent to the review of Einstein’s vierbein theory and the associated Glashow-Salam-Weinberg electroweak unification theory formulation of the relativistic wave equation should be (Glashow, 1961; Salam, 1966; Weinberg, 1967) that is helpful gathered together in one place. the cornerstone of the Standard Model. Had Einstein’s work toward unification been more widely circulated in- stead of rejected, perhaps Einstein’s original discovery of A. Similarity to Yang-Mills gauge theory n-component gauge field theory would be broadly consid- 3 ered the forefather of Yang-Mills theory. In Section I.A, This section is meant to be an outline comparing I sketch a few of the strikingly similarities between the the structure of GR and Yang-Mills (YM) theories vierbein field representation of gravity and the Yang- (Yang and Mills, 1954). There are many previous treat- Mills nonabelian gauge theory. ments of this comparison—a recent treatment by Jackiw With the hindsight of 80 years of theoretical physics is recommended (Jackiw, 2005). The actual formal re- development from the time of the first appearance of Ein- view of the vierbein theory does not begin until Sec- stein’s 1928 manuscripts, today one can see the historical tion II. rejection is a mistake. Einstein could rightly be consid- The dynamics of the metric tensor field in GR can ered one of the founding fathers of relativistic quantum be cast in the form of a YM gauge theory used to de- field theory in curved space, and these 1928 manuscripts scribe the dynamics of the quantum field in the Standard should not be forgotten. Previous attempts have been Model. In GR, dynamics is invariant under an external made to revive interest in the vierbein theory of gravita- local transformation, say Λ, of the Lorentz group SO(3,1) tion purely on the grounds of their superior use for repre- that includes rotations and boosts. Furthermore, any senting GR, regardless of unification (Kaempffer, 1968). quantum dynamics occurring within the spacetime man- Yet, this does not go far enough. One should also make ifold is invariant under internal local Lorentz transforma- the case for the requisite application to quantum fields tions, say UΛ, of the spinor representation of the SU(4) in curved space. group. Explicitly, the internal Lorentz transformation of Einstein is famous for (inadvertently) establishing an- a quantum spinor field in unitary form is other field of contemporary physics with the discovery i µν ωµν (x)S of distant quantum entanglement. Nascent quantum in- UΛ = e− 2 , (1) formation theory was borne from the seminal 1935 Ein- stein, Podolsky, and Rosen (EPR) Physical Review pa- where Sµν is the tensor generator of the transformation.7 per4, “Can Quantum-Mechanical Description of Physical Similarly, in the Standard Model, the dynamics of the Reality Be Considered Complete?” Can Einstein equally Dirac particles (leptons and quarks) is invariant under be credited for establishing the field of quantum gravity, local transformations of the internal gauge group, SU(3) posthumously? for color dynamics and SU(2) for electroweak dynamics.8 The concepts of vierbein field theory are simple, and The internal unitary transformation of the multiple com- the mathematical development is straightforward, but in the literature one can find the vierbein theory a nota- tional nightmare, making this pathway to GR appear more difficult than it really is, and hence less accessible. 5 Both Weinberg and Carroll treat GR in a traditional manner. In this manuscript, I hope to offer an accessible path- Their respective explanations of Einstein’s vierbein field theory are basically incidental. Carroll relegates his treatment to a sin- way to GR and the Dirac equation in curved space. The gle appendix. development of the vierbein field theory presented here 6 Another introduction to GR but which does not deal with borrows first from treatments given by Einstein himself the vierbein theory extensively is the treatment by D’Inverno that I have discussed above (Einstein, 1928a,b), as well (D’Inverno, 1995). 7 2 as excellent treatments by Weinberg (Weinberg, 1972) A 4 × 4 fundamental representation of SU(4) are the 4 − 1 = 15 Dirac matrices, which includes four vectors γµ, six tensors µν i µ ν 0 1 2 3 5 σ = 2 [γ , γ ], one pseudo scalar iγ γ γ γ ≡ γ , and four axial vectors γ5γµ. The generator associated with the internal µν 1 µν Lorentz transformation is S = 2 σ . 3 Einstein treated the general case of an n-Bein field. 8 A 3 × 3 fundamental representation of SU(3) are the 32 − 1 = 8 4 This is the most cited physics paper ever and thus a singular ex- Gell-Mann matrices. And, a 2 × 2 fundamental representation of ception to the general lack of interest in Einstein’s late research. SU(2) are the 22 − 1 = 3 Pauli matrices.

3 A Similarity to Yang-Mills gauge theory II MATHEMATICAL FRAMEWORK ponent YM field in unitary form is This is derived in Section VIII. A problem that is com- iΘa(x)T a monly cited regarding the gauge theory representation of U = e , (2) General Relativity is that the correction is not directly a a where the hermitian generators T = T † are in the ad- proportional to the gauge potential as would be the case joint representation of the gauge group. The common if it were strictly a YM theory (e.g. we should be able a a unitarity of (1) and (2) naturally leads to many parallels to write Γµ(x) = u kµa(x) where u is some constant between the GR and YM theories. four-vector). From (1), it seems that ωµν (x) should take the place of the gauge potential field, and in this case it is clearly a second rank field quantity. But in Einstein’s action in II. MATHEMATICAL FRAMEWORK his vierbein field representation of GR, the lowest order A. Local basis fluctuation of a second-rank vierbein field itself µ µ µ e a(x)= δa + k a(x)+ The conventional coordinate-based approach to GR ··· uses a “natural” differential basis for the tangent space plays the roles of the potential field where the gravita- µν Tp at a point p given by the partial derivatives of the tional field strength F a is related to a quantity of the coordinates at p form e µν µ ν ν µ ˆ(µ) = ∂(µ). (7) F a = ∂ k a(x) ∂ k a(x). (3) − Some 4-vector A Tp has components (3) vanishes for apparent or pseudo-gravitational fields ∈ µe that occur in rotating or accelerating local inertial frames A = A ˆ(µ) = (A0, A1, A2, A3). (8) but does not vanish for gravitational fields associated To help reinforce the construction of the frame, we use with curved space. Einstein’s expression for the La- a triply redundant notation of using a bold face symbol grangian density (that he presented as a footnote in his to denote a basis vector e, applying a caret symbol eˆ second manuscript) gives rise to a field strength of the as a hat to denote a unit basis vector, and enclosing the form of (3). An expanded version of his proof of the component subscript with parentheses eˆ µ to denote a equivalence of his vierbein-based action principle with ( ) component of a basis vector.9 It should be nearly im- gravity in the weak field limit is presented in Section VII. possible to confuse a component of an orthonormal basis In any case, a correction to the usual derivative vector in Tp with a component of any other type of ob- ∂µ µ ∂µ +Γµ ject. Also, the use of a Greek index, such as µ, denotes a → D ≡ component in a coordinate system representation. Fur- is necessary in the presence of a gravitational field. The thermore, the choice of writing the component index as local transformation has the form a superscript as in Aµ is the usual convention for indi- cating this is a component of a contravariant vector. A ψ UΛψ (4a) → contravariant vector is often just called a vector, for sim- Γµ U ΓµU † (∂µU ) U † . (4b) → Λ Λ − Λ Λ plicity of terminology. (4) is derived in Section VIII.B. This is similar to Yang- In the natural differential basis, the cotangent space, Mills gauge theory where a correction to the usual deriva- here denoted by Tp∗, is spanned by the differential ele- tive ments eˆ(µ) = dx(µ), (9) ∂µ µ ∂µ iAµ → D ≡ − is also necessary in the presence of a non-vanishing gauge which lie in the direction of the gradient of the coordinate 10 T T field. In YM, the gauge transformation has the form functions. p∗ is also called the dual space of p. Some ψ U ψ (5a) → Aµ UAµU † U∂µU †, (5b) 9 → − With eˆ(•) ∈ {e0,e1,e2,e3} and ∂(•) ∈ {∂0, ∂1, ∂2, ∂3}, some au- which is just like (4). Hence, this “gauge field theory” ap- thors write (7) concisely as proach to GR is useful in deriving the form of the Dirac eµ = ∂µ. equation in curved space. In this context, the require- I will not use this notation because I would like to reserve eµ ment of invariance of the relativistic quantum wave equa- a to represent the lattice vectors eµ ≡ γaeµ where γa are Dirac a tion to local Lorentz transformations leads to a correction matrices and eµ is the vierbein field defined below. • of the form 10 Again, with eˆ(•) ∈ {e0,e1,e2,e3} and dx( ) ∈ {dx0, dx1, dx2, dx3}, for brevity some authors write (9) as 1 β hk Γµ = e k (∂µeβh) S (6a) 2 eµ = dxµ. 1 hβ But I reserve eµ to represent anti-commuting 4-vectors, eµ ≡ = ∂µ kβhS + (6b) a µ 2 ··· γ e a, unless otherwise noted.  

4 B Vierbein field II MATHEMATICAL FRAMEWORK

dual 4-vector A Tp∗ has components the coordinate basis (whose value depends on the local ∈ curvature at a point x in the manifold) in terms of the (µ) ν (µ) A = Aµeˆ = gµν A e . (10) tetrads as the following linear combination

a Writing the component index µ as a subscript in Aµ again eˆ(µ)(x)= eµ (x) eˆ(a), (13) follows the usual convention for indicating one is dealing a with a component of a covariant vector. Again, in an at- where the functional components eµ (x) form a 4 4 × tempt to simplify terminology, a covariant vector is often invertible matrix. We will try not to blur the distinction called a 1-form, or simply a dual vector. Yet, remem- between a vector and its components. The term “vierbein bering that a vector and dual vector (1-form) refer to an field” is used to refer to the whole transformation matrix a element of the tangent space Tp and the cotangent space in (13) with 16 components, denoted by eµ (x). The e a x a , , , Tp∗, respectively, may not seem all that much easier than vierbeins µ ( ), for = 1 2 3 4, comprise four legs– remembering the prefixes contravariant and covariant in vierbein in German means four-legs. µ the first place. The inverse of the vierbein has components, e a e 1 (switched indices), that satisfy the orthonormality con- The dimension of (7) is inverse length, [ˆ(µ)]= L , and this is easy to remember because a first derivative of a ditions

function is always tangent to that function. For a basis µ a µ a µ a element, µ is a subscript when L is in the denomina- e a(x)eν (x)= δν , eµ (x)e b(x)= δb . (14) tor. Then (9), which lives in the cotangent space and as The inverse vierbein serves as a transformation matrix the dimensional inverse of (7), must have dimensions of e e(µ) that allows one to represent the tetrad basis ˆ(a)(x) in length [ˆ ] = L. So, for a dual basis element, µ is a e superscript when L is in the numerator. That they are terms of the coordinate basis ˆ(µ): dimensional inverses is expressed in the following tensor eˆ = eµ (x) eˆ . (15) product space (a) a (µ)

