Einstein's vierbein field theory of curved space
aa r X i v : . [ g r- q c ] J un Einstein’s vierbein field theory of curved space
Jeffrey Yepez
Air Force Research Laboratory, Hanscom Air Force Base, MA 01731 (Dated: January 20, 2008)
General Relativity theory is reviewed following the vierbein field theory approach proposed in1928 by Einstein. It is based on the vierbein field taken as the “square root” of the metric tensorfield. Einstein’s vierbein theory is a gauge field theory for gravity; the vierbein field playing therole of a gauge field but not exactly like the vector potential field does in Yang-Mills theory–thecorrection to the derivative (the covariant derivative) is not proportional to the vierbein field asit would be if gravity were strictly a Yang-Mills theory. Einstein discovered the spin connectionin terms of the vierbein fields to take the place of the conventional affine connection. To date,one of the most important applications of the vierbein representation is for the derivation of thecorrection to a 4-spinor quantum field transported in curved space, yielding the correct form ofthe covariant derivative. Thus, the vierbein field theory is the most natural way to represent arelativistic quantum field theory in curved space. Using the vierbein field theory, presented is aderivation of the the Einstein equation and then the Dirac equation in curved space. Einstein’soriginal 1928 manuscripts translated into English are included.
Keywords: vierbein, general relativity, gravitational gauge theory, Dirac equation in curved space
Contents
I. Introduction
II. Mathematical framework
III. Connections
IV. Curvature
V. Mathematical constructs
VI. Gravitational action
VII. Einstein’s action
VIII. Relativistic chiral matter in curved space
IX. Conclusion X. Acknowledgements References Appendices Einstein’s 1928 manuscript on distant parallelism I. n -Bein and metric II. Distant parallelism and rotational invariance III. Invariants and covariants Einstein’s 1928 manuscript on unification ofgravity and electromagnetism I. The underlying field equation II. The field equation in the first approximation INTRODUCTION
I. INTRODUCTION
The purpose of this manuscript is to provide a self-contained review of the procedure for deriving the Ein-stein equations for gravity and the Dirac equation incurved space using the vierbein field theory. This gaugefield theory approach to General Relativity (GR) wasdiscovered by Einstein in 1928 in his pursuit of a unifiedfield theory of gravity and electricity. He originally pub-lished this approach in two successive letters appearingone week apart (Einstein, 1928a,b). The first manuscript,a seminal contribution to mathematical physics, addsthe concept of distant parallelism to Riemann’s theoryof curved manifolds that is based on comparison of dis-tant vector magnitudes, which before Einstein did notincorporate comparison of distant directions.Historically there appears to have been a lack of in-terest in Einstein’s research following his discovery ofgeneral relativity, principally from the late 1920’s on-ward. Initial enthusiasm for Einstein’s unification ap-proach turned into a general rejection. In Born’s July15th, 1925 letter (the most important in the collection) toEinstein following the appearance of his student Heisen-berg’s revolutionary paper on the matrix representationof quantum mechanics, Born writes (Born, 1961):Einstein’s field theory . . . was intended tounify electrodynamics and gravitation . . . . Ithink that my enthusiam about the successof Einstein’s idea was quite genuine. In thosedays we all thought that his objective, whichhe pursued right to the end of his life, wasattainable and also very important. Many ofus became more doubtful when other types offields emerged in physics, in addition to these;the first was Yukawa’s meson field, which isa direct generalization of the electromagneticfield and describes nuclear forces, and thenthere were the fields which belong to the otherelementary particles. After that we were in-clined to regard Einstein’s ceaseless efforts asa tragic error.Weyl and Pauli’s rejection of Einstein’s thesis of distantparallelism also helped paved the way for the view thatEinstein’s findings had gone awry. Furthermore, as thebelief in the fundamental correctness of quantum theorysolidified by burgeoning experimental verifications, thetheoretical physics community seemed more inclined tolatch onto Einstein’s purported repudiation of quantummechanics: he failed to grasp the most important direc-tion of twentieth century physics.Einstein announced his goal of achieving a unified fieldtheory before he published firm results. It is already hardnot to look askance at an audacious unification agenda,but it did not help when the published version of themanuscript had a fundamental error in its opening equa-tion; even though this error was perhaps introduced by the publisher’s typist, it can cause confusion. As faras I know at the time of this writing in 2008, the two1928 manuscripts have never been translated into En-glish. English versions of these manuscripts are providedas part of this review–see the Appendix–and are includedto contribute to its completeness.In the beginning of the year 1928, Dirac introducedhis famous square root of the Klein-Gordon equation, es-tablishing the starting point for the story of relativisticquantum field theory, in his paper on the quantum theoryof the electron (Dirac, 1928). This groundbreaking pa-per by Dirac may have inspired Einstein, who completedhis manuscripts a half year later in the summer of 1928.With deep insight, Einstein introduced the vierbein field,which