(µ) µ Employing the metric tensor gµν to induce the product eˆ eˆ = 1 ν , (11) ⊗ (ν) of the vierbein field and inverse vierbein field, the inner where 1 is the identity, which is of course dimensionless. product-signature constraint is We are free to choose any orthonormal basis we like to µ ν gµν (x)e a(x) e b(x)= ηab, (16) span Tp, so long as it has the appropriate signature of the manifold on which we are working. To that end, we or using (14) equivalently we have introduce a set of basis vectors eˆ(a), which we choose as non-coordinate unit vectors, and we denote this choice by a b gµν (x)= eµ (x)eν (x)ηab. (17) using small Latin letters for indices of the non-coordinate frame. With this understanding, the inner product may So, the vierbein field is the “square root” of the metric. be expressed as Hopefully, you can already see why one should include the vierbein field theory as a member of our tribe of e e ˆ(a), ˆ(b) = ηab, (12) “square root” theories. These include the Pythagorean µ ν theorem for the distance interval ds = ηµν dx dx = where ηab = diag(1, 1, 1,
1) is the Minkowski metric − − − dt2 dx2 dy2 dz2 of flat spacetime. , the mathematicians’p beloved complex− analysis− (based− on √ 1 as the imaginary num- pber), quantum mechanics (e.g.− pathways are assigned B. Vierbein field amplitudes which are the square root of probabilities), quantum field theory (e.g. Dirac equation as the square This orthonormal basis that is independent of the co- root of the Klein Gordon equation), and quantum compu- ordinates is termed a tetrad basis.11 Although we can- tation based on the universal √swap conservative quan- not find a coordinate chart that covers the entire curved tum logic gate. To this august list we add the vierbein manifold, we can choose a fixed orthonormal basis that as the square root of the metric tensor. is independent of position. Then, from a local perspec- Now, we may form a dual orthonormal basis, which a tive, any vector can be expressed as a linear combination we denote by eˆ( ) with a Latin superscript, of 1-forms in of the fixed tetrad basis vectors at that point. Denoting the cotangent space Tp∗ that satisfies the tensor product an element of the tetrad basis by eˆ(a), we can express condition (a) a eˆ eˆ = 1 b. (18) ⊗ (b)

11 To help avoid confusion, please note that the term tetrad in the This non-coordinate basis 1-form can be expressed as a literature is often used as a synonym for the term vierbein. Here linear combination of coordinate basis 1-forms µ we use the terms to mean two distinct objects: eˆ a and e (x), ( ) (a) a (µ) respectively. eˆ = eµ (x) eˆ (x), (19)

5 B Vierbein field III CONNECTIONS where eˆ(µ) = dxµ, and vice versa using the inverse vier- III. CONNECTIONS bein field A. Affine connection (µ) µ (a) eˆ (x)= e a(x) eˆ . (20) Curvature of a Riemann manifold will cause a distor- Any vector at a spacetime point has components in the tion in a vector field, say a coordinate field Xα(x), and coordinate and non-coordinate orthonormal basis this is depicted in Figure 1. The change in the coordinate µ a V = V eˆ(µ) = V eˆ(a). (21) So, its components are related by the vierbein field trans- formation α β γ ❍ Γβγ X δX a a µ µ µ a ✻ ✻❍ V = eµ V and V = e aV . (22) ❍❥ ✁✕ The vierbeins allow us to switch back and forth between α α ¯ α ✁ X (x) X + δX ✁ Latin and Greek bases. ✁Xα + δXα Multi-index tensors can be cast in terms of mixed- ✁ index components, as for example r ✁r i j a a µ ν a a ν µ δxα V b = eµ V b = e bV ν = eµ e bV ν . (23) FIG. 1 Two spacetime points xα and xα + δxα, labeled as i and The behavior of inverse vierbeins is consistent with the j, respectively. The 4-vector at point i is Xα(x), and the 4-vector α α α conventional notion of raising and lowering indices. Here at nearby point j is X (x + δx) = X (x)+ δX (x). The parallel transported 4-vector at j is Xα(x) + δX¯ α(x) (blue). The affine is an example with the metric tensor field and the α connection is Γβγ . (For simplicity the parallel transport is rendered Minkowski metric tensor as if the space is flat). µ µν b e a = g ηab eν . (24) field from one point x to an adjacent point x + δx is The identity map has the form α α β α X (x + δx)= X (x)+ δx ∂βX . (29) a (ν) e = eν dx eˆ . (25) δXα(x) ⊗ (a) a So, the change of the coordinate field| due{z to} the manifold We can interpret eν as a set of four Lorentz 4-vectors. That is, there exists one 4-vector for each non-coordinate is defined as α β α α α index a. δX (x) δx (x)∂βX = X (x + δx) X (x). (30) We can make local Lorentz transformations (LLT) at ≡ − j any point. The signature of the Minkowski metric is pre- The difference of the two coordinate vectors at point is served by a Lorentz transformation [Xα + δXα] [Xα + δX¯ α]= δXα(x) δX¯ α(x). (31) − − a δX¯ α must vanish if either δxα vanishes or Xα vanishes. LLT: eˆ(a) eˆ(a′) =Λ a′ (x)eˆ(a), (26) → Therefore, we choose a where Λ ′ (x) is an inhomogeneous (i.e. position depen- a δX¯ α = Γα (x)Xβ(x)δxγ , (32) dent) transformation that satisfies − βγ where Γα is a multiplicative factor, called the affine con- a b βγ Λ a′ Λ b′ ηab = ηa′b′ . (27) nection. Its properties are yet to be determined. At this stage, we understand it as a way to account for the cur- A Lorentz transformation can also operate on basis 1- vature of the manifold. forms, in contradistinction to the ordinary Lorenz trans- a′ a′ The covariant derivative may be constructed as follows: formation Λ a that operates on basis vectors. Λ a a 1 transforms upper (contravariant) indices, while Λ a′ α α α α γ X (x) X (x+δx) [X (x)+δX¯ (x)] . (33) transforms lower (covariant) indices. ∇ ≡ δxγ { − } And, we can make general coordinate transformations I do not use a limit in the definition to define the deriva- (GCT) tive. Instead, I would like to just consider the situation where δxγ is a small finite quantity. We will see below µ′ ν aµ a′µ′ a′ ∂x b ∂x aµ that this quantity drops out, justifying the form of (33). GCT: T bν T b′ν′ = Λ a Λ b′ T bν . → ∂xµ ∂xν′ Inserting (29) and (32) into (33), yields

prime prime α 1 α γ α 1st 2nd γ X (x) = X (x)+ δx ∂γ X (34a) |{z} |{z} ∇ δxγ { (contra- (co- Xα(x)+Γα (x)Xβ(x)δxγ variant) variant) − βγ } α α β (28) = ∂γ X (x)+Γβγ(x)X (x). (34b)

6 B Spin connection III CONNECTIONS

So we see that δxγ cancels out and no limiting process C. Tetrad postulate to an infinitesimal size was really needed. Dropping the explicit dependence on x, as this is to be understood, we The tetrad postulate is that the covariant derivative of a have the simple expression for the covariant derivative the vierbein field vanishes, µeν = 0, and this is merely a restatement of the relation∇ we just found between the α α α β b γ X = ∂γ X +ΓβγX . (35) e ∇ affine and spin connections (41). Left multiplying by ν gives In coordinate-based differential geometry, the covariant derivative of a tensor is given by its partial derivative a b a λ b σ λ b a ωµ beν = eσ e beν Γµλ e beν ∂µeλ (42a) plus correction terms, one for each index, involving an a σ −a = eσ Γ ∂µeν . (42b) affine connection contracted with the tensor. µν − Rearranging terms, we have the tetrad postulate

a a a σ a b B. Spin connection µeν ∂µeν eσ Γ + ωµ eν =0. (43) ∇ ≡ − µν b In non-coordinate-based differential geometry, the or- Let us restate (as a reminder) the correction rules for λ dinary affine connection coefficients Γµν are replaced by applying connections. The covariant derivatives of a co- a spin connection coefficients, denoted ωµ b, but otherwise ordinate vector and 1-form are the principle is the same. Each Latin index gets a cor- ν ν ν λ µX = ∂µX +Γ X (44a) rection factor that is the spin connection contracted with ∇ µλ λ the tensor, for example µXν = ∂µXν Γ Xλ, (44b) ∇ − µν a a a c c a µX b = ∂µX b + ωµ X b ωµ X c. (36) and similarly the covariant derivatives of a non- ∇ c − b coordinate vector and 1-form are The correction is positive for a upper index and negative a a a b for a lower index. The spin connection is used to take µX = ∂µX + ωµ X (45a) ∇ b covariant derivatives of spinors, whence its name. b µXa = ∂µXa ωµ Xb. (45b) The covariant derivative of a vector X in the coordi- ∇ − a nate basis is We require a covariant derivative such as (45a) to be ν µ Lorentz invariant X = ( µX ) dx ∂ν (37a) ∇ ∇ ⊗ ν ν λ µ a′ a a′ a = ∂µX +Γ X dx ∂ν . (37b) Λ a : µX µ Λ aX (46a) µλ ⊗ ∇ → ∇
 a′ a a′ a The same object in a mixed basis, converted to the co- = µΛ a X +Λ a µX .(46b) ordinate basis, is ∇ ∇   a µ Therefore, the covariant derivative is Lorentz invariant, X =( µX ) dx eˆ(a) (38a) ∇ ∇ ⊗ ′ a a b µ a a a = ∂µX + ωµ X dx eˆ (38b) µX = Λ a µX , (47) b ⊗ (a) ∇ ∇ a ν a b λ µ σ = ∂µ (eν X )+ ωµ eλ X dx (e a∂σ) (38c)
b ⊗ so long as the covariant derivative of the Lorentz trans- σ a ν ν a a b λ µ = e a eν ∂µX + X ∂µeν + ωµ eλ X dx ∂σ formation vanishes,
b ⊗ (38d) a′
µΛ a =0. (48) σ σ a ν σ b a λ µ = ∂µX + e a∂µeν X + e aeλ ωµ X dx ∂σ. ∇ b ⊗ (38e) This imposes a constraint that allows us to see how the
spin connection behaves under a Lorentz transformation Now, relabeling indices σ ν λ gives → → a′ a′ a′ c c a′ µΛ b = ∂µΛ b + ωµ cΛ b ωµ bΛ c =0, (49) ν ν a λ ν b a λ µ ∇ − X = ∂µX + e a∂µeλ X + e aeλ ωµ X dx ∂ν ∇ b ⊗ ν ν a ν b a λ µ which we write as follows = ∂µX + e a∂µeλ + e aeλ ωµ b X
dx ∂ν . ⊗ b a′ a′ b c c b a′ (39) Λ b′ ∂µΛ b + ωµ Λ b′ Λ b ωµ Λ b′ Λ c =0. (50) 
 c − b b c c Therefore, comparing (37b) with (39), the affine connec- Now Λ b′ Λ b = δb′ , so we arrive at the transformation of tion in terms of the spin connection is the spin connection induced by a Lorentz transformation

ν ν a ν b a a′ c b a′ b a′ Γµλ = e a∂µeλ + e aeλ ωµ . (40) ωµ ′ = ωµ Λ b′ Λ c Λ b′ ∂µΛ b. (51) b b b − This can be solved for the spin connection This means that the spin connection transforms inhomo- a a a λ ν λ a geneously so that µX can transform like a Lorentz ωµ = eν e bΓ e b∂µeλ . (41) ∇ b µλ − 4-vector.

7 IV CURVATURE

The exterior derivative is defined as follows the torsion then reduces to the simple expression

a a a λ λ λ (dX)µν µXν ν Xµ (52a) T =Γ Γ . (61) ≡ ∇ − ∇ µν µν νµ a a b λ a − = ∂µXν + ωµ Xν Γ Xλ b − µν So, the torsion vanishes when the affine connection is a a b λ a ∂ν Xµ ων bXµ +Γ Xλ (52b) symmetric in its lower two indices. − − νµ a a a b a b = ∂µXν ∂ν Xµ + ωµ Xν ων Xµ . − b − b (52c) IV. CURVATURE Now, to make a remark about Cartan’s notation as writ- ten in (10), one often writes the non-coordinate basis We now derive the Riemann curvature tensor, and we 1-form (19) as do so in two ways. The first way gives us an expres- sion for the curvature in terms of the affine connection a (a) a µ and the second way gives us an equivalent expression in e eˆ = eµ dx . (53) ≡ terms of the spin connection. The structure of both ex- The spin connection 1-form is pressions are the same, so effectively the affine and spin connections can be interchanged, as long as one properly a a µ accounts for Latin and Greek indices. ω b = ωµ bdx . (54) A. Riemann curvature from the affine connection It is conventional to define a differential form In this section, we will derive the Riemann curvature dA ∂µAν ∂ν Aµ (55) tensor in terms of the affine connection. The develop- ≡ − ment in this section follows the conventional approach and a wedge product of considering parallel transport around a plaquette, as shown in Figure 2. (The term plaquette is borrowed from A B AµBν Aν Bµ, (56) ∧ ≡ − condensed matter theory and refers to a cell of a lattice.) So, this is our first pass at understanding the origin of which are both anti-symmetric in the Greek indices. the Riemann curvature tensor. In the following section, With this convention, originally due to Elie´ Cartan, the we will then re-derive the curvature tensor directly from torsion can be written concisely in terms of the frame the spin connection. and spin connection 1-forms as For the counterclockwise path (xα xα +δxα xα + α α → → a a a b δx + dx ), we have: T = de + ω b e . (57) ∧ α α ¯ α The notation is so compact that it is easy to misunder- X (x + δx) = X (x)+ δX (x) (62a) stand what it represents. For example, writing the tor- (32) = Xα(x) Γα (x)Xβ(x)δxγ . (62b) sion explicitly in the coordinate basis we have − βγ λ λ a At the end point x + δx + dx, we have Tµν = e aTµν (58a) λ a a a b a b α α α = e a ∂µeν ∂ν eµ + ωµ eν ων beµ , X (x + δx + dx) = X (x + δx)+ δX¯ (x + δx) − b − (58b)
(63a) (62b) which fully expanded gives us = Xα(x) Γα (x)Xβ(x)δxγ − βγ α λ λ a λ b a λ a λ b a +δX¯ (x + δx). Tµν = e a∂µeν + e aeν ωµ b e a∂ν eµ e aeµ ων b. − − (59) (63b) Since the affine connection is Now, we need to evaluate the last term (parallel transport λ λ a λ b a Γµν = e a∂µeν + e aeν ωµ b, (60) term) on the R.H.S.

8 A Riemann curvature from the affine connection IV CURVATURE

xα + dxα xα + δxα + dxα r r ☎ ☎ ☎ ☎ ☎ α ☎ ☎ dx ☎ ☎ ☎ ☎ α ☎ ☎r δx ☎r xα xα + δxα

α α FIG. 2 General plaquette of cell sizes δx and dx with its initial spacetime point at 4-vector xα (bottom left corner). The plaquette is a piece of a curved manifold.

(32) δX¯ α(x + δx) = Γα (x + δx)Xβ(x + δx)dxγ (64a) − βγ (62b) α α δ β β µ ν γ = Γ (x)+ ∂δΓ (x)δx X (x) Γ (x)X (x)δx dx (64b) − βγ βγ − µν α β γ α β δ γ α β µ ν γ α β µ δ ν γ = Γ X dx ∂δΓ X δx dx +Γ Γ X δx dx + ∂δΓ Γ X δx δx dx , − βγ − βγ  βγ µν  βγ µν neglect 3rd order term | {z } (64c) where for brevity in the last expression we drop the explicit functional dependence on x, as this is understood. Inserting (64c) into (63b), the 4-vector at the end point is

α α α β γ α β γ α β δ γ α β µ ν γ X (x + δx + dx)= X Γ X δx Γ X dx ∂δΓ X δx dx +Γ Γ X δx dx . (65) − βγ − βγ − βγ βγ µν Interchanging the indices of δx and dx in the last two terms, we have

α α α β γ α β γ α β γ δ α β µ γ ν X (x + δx + dx)= X Γ X δx Γ X dx ∂γΓ X δx dx +Γ Γ X δx dx . (66) − βγ − βγ − βδ βν µγ Then, replacing ν with δ in this last term, we have

α α α β γ α β γ α β γ δ α β µ γ δ X (x + δx + dx)= X Γ X δx Γ X dx ∂γΓ X δx dx +Γ Γ X δx dx . (67) − βγ − βγ − βδ βδ µγ For the clockwise path (xα xα + dxα xα + dxα + δxα), we get the same result as before with dxα and δxα interchanged: → →

α α α β γ α β γ α β γ δ α β µ γ δ X (x + δx + dx)= X Γ X dx Γ X δx ∂γΓ X dx δx +Γ Γ X dx δx . (68) − βγ − βγ − βδ βδ µγ Interchanging the indices δ and γ everywhere, we have

α α α β δ α β δ α β δ γ α β µ δ γ X (x + δx + dx) = X Γ X dx Γ X δx ∂δΓ X dx δx +Γ Γ X dx δx , (69) − βδ − βδ − βγ βγ µδ which now looks like (67) in the indices of the differentials. Hence, as we compute (67) minus (69), the zeroth and first order terms cancel, and the remaining second order terms in (67) and (69) add with the common factor δxγ dxδ (differential area)

Xα = Xα(x + δx + dx) Xα(x + dx + δx) (70a) △ − α β α β α β µ α β µ γ δ = ∂δΓ X ∂γ Γ X +Γ Γ X Γ Γ X δx dx (70b) βγ − βδ βδ µγ − βγ µδ  α α α µ α µ β γ δ  = ∂δΓ ∂γ Γ +Γ Γ Γ Γ X δx dx . (70c) βγ − βδ µδ βγ − µγ βδ  

So, the difference of transporting the vector Xα along the two separate routes around the plaquette is related

9 A Riemann curvature from the affine connection IV CURVATURE to the curvature of the manifold as follows spin connection as follows:

α α β γ δ a a a c X = R βδγ X δx dx , (71) R b = dω b + ω c ω b. (73) △ ∧ and from here we arrive at our desired result and identify Here the Greek indices are suppressed for brevity. So, the the Riemann curvature tensor as first step is to explicitly write out the curvature tensor in all its indices and then to use the vierbein field to convert α α α α µ α µ R βδγ ∂δΓβγ ∂γ Γβδ +ΓµδΓβγ Γµγ Γβδ. (72) the Latin indices to Greek indices, which gives us ≡ − − λ λ b a a a c a c Notice, from the identity (72), the curvature tensor is R σµν e aeσ ∂µων b ∂ν ωµ + ωµ ων b ων cωµ . α α ≡ − b c − b anti-symmetric in its last two indices, R βδγ = R βγδ. (74) − The quantity in parentheses is a spin curvature. Next,
we will use (41), which I restate here for convenience B. Riemann curvature from the spin connection a a τ ρ τ a ωµ b = eρ e bΓµτ e b∂µeτ . (75) In this section, we will show that the Riemann curva- − ture tensor (72) can be simply expressed in terms of the Inserting (75) into (74) gives

λ λ b a τ ρ a τ ρ τ a τ a R σµν = e aeσ ∂µ eρ e bΓ ∂ν eρ e bΓ ∂µe b∂ν eτ + ∂ν e b∂µeτ (76) ντ − µτ − a τ ρ τ a c τ ′ ρ′ τ ′ c + eρ e cΓ e c∂µeτ
eρ′ e bΓ ′
e b∂ν eτ ′ µτ − ντ −  a τ ρ τ a c τ ′ ρ′ τ ′ c eρ e cΓ e c∂ν eτ eρ′ e bΓ ′ e b∂µeτ ′ . − ντ − µτ −   i Reducing this expression is complicated to do. Since the first term in (75) depends on the affine connection Γ and the second term depends on ∂µ, we will reduce (76) in two passes, first considering terms that involve derivatives of the vierbein field and then terms that do not. So as a first pass toward reducing (76), we will consider all terms with derivatives of vierbeins, and show that these vanish. To begin with, the first order derivative terms that appear with ∂µ acting on vierbein fields are the following:

λ b a τ ρ τ a c τ ρ a τ ρ τ ′ c e aeσ [∂µ (eρ e b)Γ e c (∂µeτ ) eρ e bΓ + eρ e cΓ e b∂µ eτ ′ ] (77a) ντ − ντ ντ λ b τ a λ b a τ ρ λ b τ c τ ′ a ρ = e aeσ e b∂µeρ + e aeσ eρ ∂µe b Γ e aeσ e ceρ e b (∂µeτ )Γ ′ ντ − ντ λ b a τ τ ′ c ρ + e aeσ eρ e ce b∂µeτ ′ Γντ  (77b) λ τ a ρ λ b τ ρ τ τ ′ λ a ρ λ τ ′ τ c ρ = e aδ ∂µeρ Γ + δ eσ ∂µe bΓ δ δ e a∂µeτ Γ ′ + δ δ e c∂µeτ ′ Γ (77c) σ ντ ρ ντ − ρ σ ντ ρ σ ντ λ a ρ b τ λ λ a ρ τ b λ = e a∂µeρ Γ + eσ ∂µe bΓ e a∂µeρ Γ + e b∂µeσ Γ (77d) νσ ντ − νσ ντ b τ λ = ∂µ eσ e b Γντ (77e) τ λ = ∂µ (δσ)Γντ
(77f) = 0. (77g)