constitutes the square root of the metric tensorfield. Einstein and Dirac’s square root theories math-ematically fit well together; they even become joined atthe hip when one considers the dynamical behavior ofchiral matter in curved space.Einstein’s second manuscript represents a simple andintuitive attempt to unify gravity and electromagnetism.He originally developed the vierbein field theory ap-proach with the goal of unifying gravity and quantumtheory, a goal which he never fully attained with thisrepresentation. Nevertheless, the vierbein field theoryapproach represents progress in the right direction. Ein-stein’s unification of gravity and electromagnetism, us-ing only fields in four-dimensional spacetime, is concep-tually much simpler than the well known Kaluza-Kleinapproach to unification that posits an extra compactifiedspatial dimension. But historically it was the Kaluza-Klein notion of extra dimensions that gained popularityas it was generalized to string theory. In contradistinc-tion, Einstein’s approach requires no extra constructs,just the intuitive notion of distant parallelism. In theEinstein vierbein field formulation of the connection andcurvature, the basis vectors in the tangent space of aspacetime manifold are not derived from any coordinatesystem of that manifold.Although Einstein is considered one of the foundingfathers of quantum mechanics, he is not presently consid-ered one of the founding fathers of relativistic quantumfield theory in flat space. This is understandable since inhis theoretical attempts to discover a unified field theoryhe did not predict any of the new leptons or quarks, northeir weak or strong gauge interactions, in the StandardModel of particle physics that emerged some two decades The opening equation (1a) was originally typeset as H = h g µν , Λ µαβ , Λ νβα , · · · offered as the Hamiltonian whose variation at the end of the dayyields the Einstein and Maxwell’s equations. I have correctedthis in the translated manuscript in the appendix. The culmination of Einstein’s new field theory approach ap-peared in Physical Review in 1948 (Einstein, 1948), entitled “AGeneralized Theory of Gravitation.” INTRODUCTION following his passing. However, Einstein did employ localrotational invariance as the gauge symmetry in the first1928 manuscript and discovered what today we call the spin connection , the gravitational gauge field associatedwith the Lorentz group as the local symmetry group ( viz. local rotations and boosts).This he accomplished about three decades beforeYang and Mills discovered nonabelian gauge the-ory (Yang and Mills, 1954), the antecedent to theGlashow-Salam-Weinberg electroweak unification theory(Glashow, 1961; Salam, 1966; Weinberg, 1967) that isthe cornerstone of the Standard Model. Had Einstein’swork toward unification been more widely circulated in-stead of rejected, perhaps Einstein’s original discovery of n -component gauge field theory would be broadly consid-ered the forefather of Yang-Mills theory. In Section I.A,I sketch a few of the strikingly similarities between thevierbein field representation of gravity and the Yang-Mills nonabelian gauge theory.With the hindsight of 80 years of theoretical physicsdevelopment from the time of the first appearance of Ein-stein’s 1928 manuscripts, today one can see the historicalrejection is a mistake. Einstein could rightly be consid-ered one of the founding fathers of relativistic quantumfield theory in curved space, and these 1928 manuscriptsshould not be forgotten. Previous attempts have beenmade to revive interest in the vierbein theory of gravita-tion purely on the grounds of their superior use for repre-senting GR, regardless of unification (Kaempffer, 1968).Yet, this does not go far enough. One should also makethe case for the requisite application to quantum fieldsin curved space.Einstein is famous for (inadvertently) establishing an-other field of contemporary physics with the discoveryof distant quantum entanglement. Nascent quantum in-formation theory was borne from the seminal 1935 Ein-stein, Podolsky, and Rosen (EPR) Physical Review pa-per , “Can Quantum-Mechanical Description of PhysicalReality Be Considered Complete?” Can Einstein equallybe credited for establishing the field of quantum gravity,posthumously?The concepts of vierbein field theory are simple, andthe mathematical development is straightforward, but inthe literature one can find the vierbein theory a nota-tional nightmare, making this pathway to GR appearmore difficult than it really is, and hence less accessible.In this manuscript, I hope to offer an accessible path-way to GR and the Dirac equation in curved space. Thedevelopment of the vierbein field theory presented hereborrows first from treatments given by Einstein himselfthat I have discussed above (Einstein, 1928a,b), as wellas excellent treatments by Weinberg (Weinberg, 1972) Einstein treated the general case of an n -Bein field. This is the most cited physics paper ever and thus a singular ex-ception to the general lack of interest in Einstein’s late research. and Carroll (Carroll, 2004).
However, Weinberg andCarroll review vierbein theory as a sideline to their mainapproach to GR, which is the standard coordinate-basedapproach of differential geometry. An excellent treat-ment of quantum field theory in curved space is given byBirrell and Davies (Birrell and Davies, 1982), but againwith a very brief description of the vierbein representa-tion of GR. Therefore, it is hoped that a self-containedreview of Einstein’s vierbein theory and the associatedformulation of the relativistic wave equation should behelpful gathered together in one place.