Similarly, all the first order derivative terms that appear with ∂ν vanish as well. So, all the first order derivative terms vanish in (76). Next, we consider all second order derivatives, both ∂µ and ∂ν , acting on vierbein fields. All the terms with both ∂µ and ∂ν are the following

λ b τ τ ′ a c λ b τ τ ′ a c λ b τ a λ b τ a e aeσ e ce b (∂µeτ ) ∂ν eτ ′ e aeσ e ce b (∂ν eτ ) (∂µeτ ′ ) e aeσ (∂µe b)∂ν eτ + e aeσ (∂ν e b)∂µeτ − − λ τ a c λ τ a c λ b τ a = e ae c (∂µeτ ) ∂ν eσ e ae c (∂ν eτ ) ∂µeσ ∂ν e a ∂µeσ e beτ − − λ b τ a + ∂µe a ∂ν eσ e beτ

(78a) λ τ a c a c λ a λ a = e ae c [(∂µeτ ) ∂ν eσ (∂ν eτ ) ∂µeσ ] ∂ν e a∂µeσ + ∂µe a∂ν eσ (78b)

− − λ a τ c λ a τ c λ a λ a = e aeτ (∂µe c) ∂ν eσ + e aeτ (∂ν e c) ∂µeσ ∂ν e a∂µeσ + ∂µe a∂ν eσ (78c) − − λ τ c λ τ c λ a λ a = δ (∂µe c) ∂ν eσ + δ (∂ν e c) ∂µeσ ∂ν e a∂µeσ + ∂µe a∂ν eσ (78d) − τ τ − λ c λ c λ a λ a = ∂µe c ∂ν eσ + ∂ν e c ∂µeσ ∂ν e a∂µeσ + ∂µe a∂ν eσ (78e) − − = 0.

(78f)

10 B Riemann curvature from the spin connection V MATHEMATICAL CONSTRUCTS

Hence, all the second order derivative terms in (76) vanish, as do the first order terms. Note that we made use of the λ a fact ∂µ e aeτ = 0, so as to swap the order of differentiation,

λ a λ a ∂µ e a eτ = e a (∂µeτ ) . (79) − Finally, as a second pass toward reducing (76) to its
final form, we now consider all the remaining terms (no derivatives of the vierbein fields), and these lead to the curvature tensor expressed solely as a function of the affine connection:

λ λ b a τ ρ ρ a τ ′ ρ τ ρ τ R σµν = e aeσ eρ e b ∂µΓ ∂ν Γ + eρ e b Γ Γ ′ Γ Γ ′ (80a) ντ − µτ µτ ντ − ντ µτ λ τ h ρ ρ λ τ ′ ρ τ ρ τ i = δ δ ∂µΓ ∂ν Γ + δ δ
Γ Γ ′ Γ Γ ′ .
(80b) ρ σ ντ − µτ ρ σ µτ ντ − ντ µτ

Applying the Kronecker deltas, we arrive at the final re- An identity we will need allows us to evaluate the trace 1 sult of M − ∂µM where M is a 2-rank tensor

λ λ λ λ τ λ τ R σµν = ∂µΓνσ ∂ν Γµσ +Γµτ Γνσ Γντ Γµσ, (81) 1 − − T r[M − ∂µM]= ∂µ ln M . (82) | | which is identical to (72). If we had not already derived the curvature tensor, we could have written (81) down As an example of this identity, consider the following 2 2 by inspection because of its similarity to (74), essentially matrix and its inverse × replacing the spin connection with the affine connection.

a b 1 1 d b M = M − = − . (83) c d M c a V. MATHEMATICAL CONSTRUCTS   | | − 

Here we assemble a number of preliminary identities A demonstration of the trace identity (82) for the sim- that we will use later to derive the Einstein equation. plest case of one spatial dimension is

1 1 d b ∂xa ∂xb Tr M − ∂xM = Tr − (84a) ad bc c a ∂xc ∂xd  −      − 1 ∂xa d b∂xc ∂xb d b∂xd = Tr − − (84b) ad bc ∂xac + a∂xc ∂xbc + a∂xd  − − −  1 = [∂x(ad) ∂x(bc)] (84c) ad bc − − ∂x(ad bc) = − (84d) ad bc − = ∂x ln M . (84e) | |

This identity holds for matrices of arbitrary size. In our A. Consequence of tetrad postulate case, we shall need this identity for the case of 4 4 matrices. × The tetrad postulate of Section III.C is that the vier- The contravariant and covariant metric tensors are or- bein field is invariant under parallel transport thogonal e a =0. (88) λµ λ µ ν g gµν = δν , (85) ∇ That the metric tensor is invariant under parallel trans- so port then immediately follows µν 1 g (gµν )− . (86) a c → µgνλ = µ (eν eλ nab) (89a) ∇ ∇ We also define the negative determinant of the metric a b a b = ( µeν ) eλ nab + eν µeλ nab (89b) tensor ∇ ∇ (88) g Det gµν . (87) = 0.
(89c) ≡−

11 B Affine connection in terms of the metric tensor VI GRAVITATIONAL ACTION

This is called metric compatibility. C. Invariant volume element

Now, we consider the transport of a general 4-vector B. Affine connection in terms of the metric tensor µ µ µ λ ν V = ∂ν V +Γ V . (98) ∇ νλ Now we can make use of (89) to compute the affine Therefore, the 4-divergence of V µ is connection. Permuting indices, we can write µ µ µ λ λ λ µV = ∂µV +Γ V (99a) ρgµν = ∂ρgµν Γ gλν Γ gµλ =0 (90a) µλ ∇ − ρµ − ρν ∇ (95e) µ 1 λ = ∂µV + ∂λ√ g V (99b) λ λ √ g − µgνρ = ∂µgνρ Γ gλρ Γ gνλ = 0 (90b) ∇ − µν − µρ − 1 µ
= ∂µ √ gV . (99c) λ λ √ g − ν gρµ = ∂ν gρµ Γνρgλµ Γνµgρλ =0. (90c) − ∇ − −
If V µ vanishes at infinity, then integrating over all space Now, we take (90a) (90b) (90c): − − yields λ ∂ρgµν ∂µgνρ ∂ν gρµ + 2Γµν gλρ =0. (91) − − 4 µ 4 µ d x√ g µV = d x ∂µ √ gV =0, (100) Multiplying through by gσρ allows us to solve for the − ∇ − Z Z affine connection
which is a covariant form of Gauss’s theorem where the σ 1 σρ invariant volume element is Γ = g (∂µgνρ + ∂ν gρµ ∂ρgµν ) . (92) µν 2 − dV = √ g d4x. (101) Then, contracting the σ and µ indices, we have −

µ 1 µρ Γ = g (∂µgνρ + ∂ν gρµ ∂ρgµν ) (93a) µν 2 − D. Ricci tensor 1 µρ = g (∂ν gρµ + ∂µgρν ∂ρgµν ) (93b) The Ricci tensor is the second rank tensor formed from 2 − the Riemann curvature tensor as follows: 1 µρ 1 µρ = g ∂ν gρµ + g ∂µgρν ∂ρgµν . (93c) λ λµ 2 2 { − } Rσν R = g Rµσλν , (102) ≡ σλν Since the metric tensor is symmetric and the last term in where the Riemann tensor is brackets is anti-symmetric in the µρ indices, the product ρ ρ ρ ρ λ ρ λ must vanish. Thus R σµν = ∂µΓ ∂ν Γ +Γ Γ Γ Γ . (103) νσ − µσ µλ νσ − νλ µσ 1 Γµ = gµρ∂ g . (94) Therefore, the Ricci tensor can be written as µν 2 ν ρµ ρ ρ ρ λ ρ λ Rσν = ∂ρΓ ∂ν Γ +Γ Γ Γ Γ (104a) Furthermore, rewriting (94) as the trace of the νσ − ρσ ρλ νσ − νλ ρσ ρ λ ρ ρ λ ρ similarity transformation of the covariant derivative = ∂ρΓ Γ Γ ∂ν Γ +Γ Γ . (104b) 1 νσ νρ λσ σρ νσ λρ Tr[M − ∂ν M] = ∂ν ln Det M that we demonstrated in − − (84), we have Now, using the correction for a covariant vector λ µ 1 λρ ρAν = ∂ρAν Γρν Aλ, (105) Γ = Tr g ∂ν gρµ (95a) ∇ − µν 2 1 we can write (104b) as ∂ g
= ν ln Det ρµ (95b) ρ ρ 2 Rσν = ρΓνσ ν Γσρ. (106) 1 ∇ − ∇ = ∂ν ln( g), g Det gρµ (95c) 2 − ≡− (106) is known as the Palatini identity. The scalar cur- vature is the following contraction of the Ricci tensor = ∂ν ln √ g (95d) − µν 1 R g Rµν . (107) = ∂ν √ g. (95e) √ g − ≡ − Equating (94) to (95e), we have VI. GRAVITATIONAL ACTION

1 µρ ∂ν √ g = √ gg ∂ν gρµ. (96) A. Free field gravitational action − 2 − For generality, we write this corollary to (95e) as follows: The action for the source free gravitational field is 1 µν 1 4 δ√ g = √ gg δgµν . (97) IG = d x √ g R(x). (108) − 2 − 16πG − Z

12 A Free field gravitational action VI GRAVITATIONAL ACTION

The equation of motion for the metric tensor can be de- where we commute the variational change with the co- termined by varying (108) with respect to the metric ten- variant derivative. Now, we can expand the third term sor field. The variation is carried out in several stages. on the R.H.S. of (109) as follows: The variation of the Lagrangian density is

µν µν δ √ gR = √ gRµν δg + Rδ√ g + √ gg δRµν . − − − − (109) From the Palatini
identity (106), the change in the Ricci tensor can be written as

λ λ δRµν = ν δΓ + λδΓ , (110) −∇ µλ ∇ µν

µν (110) µν λ µν λ √ gg δRµν = √ g g ν δΓ g λδΓ − − − ∇ µλ − ∇ µν   (111a)

(89) µν λ µν λ = √ g ν g δΓ λ g δΓ , − − ∇ µλ −∇ µν  ν µ  like V
like V
   (111b) | {z } | {z } ν 1 ν since µgνρ = 0. Now from (99c) we know ν V = √ g ∂ν (√ gV ), so for the variation of the third term we have ∇ ∇ − − µν µν λ µν λ √ gg δRµν = ∂ν √ gg δΓ + ∂λ √ gg δΓ . (111c) − − − µλ − µν These surface terms drop out when integrated over all space, so
the third term on
the R.H.S. of (109) vanishes. Finally, inserting the result (97)

1 µν δ√ g = √ gg δgµν − 2 − into the second term on the R.H.S. of (109) we can write the variation of the gravitational action (108) entirely in terms of the variation of the metric tensor field

1 4 µν 1 µν δIG = d x √ g Rµν δg + g Rδgµν . (112) 16πG − 2 Z   The variation of identity vanishes, µ µτ µτ µτ δ[δλ ]= δ [g gτλ] = (δg ) gτλ + g δgτλ =0, (113) from which we find the following useful identity µτ νλ νλ µτ δg g gτλ + g g δgτλ =0 (114) or µν νλ µτ δg = g g δgτλ. (115) − With this identity, we can write the variation of the free gravitational action as

1 4 µτ νλ 1 µν δIG = d x √ g Rµν g g δgτλ g Rδgµν −16πG − − 2 Z   (116a)

1 4 µν 1 µν = d x √ g R g R δgµν . (116b) −16πG − − 2 Z   Since the variation of the metric does not vanish in general, for the gravitational action to vanish the quantity in square brackets must vanish. This quantity is call the Einstein tensor 1 Gµν Rµν gµν R. (117) ≡ − 2 So, the equation of motion for the free gravitation field is simply Gµν =0. (118)

13 A Free field gravitational action VI GRAVITATIONAL ACTION

B. Variation with respect to the vierbein field

In terms of the vierbein field, the metric tensor is

a b gµν (x)= eµ (x)eν (x)ηab, (119) so its variation can be directly written in terms of the variation of the vierbein

a b a b δgµν = δeµ eν ηab + eµ δeν ηab (120a) a a = δeµ eνa + eµaδeν (120b)

(79) a a = eµ δeνa δeµaeν . (120c) − −

Now, we can look upon δeµa as a field quantity whose indices we can raise or lower with the appropriate use of the metric tensor. Thus, we can write the variation of the metric as

a λ λ a δgµν = gνλeµ δe a gµλδe aeν (120d) − − a a λ = (gµλeν + gνλeµ ) δe a. (120e) − Therefore, inserting this into (116b), the variation of the source-free gravitational action with respect to the vierbein field is

1 4 µν 1 µν a a λ δIG = d x√ g R g R gµλeν + gνλeµ δe a (121a) 16πG − − 2 Z    1 4 ν a 1 ν a µ a 1 µ a λ = d x√ g Rλ eν δ Reν + R λeµ δ Reµ δe a (121b) 16πG − − 2 λ − 2 λ Z   1 4 µ 1 µ a λ = d x√ g R λ δ R eµ δe a. (121c) 8πG − − 2 λ Z   

Since the variation of the vierbein field does not vanish in is as a functional derivative with respect to the metric general, for the gravitational action to vanish the quan- tity in square brackets must vanish. Multiplying this by δIM 1 λν d4x√ gT µν. (124) g , the equation of motion is δg ≡ 2 − µν Z µν a G eµ =0, (122) So, in consideration of (123) and (124), we should write a which leads to (118) since eµ = 0. Yet (122) is a more 6 general equation of motion since it allows cancelation a λ 1 µν uλ δe a = T δgµν (125a) across components instead of the simplest case where 2 µν each component of G vanishes separately. (120e) 1 µν a a λ = T (gµλeν + gνλeµ ) δe a,(125b) −2

µν λ C. Action for a gravitational source which we can solve for T . Dividing out δe a and then multiplying through by gλβ we get The fundamental principle in general relativity is that the presence of matter warps the spacetime manifold in βa 1 µν β a β a u = T δ eν + δ eµ (126a) the vicinity of the source. The vierbein field allows us to −2 µ ν quantify this principle in a rather direct way. The varia- 1 βν a µβ a
= T eν + T eµ (126b) tion of the action for the matter source to lowest order is −2 linearly proportional to the variation of the vierbein field βν a = T eν ,
(126c) − 4 a λ δIM = d x√ guλ δe a, (123) since the energy-momentum tensor is symmetric. Thus, − Z we have a where the components uλ are constants of proportion- µν µa ν ality. However, the usual definition of the matter action T = u e a. (127) −

14 D Full gravitational action VII EINSTEIN’S ACTION

Alternatively, again in consideration of (123) and (124), VII. EINSTEIN’S ACTION we could also write

a λ 1 µν uλ δe a = T δgµν (128a) In this section, we review the derivation of the equa- 2 tion of motion of the metric field in the weak field ap- (120b) 1 µν a a proximation. We start with a form of the Lagrangian = T (δe e + e δe ) (128b) 2 µ νa µa ν density presented in (Einstein, 1928a) for the vierbein 1 field theory. Einstein’s intention was the unification of = T (δeµ eνa + eµaδeν ) (128c) 2 µν a a electromagnetism with gravity. 1 λ νa µa λ = Tλν δe ae + Tµλe δe a (128d) With h denoting the determinant of eµa (i.e. h 2 √ g), the useful identity | | ≡ 1 νa µa λ
− = (T e + T e ) δe (128e) 2 λν µλ a νa λ = Tλν e δe a. (128f) 1 µν δ√ g = √ gg δgµν − 2 − This implies that the energy-stress tensor is proportional to the vierbein field T = e u a. (129) can be rewritten strictly in terms of the vierbein field as µν µa ν follows Consequently, the variation of the energy-stress tensor is then 1 µν a λ a δh = hg δgµν (138a) δTµν = δeµauν = gµλδe auν . (130) 2 1 µν a b This can also be written as = hg δ(eµ eν ) ηab (138b) λ λ a λ λ a 2 T ν = e auν δT ν = δe auν . (131) 1 µν a b 1 µν a b → = hg δeµ eν ηab + hg eµ δeν ηab (138c) Inserting this back into the action for a graviational 2 2 1 1 source (123) we have = hδe aeµ + heν δe b (138d) 2 µ a 2 b ν 4 λ a µ IM = d x√ gT λ (132a) = hδeµ e a. (138e) − Z 4 µν = d x√ gg Tµν . (132b) − Z

D. Full gravitational action A. Lagrangian density form 1

The variation of the full gravitational action is the sum of variations of the source-free action and gravitational With the following definition action for matter

δI = δIG + δIM . (133) ν 1 νa Λ e (∂β eαa ∂αeβa), (139) αβ ≡ 2 − Inserting (121c) and (132b) into (133) then gives

4 1 µ 1 µ a a λ δIG = d x√ g R λ δ R eµ + uλ δe a. the first Lagrangian density that we consider is the fol- − 8πG − 2 λ     lowing: Z (134) Therefore, with the requirement that δIG = 0, we obtain the equation of motion of the vierbein field µν α β = hg Λµ Λν , (140a) L β α µ 1 µ a a h R λ δ R eµ = 8πGuλ . (135) µν αa βb − 2 λ − = g e e (∂βeµa ∂µeβa)(∂αeνb ∂ν eαb).   4 − − Multiplying through by eνa (140b)

µ 1 µ a a R λ δ R eµ eνa = 8πGeνauλ (136) − 2 λ − For a weak field, we have the following first-order expan-   gives the well known Einstein equation sion 1 Rνλ gνλR = 8πGTνλ. (137) − 2 − eµa = δµa kµa . (141) − ···

15 B Lagrangian density form 2 VII EINSTEIN’S ACTION

The lowest-order change (2nd order in δh) is This implies the following:

h µν αa βb ∂2kαν ∂ ∂αkβν δ = η δ δ (∂βkµa ∂µkβa)(∂αkνb ∂ν kαb) β =0 (146a) L 4 − − − (142a) or

h µν α α β β 2 α µα = η (∂βkµ ∂µkβ )(∂αkν ∂ν kα ) (142b) ∂ kβ ∂µ∂βk =0. (146b) 4 − − − h µν α β α β = η (∂βkµ ∂αkν ∂βkµ ∂ν kα (142c) These are the first two terms in Einstein’s Eq. (5). 4 − Notice that we started with a Lagrangian density with α β α β ∂µkβ ∂αkν + ∂µkβ ∂ν kα ) the usual quadratic form in the field strength of the form − h µα ν β µν α β (145c), which is = η ∂βkµ ∂ν kα η ∂βkµ ∂ν kα (142d) 4 − h αβν µα ν β µν α β = Fαβν F , (147) η ∂µkβ ∂ν kα + η ∂µkβ ∂ν kα − L 4  h µα ν µν α where the field strength is = ( η ∂β∂ν kµ + η ∂β∂ν kµ (142e) 4 − µα ν µν α β αβν α βν β αν + η ∂µ∂ν kβ η ∂µ∂ν kβ ) kα . (142f) F = ∂ k ∂ k . (148) − − So δ d4x = 0 implies the equation of motion If we had varied the action with respect to kβν , then we L αν να α ν 2 α would have obtained the same equation of motion (146b). ∂Rβ∂ν k + ∂β∂ν k + ∂ ∂ν kβ ∂ kβ = 0 (143a) − − or C. First-order fluctuation in the metric tensor 2 α µα αµ α µ ∂ kβ ∂µ∂βk + ∂β∂µk ∂µ∂ kβ =0. (143b) − − The metric tensor expressed in terms of the vierbein The above equation of motion (143b) is identical to field is Eq. (5) in Einstein’s second paper. a a a gαβ = eα eβa = (δα + kα ) (δβa + kβa) . (149) B. Lagrangian density form 2 So the first order fluctuation of the metric tensor field is the symmetric tensor Now the second Lagrangian density we consider is the following: gαβ gαβ δαβ = kαβ + kβα . (150) ≡ − ··· ασ βτ µ ν = hgµν g g Λ Λ (144a) L αβ στ We define the electromagnetic four-vector by contracting h ασ βτ µa νb the field strength tensor = gµν g g e e (∂βeαa ∂αeβa) (∂τ eσb ∂σeτb) . 4 − − α 1 αa (144b) ϕµ Λ = e (∂αeµa ∂µeαa) . (151) ≡ µα 2 − We will see this leads to the same equation of motion that we got from the first form of the Lagrangian density. The This implies lowest-order change is 1 αa ϕµ = δ (∂αkµa ∂µkαa) , (152) h ασ βτ µa νb 2 − δ = ηµν η η δ δ (∂βkαa ∂αkβa) (∂τ eσb ∂σeτb) L 4 − − so we arrive at (145a) α α h ασ βτ ν ν 2ϕµ = ∂αkµ ∂µkα . (153) = η η (∂βkαν ∂αkβν ) (∂τ kσ ∂σkτ ) (145b) − 4 − − h β αν α βν = (∂βkαν ∂αkβν ) ∂ k ∂ k (145c) 4 − − D. Field equation in the weak field limit h β αν α βν
β αν = ∂βkαν ∂ k ∂βkαν ∂ k ∂αkβν ∂ k The equation of motion for the fluctuation of the met- 4 − − ric tensor from the first form of the Lagrangian density α βν + ∂αkβν ∂ k (145d) is obtained by adding (143b) to itself but with α and β h β αν α βν α βν exchanged: = ∂βkαν ∂ k
∂βkαν ∂ k ∂βkαν ∂ k 4 − − 2 µ µ µ β αν ∂ kβα ∂ ∂βkµα + ∂β∂ kαµ ∂ ∂αkβµ +∂βkαν ∂ k (145e) − − 2 µ µ µ +∂ kαβ ∂ ∂αkµβ + ∂α∂ kβµ ∂ ∂βkαµ =0, h 2 αν α βν = ∂ k + ∂
β∂ k kαν . (145f) − − 2 − (154)
16 A Invarianceinflatspace VIII RELATIVISTICCHIRALMATTERINCURVED SPACE which has a cancellation of four terms leaving field

2 µ µ ∂ gαβ ∂ ∂αkµβ ∂ ∂βkµα =0. (155) 1 2 µ µ µ ∂ gαβ + ∂ ∂αgµβ + ∂ ∂βgµα ∂α∂βgµ − − 2 − − Similarly, we arrive at the same result starting with the   = ∂βϕα + ∂αϕβ . (161) equation of motion for the fluctuation of the metric ten- sor obtained from the second form of the Lagrangian den- This result is the same as Eq. (7) in Einstein’s second sity, again by adding (146a) to itself but with α and β paper. In the case of the vanishing of φα, (161) agrees to exchanged: first order with the equation of General Relativity

2 µ ∂ kβα ∂ ∂βkµα R . − αβ =0 (162) 2 µ (156) + ∂ kαβ ∂ ∂αkµβ =0. − Thus, Einstein’s action expressed explicitly in terms of In this case, the sum is exactly the same as what we just the vierbein field reproduces the law of the pure gravita- obtained in (155) but with no cancellation of terms tional field in weak field limit.