A. Similarity to Yang-Mills gauge theory
This section is meant to be an outline comparingthe structure of GR and Yang-Mills (YM) theories(Yang and Mills, 1954). There are many previous treat-ments of this comparison—a recent treatment by Jackiwis recommended (Jackiw, 2005). The actual formal re-view of the vierbein theory does not begin until Sec-tion II.The dynamics of the metric tensor field in GR canbe cast in the form of a YM gauge theory used to de-scribe the dynamics of the quantum field in the StandardModel. In GR, dynamics is invariant under an externallocal transformation, say Λ, of the Lorentz group SO(3,1)that includes rotations and boosts. Furthermore, anyquantum dynamics occurring within the spacetime man-ifold is invariant under internal local Lorentz transforma-tions, say U Λ , of the spinor representation of the SU(4)group. Explicitly, the internal Lorentz transformation ofa quantum spinor field in unitary form is U Λ = e − i ω µν ( x ) S µν , (1)where S µν is the tensor generator of the transformation. Similarly, in the Standard Model, the dynamics of theDirac particles (leptons and quarks) is invariant underlocal transformations of the internal gauge group, SU(3)for color dynamics and SU(2) for electroweak dynamics. The internal unitary transformation of the multiple com- Both Weinberg and Carroll treat GR in a traditional manner.Their respective explanations of Einstein’s vierbein field theoryare basically incidental. Carroll relegates his treatment to a sin-gle appendix. Another introduction to GR but which does not deal withthe vierbein theory extensively is the treatment by D’Inverno(D’Inverno, 1995). A 4 × − γ µ , six tensors σ µν = i [ γ µ , γ ν ], one pseudo scalar iγ γ γ γ ≡ γ , and fouraxial vectors γ γ µ . The generator associated with the internalLorentz transformation is S µν = σ µν . A 3 × − × − Similarity to Yang-Mills gauge theory II MATHEMATICAL FRAMEWORK ponent YM field in unitary form is U = e i Θ a ( x ) T a , (2)where the hermitian generators T a = T a † are in the ad-joint representation of the gauge group. The commonunitarity of (1) and (2) naturally leads to many parallelsbetween the GR and YM theories.From (1), it seems that ω µν ( x ) should take the placeof the gauge potential field, and in this case it is clearlya second rank field quantity. But in Einstein’s action inhis vierbein field representation of GR, the lowest orderfluctuation of a second-rank vierbein field itself e µa ( x ) = δ µa + k µa ( x ) + · · · plays the roles of the potential field where the gravita-tional field strength F µνa is related to a quantity of theform F µνa = ∂ µ k νa ( x ) − ∂ ν k µa ( x ) . (3)(3) vanishes for apparent or pseudo-gravitational fieldsthat occur in rotating or accelerating local inertial framesbut does not vanish for gravitational fields associatedwith curved space. Einstein’s expression for the La-grangian density (that he presented as a footnote in hissecond manuscript) gives rise to a field strength of theform of (3). An expanded version of his proof of theequivalence of his vierbein-based action principle withgravity in the weak field limit is presented in Section VII.In any case, a correction to the usual derivative ∂ µ → D µ ≡ ∂ µ + Γ µ is necessary in the presence of a gravitational field. Thelocal transformation has the form ψ → U Λ ψ (4a)Γ µ → U Λ Γ µ U † Λ − ( ∂ µ U Λ ) U † Λ . (4b)(4) is derived in Section VIII.B. This is similar to Yang-Mills gauge theory where a correction to the usual deriva-tive ∂ µ → D µ ≡ ∂ µ − iA µ is also necessary in the presence of a non-vanishing gaugefield. In YM, the gauge transformation has the form ψ → U ψ (5a) A µ → U A µ U † − U ∂ µ U † , (5b)which is just like (4). Hence, this “gauge field theory” ap-proach to GR is useful in deriving the form of the Diracequation in curved space. In this context, the require-ment of invariance of the relativistic quantum wave equa-tion to local Lorentz transformations leads to a correctionof the form Γ µ = 12 e βk ( ∂ µ e βh ) S hk (6a)= ∂ µ (cid:18) k βh S hβ (cid:19) + · · · (6b) This is derived in Section VIII. A problem that is com-monly cited regarding the gauge theory representation ofGeneral Relativity is that the correction is not directlyproportional to the gauge potential as would be the caseif it were strictly a YM theory ( e.g. we should be ableto write Γ µ ( x ) = u a k µa ( x ) where u a is some constantfour-vector). II. MATHEMATICAL FRAMEWORKA. Local basis