2 µ µ ∂ gαβ ∂ ∂αkµβ ∂ ∂βkµα =0. (157) − − VIII. RELATIVISTIC CHIRAL MATTER IN CURVED Using (153) above, just with relabeled indices, SPACE

µ µ 2ϕα = ∂µkα ∂αkµ . (158) A. Invariance in flat space − Taking derivatives of (158) we have ancillary equations The external Lorentz transformations, Λ that act on of motion: 4-vectors, commute with the internal Lorentz transfor- U i.e. µ µ mations, (Λ) that act on spinor wave functions, ∂µ∂βkα + ∂α∂βkµ = 2∂βϕα (159a) − − µ [Λ ν ,U(Λ)] = 0. (163) and Note that we keep the indices on U(Λ) suppressed, just µ µ as we keep the indices of the Dirac matrices and the com- ∂µ∂αkβ + ∂α∂βkµ = 2∂αϕβ. (159b) − − ponent indices of ψ suppressed as is conventional when writing matrix multiplication. Only the exterior space- Adding the ancilla (159) to our equation of motion (157) time indices are explicitly written out. With this con- gives vention, the Lorentz transformation of a Dirac gamma 2 µ µ matrix is expressed as follows: ∂ gαβ + ∂ ∂α(kµβ + kβµ)+ ∂ ∂β(kµα + kαµ) − µ 2∂α∂βk = 2(∂βϕα + ∂αϕβ). 1 µ µ σ − µ U(Λ)− γ U(Λ) = Λ σγ . (164) (160) The invariance of the Dirac equation in flat space Then making use of (150) this can be written in terms of under a Lorentz transformation is well known the symmetric first-order fluctuation of the metric tensor (Peskin and Schroeder, 1995):

µ LLT µ 1 ν 1 [iγ ∂µ m] ψ(x) iγ Λ− ∂ν m U(Λ)ψ Λ− x (165a) − −→ µ − h 1 µ i 1 ν 1 = U(Λ)U(Λ)
− iγ Λ− ∂ν m U
(Λ)ψ Λ− x (165b) µ − h i (163) 1 µ
1 ν 1
= U(Λ) iU(Λ)− γ U(Λ) Λ− ∂ν m ψ Λ− x (165c) µ − h i (164) µ σ 1 ν
1
= U(Λ) i Λ γ Λ− ∂ν m ψ Λ− x (165d) σ µ − h µ 1 ν σ i 1 = U(Λ) i Λ Λ−
γ ∂ν m ψ Λ− x
(165e) σ µ − h ν σ 1 i = U(Λ) [iδ γ ∂ν
m] ψ Λ− x
(165f) σ − ν 1 = U(Λ) [iγ ∂ν m] ψ Λ− x . (165g) −

B. Invariance in curved space where I put a minus on the subscript to indicate the in- 1 verse transformation, i.e. Λ 1 U(Λ)− . Of course, 2 Switching to a compact notation for the interior − ≡ 17 Lorentz transformation, Λ 1 U(Λ), (164) is 2 ≡ µ µ σ Λ 1 γ Λ 1 =Λ σγ , (166) − 2 2 B Invarianceincurvedspace VIII RELATIVISTICCHIRALMATTERIN CURVED SPACE exterior Lorentz transformations can be used as a simi- C. Covariant derivative of a spinor field larity transformation on the Dirac matrices The Lorentz transformation for a spinor field is µ σ 1 ν ν Λ σγ Λ− µ = γ . (167) 1 ab
Λ 1 =1+ λab S , (173) Below we will need the following identity: 2 2

µ λ a 1 ν (163) µ λ a 1 ν where the generator of the transformation is anti- Λ λe aγ (Λ− ) Λ 1 = Λ 1 Λ 1 Λ λe aγ Λ 1 (Λ− ) ab ba µ 2 2 2 2 µ symmetric S = S . The generator satisfies the fol- − −
(168a) lowing commutator (166) ν µ σ λ a 1 Shk,Sij ηhj Ski ηkiShj ηhiSkj ηkj Shi. = Λ 1 Λ σ Λ λe aγ (Λ− ) µ [ ]= + (174) 2 − −
(168b) Thus, the local Lorentz transformations (LLT) of a a (167) ν λ a Lorentz 4-vector, x say, and a Dirac 4-spinor, ψ say, = Λ 1 Λ λe aγ . (168c) 2 are respectively:

We require the Dirac equation in curved space be invari- a a a b LLT: x x′ =Λ b x (175) ant under Lorentz transformation when the curvature of → space causes a correction Γµ. That is, we require and

µ a e a γ (∂µ +Γµ) ψ(x) LLT: ψ ψ′ =Λ 1 ψ. (176) 2 LLT µ λ a 1 ν 1 → Λ λe aγ (Λ− ) µ (∂ν +Γν′ )Λ 1 ψ(Λ− x) (169a) −→ 2 The covariant derivative of a 4-vector is µ λ a 1 ν 1 = Λ λe aγ (Λ− ) µΛ 1 ∂ν +Λ 1 Γν′ Λ 1 ψ(Λ− x) α α α β 2 − 2 2 γ X = ∂γ X +Γ X , (177) ∇ βγ µ λ a 1 ν  1  +Λ λe aγ Λ− ∂ν Λ 1 ψ(Λ− x) (169b) µ 2 and the 4-vector at the nearby location is changed by the   (168c) ν λ a
1 curvature of the manifold. So we write it in terms of the = Λ 1 Λ λe aγ ∂ν +Λ 1 Γν′ Λ 1 ψ(Λ− x) original 4-vector with a correction 2 − 2 2

h 1  α α α αk β γ +Λ 1 ∂ν Λ 1 ψ(Λ− x) (169c) X k (x + δx )= X k (x) Γ (x)X (x)δx , (178) − 2 2 − βγ ν λ a  i = Λ 1 Λ λe aγ [(∂ν +Γν ) 2 as depicted in Fig. 3. 1 Γν +Λ 1 Γ′ Λ 1 +Λ 1 ∂ν Λ 1 ψ(Λ− x). 2 ν 2 2 2 − − − α k x Xβ x δxγ =0    Γβγ ( ✛) ( ) (169d) ❇▼ ❇▼ ✂✍ | {z } Xαk (x)❇ ❇ ✂Xαk (x + δx) In the last line we added and subtracted Γ . To achieve ❇ ❇ ✂ ν ❇r ❇✂r invariance, the last three terms in the square brackets α α α must vanish. Thus we find the form of the local “gauge” x x + δx transformation requires the correction field to transform as follows: FIG. 3 Depiction of the case of an otherwise constant field dis- torted by curved space. The field value Xα(x) is parallel trans- α Γν +Λ 1 Γ′ Λ 1 +Λ 1 ∂ν Λ 1 =0 (170a) ported along the curved manifold (blue curve) by the distance δx 2 ν 2 2 2 − − − going from point xα to xα + δxα. or   Likewise, the correction to the vierbein field due the Γν′ =Λ 1 Γν Λ 1 ∂ν Λ 1 Λ 1 . (170b) 2 − 2 − 2 − 2 curvature of space is   µ α µ µk β α Therefore, the Dirac equation in curved space e k k(x + δx )= e k k(x) Γ (x)e k(x)δx . (179) − βα a µ iγ e a(x) µ ψ m ψ =0 (171) D − The Lorentz transformation of a 2-rank tensor field is a b is invariant under a Lorentz transformation provided the Λ a′ Λ b′ ηab = ηa′b′ . (180) generalized derivative that we use is Moreover, the Lorentz transformation is invertible µ = ∂µ +Γµ, (172) D i a i a i Λ aΛj = δj =Λ j Λa , (181) where Γµ transforms according to (170b). This is anal- ogous to a gauge correction; however, in this case Γµ is where the inverse is obtained by exchanging index labels, not a vector potential field. changing covariant indices to contravariant indices and

18 C Covariant derivative of a spinor field VIII RELATIVISTIC CHIRAL MATTER IN CURVED SPACE contravariant to covariant. In the case of infinitesimal is seen to have the physical interpretation of generalizing transformations we have the infinitesimal transformation (182) to the case of in- finitesimal transport in curved space. Relabeling indices, i i i Λ j (x)= δj + λ j (x), (182) we have a ν a σ a ν where ωµ = e b∂µeν +Γ eσ e b (189a) b − µν ν a σ a ij ji = e b ∂µeν Γ eσ (189b) 0= λij + λji = λ + λ . (183) µν − ν a − = e b µeν , (189c) Lorentz and inverse Lorentz transformations of the − ∇
vierbein fields are where here the covariant derivative of the vierbien 4- vector is not zero.12 Writing the Lorentz transformation µ a µ e¯ h′ (x)=Λh′ (x)e a(x) (184) in the usual infinitesimal form h h h and Λ k = δk + λ k (190)

µ a′ µ implies e h(x)=Λ h(x)¯e a′ (x), (185) h h α λ k = ωα k δx (191a) where temporarily I am putting a bar over the trans- −ν h α = e k αeν δx (191b) formed vierbein field as a visual aid. Since the vierbein ∇ field is invertible, we can express the Lorentz tranforma- or
tion directly in terms of the vierbeins themselves β α λhk = e k ( αeβh) δx . (191c) a′ µ a′ ∇ e¯µ (x)e h(x)=Λ h(x). (186) Using (173), the Lorentz transformation of the spinor Now, we transport the Lorentz transformation tensor it- field is self. The L.H.S. of (186) has two upper indices, the Latin 1 hk Λ 1 ψ = 1+ λhk S ψ = ψ + δψ. (192) index a′ and the Greek index µ, and we choose to use the 2 2 upper indices to connect the Lorentz transformation ten-   This implies the change of the spinor is sor between neighboring points. These indices are treated differently: a Taylor expansion can be used to connect a 1 β α hk δψ = e k ( αeβh) δx S ψ (193a) quantity in its Latin non-coordinate index at one point 2 ∇ α to a neighboring point, but the affine connection must be = Γαψ δx , (193b) used for the Greek coordinate index. Thus, we have where the correction to the spinor field is found to be ′ ′ Λh (x + δxα) =e ¯ h (x + δxα)eµk (x + δxα) (187a) k µk k 1 β hk Γα = e k ( αeβh) S (194a) ∂e¯ h 2 ∇ h µ α µk α = e¯µ (x)+ δx e k(x + δx ) 1 β σ hk ∂xα = e k ∂αeβh Γ eσh S (194b) ! 2 − µβ (187b) 1 β hk 1
σ β hk = e k (∂αeβh) S Γµβ(eσhe k)S (194c) h 2 − 2 (179) h ∂e¯µ α = e¯µ (x)+ δx (187c) 1 β hk ∂xα = e k (∂αeβh) S , (194d) ! 2 µ µ β α β e k Γβα(x)e k(x)δx where the last term in (194c) vanishes because eσhe k is × − symmetric whereas Shk is anti-symmetric in the indices  ∂e¯ h  h µ α µ µ β α h h and k. Thus, we have derived the form of the covariant = δk + δx e k Γ e kδx e¯µ ∂xα − βα derivative of the spinor wave function (187d) µψ = ∂µψ +Γµψ (195a) ∂e¯ h D h µ µ µ h β α 1 = δk + δβ Γβαe¯µ e kδx β hk ∂xα − = ∂µ + e k µeβh S ψ (195b) ! 2 ∇   (187e) 1 = ∂ + eβ ∂ e Shk ψ. (195c) h µ h µ h β α µ k µ βh = δ + e k∂αe¯ Γ e¯ e k δx 2 k µ − βα µ    (187f) h h α = δk ωα kδx , (187g) − 12 Remember the Tetrad postulate that we previously derived where the spin connection a a a σ a b ∇µeν = ∂µeν − eσ Γµν + ωµ beν = 0 h µ h µ h β ωα = e k∂αe¯ +Γ e¯ e k (188) is exactly (189b). k − µ βα µ

19 X ACKNOWLEDGEMENTS

This is the generalized derivative that is needed to cor- Carroll, S. M. (2004). An Introduction to General Relativity, rectly differentiate a Dirac 4-spinor field in curved space. Spacetime and Geometry. Addison Wesley. D’Inverno, R. (1995). Introducing Einstein’s relativity. Ox- ford. IX. CONCLUSION Dirac, P. A. M. (1928). The Quantum Theory of the Electron. Royal Society of London Proceedings Series A, 117:610–624. A detailed derivation of the Einstein equation from the Einstein, A. (1928a). New possibility for a unified field Sitzungsberichte der least action principle and a derivation of the relativistic theory of gravity and electricity. Preussischen Akademie der Wissenschaften. Physikalisch- Dirac equation in curved space from considerations of Mathematische Klasse., pages 223–227. invariance with respect to Lorentz transformations have Einstein, A. (1928b). Riemann geometry with preservation been presented. The field theory approach that was pre- of the concept of distant parallelism. Sitzungsberichte der sented herein relied on a factored decomposition of the Preussischen Akademie der Wissenschaften. Physikalisch- metric tensor field in terms of a product of vierbein fields Mathematische Klasse., pages 217–221. that Einstein introduced in 1928. In this sense, the vier- Einstein, A. (1948). A generalized theory of gravitation. Rev. bein field is considered the square root of the metric ten- Mod. Phys., 20(1):35–39. sor. The motivation for this decomposition follows natu- Glashow, S. L. (1961). Partial-symmetries of weak interac- rally from the anti-commutator tions. Nuclear Physics, 22(4):579 – 588. Jackiw, R. (2005). 50 years of Yang-Mills theory, chapter 10, µ a ν b µν e a(x)γ ,e b(x)γ =2g (x), (196) pages 229–251. World Scientific, Singapore. Edited by Ger- { } ardus t’Hooft. a where γ are the Dirac matrices. Dirac originally dis- Kaempffer, F. A. (1968). Vierbein field theory of gravitation. covered an aspect of this important identity when he Phys. Rev., 165(5):1420–1423. successfully attempted to write down a linear quantum Peskin, M. E. and Schroeder, D. V. (1995). An Introduction wave equation that when squared gives the well known to Quantum Field Theory. Westview Press of the Perseus Klein-Gordon equation. Thus, dealing with relativistic Books Group, New York, 1st edition. quantum mechanics in flat space, Dirac wrote this iden- Salam, A. (1966). Magnetic monopole and two photon theo- tity as ries of c-violation. Physics Letters, 22(5):683 – 684. Weinberg, S. (1967). A model of leptons. Phys. Rev. Lett., γa,γb =2ηab, (197a) 19(21):1264–1266. { } Weinberg, S. (1972). Gravitation and Cosmology, Principles where η = diag(1, 1, 1, 1). Einstein had the brilliant and applications of the general theory of relativity. Wiley. insight to write the− part− of− the identity that depends on Yang, C. N. and Mills, R. L. (1954). Conservation of isotopic the spacetime curvature as spin and isotopic gauge invariance. Phys. Rev., 96(1):191– 195. µ ν ab µν e a(x)e b(x)η = g (x). (197b) Combining (197a) and (197b) into (196) is essential to correctly develop a relativistic quantum field theory in curved space. However, (197b) in its own right is a suf- ficient point of departure if one seeks to simply derive the Einstein equation capturing the dynamical behavior of spacetime.

X. ACKNOWLEDGEMENTS

I would like to thank Carl Carlson for checking the derivations presented above. I would like to thank Hans C. von Baeyer for his help searching for past English translations of the 1928 Einstein manuscripts (of which none were found) and his consequent willingness to trans- late the German text into English.

References

Birrell, N. and Davies, P. (1982). Quantum fields in curved space. Cambridge University Press. Born, M. (1961). The Born-Einstein letters 1916-1955. Inter- national series in pure and applied physics. Macmillan, New York. ISBN-13: 978-1-4039-4496-2, republished in English in 2005.

20 Appendices

The following two manuscripts, translated here in English by H.C. von Baeyer and into LATEX by the author, originally appeared in German in Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse in the summer of 1928.

Einstein’s 1928 manuscript on distant parallelism

21 Riemann geometry with preservation of the concept of distant parallelism

A. Einstein June 7, 1928

Riemann geometry led in general relativity to a phys- given by ical description of the gravitational field, but does not 2 2 µ ν yield any concepts that can be applied to the electromag- A = Aa = hµahνaA A . (2) netic field. For this reason the aim of theoreticians is to ··· X find natural generalizations or extensions of Riemann ge- The metric tensor components gµν are given by the for- ometry that are richer in content, in hopes of reaching a mula logical structure that combines all physical field concepts from a single point of view. Such efforts have led me to gµν = hµahνa, (3) a theory which I wish to describe without any attempt ··· at physical interpretation because the naturalness of its where, of course, the index a is summed over. With fixed ν concepts lends it a certain interest in its own right. a, the ha are the components of a contravariant vector. Riemannian geometry is characterized by the facts that The following relations also hold: the infinitesimal neighborhood of every point P has a ν ν Euclidian metric, and that the magnitudes of two line el- hµaha = δµ (4) ements that belong to the infinitesimal neighborhoods of ··· two finitely distant points P and Q are comparable. How- µ ever, the concept of parallelism of these two line elements hµah = δab, (5) b ··· is missing; for finite regions the concept of direction does not exist. The theory put forward in the following is char- where δ =1 or δ = 0 depending on whether the two in- acterized by the introduction, in addition to the Riemann dices are equal or different. The correctness of (4) and metric, of a “direction,” or of equality of direction, or of (5) follows from the above definition of hνa as normal- ν “parallelism” for finite distances. Correspondingly, new ized subdeterminants of ha. The vector character of hνa invariants and tensors will appear in addition to those of follows most easily from the fact that the left hand side, Riemann geometry. and hence also the right hand side, of (1a) is invariant un- der arbitrary coordinate transformations for any choice of the vector (A). 2 ν I. n-BEIN AND METRIC The n-Bein field is determined by n functions ha, n(n+1) while the Riemann metric is determined by merely 2 At the arbitrary point P of the n-dimensional con- quantities gµν . According to (3) the metric is given by tinuum erect an orthogonal n-Bein from n unit vectors the n-Bein field, but not vice versa. representing an orthogonal coordinate system. Let Aa be the components of a line element, or of any other vector, w.r.t. this local system (n-Bein). For the description of II. DISTANT PARALLELISM AND ROTATIONAL a finite region introduce furthermore the Gaussian coor- INVARIANCE dinate system xν . Let Aν be the ν-components of the ν vector (A) w.r.t. the latter, furthermore let ha be the By positing the n-Bein field, the existence of the Rie- ν components of the unit vectors that form the n-Bein. mann metric and of distant parallelism are expressed si- Then13 multaneously. If (A) and (B) are two vectors at the