The conventional coordinate-based approach to GRuses a “natural” differential basis for the tangent space T p at a point p given by the partial derivatives of thecoordinates at p ˆ e ( µ ) = ∂ ( µ ) . (7)Some 4-vector A ∈ T p has components A = A µ ˆ e ( µ ) = ( A , A , A , A ) . (8)To help reinforce the construction of the frame, we usea triply redundant notation of using a bold face symbolto denote a basis vector e , applying a caret symbol ˆ e as a hat to denote a unit basis vector, and enclosing thecomponent subscript with parentheses ˆ e ( µ ) to denote acomponent of a basis vector. It should be nearly im-possible to confuse a component of an orthonormal basisvector in T p with a component of any other type of ob-ject. Also, the use of a Greek index, such as µ , denotes acomponent in a coordinate system representation. Fur-thermore, the choice of writing the component index asa superscript as in A µ is the usual convention for indi-cating this is a component of a contravariant vector. Acontravariant vector is often just called a vector, for sim-plicity of terminology.In the natural differential basis, the cotangent space,here denoted by T ∗ p , is spanned by the differential ele-ments ˆ e ( µ ) = dx ( µ ) , (9)which lie in the direction of the gradient of the coordinatefunctions. T ∗ p is also called the dual space of T p . Some With ˆ e ( • ) ∈ { e , e , e , e } and ∂ ( • ) ∈ { ∂ , ∂ , ∂ , ∂ } , some au-thors write (7) concisely as e µ = ∂ µ . I will not use this notation because I would like to reserve e µ to represent the lattice vectors e µ ≡ γ a e µa where γ a are Diracmatrices and e µa is the vierbein field defined below. Again, with ˆ e ( • ) ∈ { e , e , e , e } and dx ( • ) ∈{ dx , dx , dx , dx } , for brevity some authors write (9) as e µ = dx µ . But I reserve e µ to represent anti-commuting 4-vectors, e µ ≡ γ a e µa , unless otherwise noted. Vierbein field II MATHEMATICAL FRAMEWORK dual 4-vector A ∈ T ∗ p has components A = A µ ˆ e ( µ ) = g µν A ν e ( µ ) . (10)Writing the component index µ as a subscript in A µ againfollows the usual convention for indicating one is dealingwith a component of a covariant vector. Again, in an at-tempt to simplify terminology, a covariant vector is oftencalled a 1-form, or simply a dual vector. Yet, remem-bering that a vector and dual vector (1-form) refer to anelement of the tangent space T p and the cotangent space T ∗ p , respectively, may not seem all that much easier thanremembering the prefixes contravariant and covariant inthe first place.The dimension of (7) is inverse length, [ˆ e ( µ ) ] = L , andthis is easy to remember because a first derivative of afunction is always tangent to that function. For a basiselement, µ is a subscript when L is in the denomina-tor. Then (9), which lives in the cotangent space and asthe dimensional inverse of (7), must have dimensions oflength [ˆ e ( µ ) ] = L . So, for a dual basis element, µ is asuperscript when L is in the numerator. That they aredimensional inverses is expressed in the following tensorproduct space ˆ e ( µ ) ⊗ ˆ e ( ν ) = µν , (11)where is the identity, which is of course dimensionless.We are free to choose any orthonormal basis we like tospan T p , so long as it has the appropriate signature ofthe manifold on which we are working. To that end, weintroduce a set of basis vectors ˆ e ( a ) , which we choose asnon-coordinate unit vectors, and we denote this choice byusing small Latin letters for indices of the non-coordinateframe. With this understanding, the inner product maybe expressed as (cid:0) ˆ e ( a ) , ˆ e ( b ) (cid:1) = η ab , (12)where η ab = diag(1 , − , − , −
1) is the Minkowski metricof flat spacetime.
B. Vierbein field