ν ν points P and Q respectively, which w.r.t. the local n- A = h Aa . (1) a ··· Beins have equal local coordinates (i.e. Aa = Ba) they are to be regarded as equal (on account of (2)) and as By inverting (1) and calling h the normalized sub- νa “parallel.” determinants (cofactor) of h ν we obtain a If we consider only the essential, i.e. the objectively Aa = hµAµ . (1a) meaningful, properties to be the metric and distant par- a ··· allelism, we recognize that the n-Bein field is not yet com- The magnitude A of the vector (A), on account of the pletely determined by these demands. Both the metric Euclidian property of the infinitesimal neighborhoods, is and distant parallelism remain intact if one replaces the n-Beins of all points of the continuum by others which result from the original ones by a common rotation. We call this replaceability of the n-Bein field rotational in- 13 We use Greek letters for the coordinate indices, Latin letters for variance and assume: Only rotationally invariant math- Bein indices. ematical relationships can have real meaning. Keeping a fixed coordinate system, and given a metric Besides this parallel transport law there is another µ as well as a distant parallelism relationship, the ha are (nonintegrable) symmetrical law of transport that be- ν not yet fully determined; a substitution of the ha is still longs to the Riemann metric according to (2) and (3). possible which corresponds to rotational invariance, i.e. It is given by the well-known equations the equation dAν = Γν Aµdxτ − µσ A∗ = d A (6) a am m  (8) ··· ν 1 νa ∂gµα ∂gτα ∂gµσ Γµσ = 2 g ∂xτ + ∂xµ ∂xα .  where the dam is chosen to be orthogonal and indepen- −   dent of the coordinates. (Aa) is an arbitrary vector w.r.t. ν The Γµσ symbols are given in terms of the n-Bein field h the local coordinate system; (Aa∗) is the same one in terms according to (3). It should be noted that of the rotated local system. According to (1a), equation (6) yields gµν = hµhν . (9) a a ··· µ µ Equations (4) and (5) imply hµa∗ A = damhµmA µλ µ or g gνλ = δν µν which defines g in terms of gµν . This law of transport h∗ = damhµm, (6a) µa ··· based on the metric is of course also rotationally invariant in the sense defined above. where

damdbm = dmadmb = δab, (6b) ··· III. INVARIANTS AND COVARIANTS ∂d am = 0. (6c) ∂xν ··· In the manifold we have been studying, there exist, in addition to the tensors and invariants of Riemann geom- The assumption of rotational invariance then requires etry, which contain the quantities h only in the combi- that equations containing h are to be regarded as mean- nations given by (3), further tensors and invariants, of ingful only if they retain their form when they are ex- which we want to consider only the simplest. pressed in terms of h according to (6). Or: n-Bein fields ∗ Starting from a vector (Aν ) at the point (xν ), the two related by local uniform rotations are equivalent. The transports d and d¯ to the neighboring point (xν + dxν ) law of infinitesimal parallel transport of a vector in going result in the two vectors from a point (xν ) to a neighboring point (xν + dxν ) is evidently characterized by the equation Aν + dAν

dAa =0 (7) and ··· ν ν which is to say the equation A + dA .

ν ∂hµa µ τ µ The difference 0= d(hµaA )= τ A dx + hµadA =0. ∂x dAν dAν = (Γν ∆ν )Aαdxβ − αβ − αβ Multiplying by hν and using (5), this equation becomes a is also a vector. Hence dAν = ∆ν Aµdxτ ν ν − µσ Γαβ ∆αβ where (7a) − ν νa ∂hµa  is a tensor, and so is its antisymmetric part ∆µσ = h ∂xσ .  1 ν ν ν This parallel transport law is rotationally invariant and is (∆αβ ∆βα)=Λαβ. ν 2 − ··· unsymmetrical with respect to the lower indices of ∆µσ. If the vector (A) is moved along a closed path accord- The fundamental meaning of this tensor in the theory ing to this law, it returns to itself; this means that the here developed emerges from the following: If this tensor Riemann tensor R, defined in terms of the transport co- vanishes, the continuum is Euclidian. For if efficients ∆ν , µσ ∂h ∂h 0=2Λν = ha αa + βa , (10) i i αβ ∂xβ ∂xα i ∂∆kl ∂∆km i α i α   Rk,lm = m + l + ∆αl∆km ∆αm∆kl − ∂x ∂x − then multiplication by hνb yields will vanish identically because of (7a)—as can be verified ∂h ∂h 0= αb + βb . easily. ∂xβ ∂xα

2 We can therefore put with the invariant volume element ∂Ψ h = b . h dτ, αb ∂xα h h dτ The field is therefore derivable from n scalars Ψ . We where is the determinant of µa , and is the product b dx . . . dx | | choose the coordinates according to the equation 1 n. The assumption

b δJ Ψb = x . (11) =0

ν Then, according to (7a) all ∆αβ vanish, and the hµa as yields 16 differential equations for the 16 values of hµa. well as the gµν are constant. Whether one can get physically meaningful laws in this ν Since the tensor Λαβ is evidently also formally the sim- way will be investigated later. plest one allowed by our theory, the simplest character- It is helpful to compare Weyl’s modification of Rie- ν ization of the continuum will be tied to Λαβ, not to the mann’s theory with the theory developed here: more complicated Riemann curvature tensor. The sim- plest forms that can come into play here are the vector WEYL: Comparison neither of distant vector magnitudes nor of directions; α Λµα RIEMANN: Comparison of distant vector as well as the invariants magnitudes, but not of distant directions; µν α β ασ βτ µ ν g ΛµβΛνα and gµν g g Λ Λστ . αβ THIS THEORY: Comparison of distant vec- From one of the latter (or from linear combinations) an tor magnitude and directions. invariant integral J can be constructed by multiplication

3 Einstein’s 1928 manuscript on unification of gravity and electromagnetism New possibility for a unified field theory of gravity and electricity

A. Einstein June 14, 1928

µν α A few days ago I explained in a short paper in these where the quantities h (= det hµα), g , and Λµν are Reports how it is possible to use an n-Bein-Field to for- defined in (9) and (10) of the previous paper. mulate a geometric theory based on the fundamental Let the h field describe the electrical and the gravita- concepts of the Riemann metric and distant parallelism. tional field simultaneously. A “purely gravitational field” At the time I left open the question whether this the- results when equation (1) is fulfilled and, in addition, ory could serve to represent physical relationships. Since then I have discovered that this theory—at least in first α φµ =Λµα (2) approximation—yields the field equations of gravity and ··· electromagnetism very simply and naturally. It is there- vanish, which represents a covariant and rotationally in- fore conceivable that this theory will replace the original variant subsidiary condition.15 version of the theory of relativity. The introduction of distant parallelism implies that ac- cording to this theory there is something like a straight line, i.e. a line whose elements are all parallel to each II. THE FIELD EQUATION IN THE FIRST APPROXIMATION other; of course such a line is in no way identical to a geodesic. Furthermore, in contrast to the usual general theory of relativity, there is the concept of relative rest of If the manifold is the Minkowski world of special rela- two mass points (parallelism of two line elements which tivity, one can choose the coordinates in such a way that √ belong to two different worldlines.) h11 = h22 = h33 = 1,h44 = j (= 1), and that all h − In order for the general theory to be useful immediately other ’s vanish. This set of values is somewhat inconve- as field theory one must assume the following: nient for calculation. For that reason we prefer to choose the x coordinate in this to be pure imaginary; in that 4 § 1. The number of dimensions is 4 (n = 4). case the Minkowski world (absence of any field when the coordinates are chosen appropriately) can be described 2. The fourth local component Aa (a = 4) of a vector by is pure imaginary, and hence so are the components µ of the four legs of the Vier-Bein, the quantities h 4 hµa = δµa (3) 14 ··· and hµ4. The case of infinitely weak fields can be represented suit- The coefficients gµν (= hµαhνα) of course all become real. ably by Accordingly, we choose the square of the magnitude of a timelike vector to be negative. hµa = δµa + kµα (4) ···

where the kµα are small quantities of first order. Neglect- I. THE UNDERLYING FIELD EQUATION ing quantities of third or higher order we must replace (1a) by (1b), considering (10) and (7a) of the previous Let the variation of a Hamiltonian integral vanish for paper: µ variations of the field potentials hµα (or hα ) that vanish on the boundary of a domain: 1 ∂k ∂k ∂k ∂k H = µα βα µβ αβ . (1b) −4 ∂x − ∂x ∂x − ∂x ···  β µ   α µ  δ Hdτ =0. (1) ··· Z  After variation one obtains the field equations in the first approximation H hgµν α β , = Λµβ Λν α (1a) ∂2k ∂2k ∂2k ∂2k ··· βα µα αµ βµ . 2 + =0 (5) ∂xµ − ∂xµ∂xβ ∂xβ∂xµ − ∂xµ∂xα ···

14 Instead one could also define the square of the magnitude of the local vector A to be A2 + A2 + A2 − A2 and introduce Lorentz 1 2 3 4 15 transformations instead of rotations of the local n-Bein. In that Here there remains a certain ambiguity of interpretation, because case all the h’s would be real, but the immediate connection with one could also characterize the pure gravitational field by the ∂φ the general theory would be lost. vanishing of µ − ∂φν . ∂xν ∂xµ 16 These are 16 equations for the 16 components kαβ . Our (where Rαβ is the once reduced Riemann tensor.) Thus task now is to see whether this system of equations con- it is proved that our new theory correctly reproduces the tains the known laws of the gravitational and electromag- law of the pure gravitational field in the first approxima- netic fields. For this purpose we must introduce gαβ and tion. φα in (5) in place of kαβ. We must put By differentiating (2a) with respect to xα and taking into account the equation obtained from (5) by reducing gαβ = hαahβa = (δαa + kαa)(δβa + kβa). with respect to α and β one obtains

Or, exact to first order, ∂φ α =0. (8) ∂xα ··· gαβ δαβ = gαβ = kαβ + kβα. (6) − ··· Noting that the left side Lαβ of (7) obeys the identity From (2) one also gets the quantities to first order exactly ∂ 1 Lαβ δαβLσσ =0, ∂x − 2 ∂kαµ ∂kµµ β   2φα = (2a) ∂x − ∂x µ α we find from (7) that By exchanging α and β in (5) and adding to (5) one gets ∂2φ ∂2φ ∂ ∂φ α + β σ =0 2 2 2 2 ∂ g ∂ k ∂ k ∂x ∂xα∂xβ − ∂xα ∂xσ αβ µα µβ =0. β   ∂x2 − ∂x ∂x − ∂x ∂x µ µ β µ α or

If one adds to this equation the two following equations 2 ∂ φα which follow from (2a) 2 =0. (9) ∂xβ ··· 2 2 ∂ kαµ ∂ kµµ ∂φα + = 2 Equations (8) and (9) together are known to be equiv- −∂x ∂x ∂x ∂x − ∂x µ β β α β alent to Maxwell’s equations for empty space. The new ∂2k ∂2k ∂φ βµ + µµ = 2 β , theory therefore yields Maxwell’s equations in first ap- −∂xµ∂xα ∂xα∂xβ − ∂xα proximation. The separation of the gravitational field from the elec- then one obtains, in view of (6), tromagnetic field appears artificial according to this the- ory. And it is clear that equations (5) imply more than 1 ∂2g ∂2g ∂2g ∂2g αβ + µα + µβ µµ (7), (8), and (9) together. Furthermore, it is remarkable 2 − ∂x2 ∂x ∂x ∂x ∂x − ∂x ∂x  µ µ β µ α α β  that in this theory the electric field does not enter the ∂φ ∂φ field equations quadratically. = α + β . (7) ∂xβ ∂xα ··· Note added in proof: One obtains very similar results by starting with the Hamiltonian The case of vanishing electromagnetic fields is character- ασ βτ µ ν ized by the vanishing of φα. In that case, (7) agrees to H = hgµν g g ΛαβΛστ . first order with the equation of General Relativity There is therefore at this time a certain uncertainty with Rαβ =0 respect to the choice of H.

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