This orthonormal basis that is independent of the co-ordinates is termed a tetrad basis . Although we can-not find a coordinate chart that covers the entire curvedmanifold, we can choose a fixed orthonormal basis thatis independent of position. Then, from a local perspec-tive, any vector can be expressed as a linear combinationof the fixed tetrad basis vectors at that point. Denotingan element of the tetrad basis by ˆ e ( a ) , we can express To help avoid confusion, please note that the term tetrad in theliterature is often used as a synonym for the term vierbein . Herewe use the terms to mean two distinct objects: ˆ e ( a ) and e µ ( x ),respectively. the coordinate basis (whose value depends on the localcurvature at a point x in the manifold) in terms of thetetrads as the following linear combinationˆ e ( µ ) ( x ) = e µa ( x ) ˆ e ( a ) , (13)where the functional components e µa ( x ) form a 4 × e µa ( x ). Thevierbeins e µa ( x ), for a = 1 , , ,
4, comprise four legs– vierbein in German means four-legs.The inverse of the vierbein has components, e µa (switched indices), that satisfy the orthonormality con-ditions e µa ( x ) e ν a ( x ) = δ µν , e µa ( x ) e µb ( x ) = δ ab . (14)The inverse vierbein serves as a transformation matrixthat allows one to represent the tetrad basis ˆ e ( a ) ( x ) interms of the coordinate basis ˆ e ( µ ) :ˆ e ( a ) = e µa ( x ) ˆ e ( µ ) . (15)Employing the metric tensor g µν to induce the productof the vierbein field and inverse vierbein field, the innerproduct-signature constraint is g µν ( x ) e µa ( x ) e νb ( x ) = η ab , (16)or using (14) equivalently we have g µν ( x ) = e µa ( x ) e ν b ( x ) η ab . (17)So, the vierbein field is the “square root” of the metric.Hopefully, you can already see why one should includethe vierbein field theory as a member of our tribe of“square root” theories. These include the Pythagoreantheorem for the distance interval ds = p η µν dx µ dx ν = p dt − dx − dy − dz , the mathematicians’ belovedcomplex analysis (based on √− √ swap conservative quan-tum logic gate. To this august list we add the vierbeinas the square root of the metric tensor.Now, we may form a dual orthonormal basis, whichwe denote by ˆ e ( a ) with a Latin superscript, of 1-forms inthe cotangent space T ∗ p that satisfies the tensor productcondition ˆ e ( a ) ⊗ ˆ e ( b ) = ab . (18)This non-coordinate basis 1-form can be expressed as alinear combination of coordinate basis 1-formsˆ e ( a ) = e µa ( x ) ˆ e ( µ ) ( x ) , (19)5 Vierbein field III CONNECTIONS where ˆ e ( µ ) = dx µ , and vice versa using the inverse vier-bein field ˆ e ( µ ) ( x ) = e µa ( x ) ˆ e ( a ) . (20)Any vector at a spacetime point has components in thecoordinate and non-coordinate orthonormal basis V = V µ ˆ e ( µ ) = V a ˆ e ( a ) . (21)So, its components are related by the vierbein field trans-formation V a = e µa V µ and V µ = e µa V a . (22)The vierbeins allow us to switch back and forth betweenLatin and Greek bases.Multi-index tensors can be cast in terms of mixed-index components, as for example V ab = e µa V µb = e νb V aν = e µa e νb V µν . (23)The behavior of inverse vierbeins is consistent with theconventional notion of raising and lowering indices. Hereis an example with the metric tensor field and theMinkowski metric tensor e µa = g µν η ab e νb . (24)The identity map has the form e = e νa dx ( ν ) ⊗ ˆ e ( a ) . (25)We can interpret e νa as a set of four Lorentz 4-vectors.That is, there exists one 4-vector for each non-coordinateindex a .We can make local Lorentz transformations (LLT) atany point. The signature of the Minkowski metric is pre-served by a Lorentz transformationLLT: ˆ e ( a ) → ˆ e ( a ′ ) = Λ aa ′ ( x )ˆ e ( a ) , (26)where Λ aa ′ ( x ) is an inhomogeneous ( i.e. position depen-dent) transformation that satisfiesΛ aa ′ Λ bb ′ η ab = η a ′ b ′ . (27)A Lorentz transformation can also operate on basis 1-forms, in contradistinction to the ordinary Lorenz trans-formation Λ a ′ a that operates on basis vectors. Λ a ′ a transforms upper (contravariant) indices, while Λ aa ′ transforms lower (covariant) indices.And, we can make general coordinate transformations(GCT)GCT: T aµbν → T a ′ µ ′ b ′ ν ′ = Λ a ′ a |{z} prime1st(contra-variant) ∂x µ ′ ∂x µ Λ bb ′ |{z} prime2nd(co-variant) ∂x ν ∂x ν ′ T aµbν . (28) III. CONNECTIONSA. Affine connection
Curvature of a Riemann manifold will cause a distor-tion in a vector field, say a coordinate field X α ( x ), andthis is depicted in Figure 1. The change in the coordinate ✻ ✻✁✁✁✁✁✁✕❍❍❍❥ i jδx α X α + ¯ δX α X α ( x ) X α + δX α Γ αβγ X β δX γ rr FIG. 1
Two spacetime points x α and x α + δx α , labeled as i and j , respectively. The 4-vector at point i is X α ( x ), and the 4-vectorat nearby point j is X α ( x + δx ) = X α ( x ) + δX α ( x ). The paralleltransported 4-vector at j is X α ( x ) + ¯ δX α ( x ) (blue). The affineconnection is Γ αβγ . (For simplicity the parallel transport is renderedas if the space is flat). field from one point x to an adjacent point x + δx is X α ( x + δx ) = X α ( x ) + δx β ∂ β X α | {z } δX α ( x ) . (29)So, the change of the coordinate field due to the manifoldis defined as δX α ( x ) ≡ δx β ( x ) ∂ β X α = X α ( x + δx ) − X α ( x ) . (30)The difference of the two coordinate vectors at point j is[ X α + δX α ] − [ X α + ¯ δX α ] = δX α ( x ) − ¯ δX α ( x ) . (31)¯ δX α must vanish if either δx α vanishes or X α vanishes.Therefore, we choose¯ δX α = − Γ αβγ ( x ) X β ( x ) δx γ , (32)where Γ αβγ is a multiplicative factor, called the affine con-nection. Its properties are yet to be determined. At thisstage, we understand it as a way to account for the cur-vature of the manifold.The covariant derivative may be constructed as follows: ∇ γ X α ( x ) ≡ δx γ { X α ( x + δx ) − [ X α ( x )+ ¯ δX α ( x )] } . (33)I do not use a limit in the definition to define the deriva-tive. Instead, I would like to just consider the situationwhere δx γ is a small finite quantity. We will see belowthat this quantity drops out, justifying the form of (33).Inserting (29) and (32) into (33), yields ∇ γ X α ( x ) = 1 δx γ { X α ( x ) + δx γ ∂ γ X α (34a) − X α ( x ) + Γ αβγ ( x ) X β ( x ) δx γ } = ∂ γ X α ( x ) + Γ αβγ ( x ) X β ( x ) . (34b)6 Spin connection III CONNECTIONS
So we see that δx γ cancels out and no limiting processto an infinitesimal size was really needed. Dropping theexplicit dependence on x , as this is to be understood, wehave the simple expression for the covariant derivative ∇ γ X α = ∂ γ X α + Γ αβγ X β . (35)In coordinate-based differential geometry, the covariantderivative of a tensor is given by its partial derivativeplus correction terms, one for each index, involving anaffine connection contracted with the tensor. B. Spin connection
In non-coordinate-based differential geometry, the or-dinary affine connection coefficients Γ λµν are replaced byspin connection coefficients, denoted ω µab , but otherwisethe principle is the same. Each Latin index gets a cor-rection factor that is the spin connection contracted withthe tensor, for example ∇ µ X ab = ∂ µ X ab + ω µac X cb − ω µcb X ac . (36)The correction is positive for a upper index and negativefor a lower index. The spin connection is used to takecovariant derivatives of spinors, whence its name.The covariant derivative of a vector X in the coordi-nate basis is ∇ X = ( ∇ µ X ν ) dx µ ⊗ ∂ ν (37a)= (cid:0) ∂ µ X ν + Γ νµλ X λ (cid:1) dx µ ⊗ ∂ ν . (37b)The same object in a mixed basis, converted to the co-ordinate basis, is ∇ X = ( ∇ µ X a ) dx µ ⊗ ˆ e ( a ) (38a)= (cid:0) ∂ µ X a + ω µab X b (cid:1) dx µ ⊗ ˆ e ( a ) (38b)= (cid:0) ∂ µ ( e νa X ν ) + ω µab e λb X λ (cid:1) dx µ ⊗ ( e σa ∂ σ ) (38c)= e σa (cid:0) e νa ∂ µ X ν + X ν ∂ µ e νa + ω µab e λb X λ (cid:1) dx µ ⊗ ∂ σ (38d)= (cid:0) ∂ µ X σ + e σa ∂ µ e νa X ν + e σa e λb ω µab X λ (cid:1) dx µ ⊗ ∂ σ . (38e)Now, relabeling indices σ → ν → λ gives ∇ X = (cid:0) ∂ µ X ν + e νa ∂ µ e λa X λ + e νa e λb ω µab X λ (cid:1) dx µ ⊗ ∂ ν = (cid:2) ∂ µ X ν + (cid:0) e νa ∂ µ e λa + e νa e λb ω µab (cid:1) X λ (cid:3) dx µ ⊗ ∂ ν . (39)Therefore, comparing (37b) with (39), the affine connec-tion in terms of the spin connection isΓ νµλ = e νa ∂ µ e λa + e νa e λb ω µab . (40)This can be solved for the spin connection ω µab = e νa e λb Γ νµλ − e λb ∂ µ e λa . (41) C. Tetrad postulate
The tetrad postulate is that the covariant derivative ofthe vierbein field vanishes, ∇ µ e νa = 0, and this is merelya restatement of the relation we just found between theaffine and spin connections (41). Left multiplying by e νb gives ω µab e νb = e σa e λb e νb Γ σµλ − e λb e νb ∂ µ e λa (42a)= e σa Γ σµν − ∂ µ e νa . (42b)Rearranging terms, we have the tetrad postulate ∇ µ e νa ≡ ∂ µ e νa − e σa Γ σµν + ω µab e νb = 0 . (43)Let us restate (as a reminder) the correction rules forapplying connections. The covariant derivatives of a co-ordinate vector and 1-form are ∇ µ X ν = ∂ µ X ν + Γ νµλ X λ (44a) ∇ µ X ν = ∂ µ X ν − Γ λµν X λ , (44b)and similarly the covariant derivatives of a non-coordinate vector and 1-form are ∇ µ X a = ∂ µ X a + ω µab X b (45a) ∇ µ X a = ∂ µ X a − ω µba X b . (45b)We require a covariant derivative such as (45a) to beLorentz invariantΛ a ′ a : ∇ µ X a → ∇ µ (cid:16) Λ a ′ a X a (cid:17) (46a)= (cid:16) ∇ µ Λ a ′ a (cid:17) X a + Λ a ′ a ∇ µ X a . (46b)Therefore, the covariant derivative is Lorentz invariant, ∇ µ X a = Λ a ′ a ∇ µ X a , (47)so long as the covariant derivative of the Lorentz trans-formation vanishes, ∇ µ Λ a ′ a = 0 . (48)This imposes a constraint that allows us to see how thespin connection behaves under a Lorentz transformation ∇ µ Λ a ′ b = ∂ µ Λ a ′ b + ω µa ′ c Λ cb − ω µcb Λ a ′ c = 0 , (49)which we write as followsΛ bb ′ ∂ µ Λ a ′ b + ω µa ′ c Λ bb ′ Λ cb − ω µcb Λ bb ′ Λ a ′ c = 0 . (50)Now Λ bb ′ Λ cb = δ cb ′ , so we arrive at the transformation ofthe spin connection induced by a Lorentz transformation ω µa ′ b ′ = ω µcb Λ bb ′ Λ a ′ c − Λ bb ′ ∂ µ Λ a ′ b . (51)This means that the spin connection transforms inhomo-geneously so that ∇ µ X a can transform like a Lorentz4-vector.7 V CURVATURE
The exterior derivative is defined as follows( dX ) µν a ≡ ∇ µ X νa − ∇ ν X µa (52a)= ∂ µ X νa + ω µab X νb − Γ λµν X λa − ∂ ν X µa − ω νab X µb + Γ λνµ X λa (52b)= ∂ µ X νa − ∂ ν X µa + ω µab X νb − ω νab X µb . (52c)Now, to make a remark about Cartan’s notation as writ-ten in (10), one often writes the non-coordinate basis1-form (19) as e a ≡ ˆ e ( a ) = e µa dx µ . (53)The spin connection 1-form is ω ab = ω µab dx µ . (54)It is conventional to define a differential form dA ≡ ∂ µ A ν − ∂ ν A µ (55)and a wedge product A ∧ B ≡ A µ B ν − A ν B µ , (56)which are both anti-symmetric in the Greek indices.With this convention, originally due to ´Elie Cartan, thetorsion can be written concisely in terms of the frameand spin connection 1-forms as T a = de a + ω ab ∧ e b . (57)The notation is so compact that it is easy to misunder-stand what it represents. For example, writing the tor-sion explicitly in the coordinate basis we have T µνλ = e λa T µνa (58a)= e λa (cid:0) ∂ µ e νa − ∂ ν e µa + ω µab e νb − ω νab e µb (cid:1) , (58b)which fully expanded gives us T µνλ = e λa ∂ µ e νa + e λa e νb ω µab − e λa ∂ ν e µa − e λa e µb ω νab . (59)Since the affine connection isΓ λµν = e λa ∂ µ e νa + e λa e νb ω µab , (60) the torsion then reduces to the simple expression T µνλ = Γ λµν − Γ λνµ . (61)So, the torsion vanishes when the affine connection issymmetric in its lower two indices. IV. CURVATURE
We now derive the Riemann curvature tensor, and wedo so in two ways. The first way gives us an expres-sion for the curvature in terms of the affine connectionand the second way gives us an equivalent expression interms of the spin connection. The structure of both ex-pressions are the same, so effectively the affine and spinconnections can be interchanged, as long as one properlyaccounts for Latin and Greek indices.
A. Riemann curvature from the affine connection
In this section, we will derive the Riemann curvaturetensor in terms of the affine connection. The develop-ment in this section follows the conventional approachof considering parallel transport around a plaquette, asshown in Figure 2. (The term plaquette is borrowed fromcondensed matter theory and refers to a cell of a lattice.)So, this is our first pass at understanding the origin ofthe Riemann curvature tensor. In the following section,we will then re-derive the curvature tensor directly fromthe spin connection.For the counterclockwise path ( x α → x α + δx α → x α + δx α + dx α ), we have: X α ( x + δx ) = X α ( x ) + ¯ δX α ( x ) (62a) ( ) = X α ( x ) − Γ αβγ ( x ) X β ( x ) δx γ . (62b)At the end point x + δx + dx , we have X α ( x + δx + dx ) = X α ( x + δx ) + ¯ δX α ( x + δx )(63a) ( ) = X α ( x ) − Γ αβγ ( x ) X β ( x ) δx γ +¯ δX α ( x + δx ) . (63b)Now, we need to evaluate the last term (parallel transportterm) on the R.H.S.8 Riemann curvature from the affine connection IV CURVATURE ☎☎☎☎☎☎☎☎ ☎☎☎☎☎☎☎☎ x α x α + δx α x α + δx α + dx α x α + dx α δx α dx α rrrr FIG. 2
General plaquette of cell sizes δx α and dx α with its initial spacetime point at 4-vector x α (bottom left corner). The plaquette isa piece of a curved manifold. ¯ δX α ( x + δx ) ( ) = − Γ αβγ ( x + δx ) X β ( x + δx ) dx γ (64a) ( ) = − (cid:2) Γ αβγ ( x ) + ∂ δ Γ αβγ ( x ) δx δ (cid:3) (cid:2) X β ( x ) − Γ βµν ( x ) X µ ( x ) δx ν (cid:3) dx γ (64b)= − Γ αβγ X β dx γ − ∂ δ Γ αβγ X β δx δ dx γ + Γ αβγ Γ βµν X µ δx ν dx γ + ∂ δ Γ αβγ Γ βµν X µ δx δ δx ν dx γ | {z } neglect 3 rd order term , (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 X α ( x + δx + dx ) = X α − Γ αβγ X β δx γ − Γ αβγ X β dx γ − ∂ δ Γ αβγ X β δx δ dx γ + Γ αβγ Γ βµν X µ δx ν dx γ . (65)Interchanging the indices of δx and dx in the last two terms, we have X α ( x + δx + dx ) = X α − Γ αβγ X β δx γ − Γ αβγ X β dx γ − ∂ γ Γ αβδ X β δx γ dx δ + Γ αβν Γ βµγ X µ δx γ dx ν . (66)Then, replacing ν with δ in this last term, we have X α ( x + δx + dx ) = X α − Γ αβγ X β δx γ − Γ αβγ X β dx γ − ∂ γ Γ αβδ X β δx γ dx δ + Γ αβδ Γ βµγ X µ δx γ dx δ . (67)For the clockwise path ( x α → x α + dx α → x α + dx α + δx α ), we get the same result as before with dx α and δx α interchanged: X α ( x + δx + dx ) = X α − Γ αβγ X β dx γ − Γ αβγ X β δx γ − ∂ γ Γ αβδ X β dx γ δx δ + Γ αβδ Γ βµγ X µ dx γ δx δ . (68)Interchanging the indices δ and γ everywhere, we have X α ( x + δx + dx ) = X α − Γ αβδ X β dx δ − Γ αβδ X β δx δ − ∂ δ Γ αβγ X β dx δ δx γ + Γ αβγ Γ βµδ X µ dx δ δx γ , (69)which now looks like (67) in the indices of the differentials. Hence, as we compute (67) minus (69), the zeroth andfirst order terms cancel, and the remaining second order terms in (67) and (69) add with the common factor δx γ dx δ (differential area) △ X α = X α ( x + δx + dx ) − X α ( x + dx + δx ) (70a)= (cid:16) ∂ δ Γ αβγ X β − ∂ γ Γ αβδ X β + Γ αβδ Γ βµγ X µ − Γ αβγ Γ βµδ X µ (cid:17) δx γ dx δ (70b)= (cid:16) ∂ δ Γ αβγ − ∂ γ Γ αβδ + Γ αµδ Γ µβγ − Γ αµγ Γ µβδ (cid:17) X β δx γ dx δ . (70c)So, the difference of transporting the vector X α along the two separate routes around the plaquette is related9 Riemann curvature from the affine connection IV CURVATURE to the curvature of the manifold as follows △ X α = R αβδγ X β δx γ dx δ , (71)and from here we arrive at our desired result and identifythe Riemann curvature tensor as R αβδγ ≡ ∂ δ Γ αβγ − ∂ γ Γ αβδ + Γ αµδ Γ µβγ − Γ αµγ Γ µβδ . (72)Notice, from the identity (72), the curvature tensor isanti-symmetric in its last two indices, R αβδγ = − R αβγδ . B. Riemann curvature from the spin connection
In this section, we will show that the Riemann curva-ture tensor (72) can be simply expressed in terms of the spin connection as follows: R ab = dω ab + ω ac ∧ ω cb . (73)Here the Greek indices are suppressed for brevity. So, thefirst step is to explicitly write out the curvature tensor inall its indices and then to use the vierbein field to convertthe Latin indices to Greek indices, which gives us R λσµν ≡ e λa e σb (cid:0) ∂ µ ω νab − ∂ ν ω µab + ω µac ω νcb − ω νac ω µcb (cid:1) . (74)The quantity in parentheses is a spin curvature. Next,we will use (41), which I restate here for convenience ω µab = e ρa e τ b Γ ρµτ − e τ b ∂ µ e τ a . (75)Inserting (75) into (74) gives R λσµν = e λa e σb (cid:2) ∂ µ (cid:0) e ρa e τ b Γ ρντ (cid:1) − ∂ ν (cid:0) e ρa e τ b Γ ρµτ (cid:1) − ∂ µ e τ b ∂ ν e τ a + ∂ ν e τ b ∂ µ e τ a (76)+ (cid:16) e ρa e τ c Γ ρµτ − e τ c ∂ µ e τ a (cid:17)(cid:16) e ρ ′ c e τ ′ b Γ ρ ′ ντ ′ − e τ ′ b ∂ ν e τ ′ c (cid:17) − (cid:16) e ρa e τ c Γ ρντ − e τ c ∂ ν e τ a (cid:17)(cid:16) e ρ ′ c e τ ′ b Γ ρ ′ µτ ′ − e τ ′ b ∂ µ e τ ′ c (cid:17)i . Reducing this expression is complicated to do. Since the first term in (75) depends on the affine connection Γ andthe second term depends on ∂ µ , we will reduce (76) in two passes, first considering terms that involve derivatives ofthe 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 thesevanish. To begin with, the first order derivative terms that appear with ∂ µ acting on vierbein fields are the following: e λa e σb [ ∂ µ ( e ρa e τ b ) Γ ρντ − e τ c ( ∂ µ e τ a ) e ρc e τ b Γ ρντ + e ρa e τ c Γ ρντ e τ ′ b ∂ µ e τ ′ c ] (77a)= (cid:2) e λa e σb e τ b ∂ µ e ρa + e λa e σb e ρa ∂ µ e τ b (cid:3) Γ ρντ − e λa e σb e τ c e ρc e τ ′ b ( ∂ µ e τ a ) Γ ρντ ′ + e λa e σb e ρa e τ c e τ ′ b ∂ µ e τ ′ c Γ ρντ (77b)= e λa δ τσ ∂ µ e ρa Γ ρντ + δ λρ e σb ∂ µ e τ b Γ ρντ − δ τρ δ τ ′ σ e λa ∂ µ e τ a Γ ρντ ′ + δ λρ δ τ ′ σ e τ c ∂ µ e τ ′ c Γ ρντ (77c)= e λa ∂ µ e ρa Γ ρνσ + e σb ∂ µ e τ b Γ λντ − e λa ∂ µ e ρa Γ ρνσ + e τ b ∂ µ e σb Γ λντ (77d)= ∂ µ (cid:0) e σb e τ b (cid:1) Γ λντ (77e)= ∂ µ ( δ τσ ) Γ λντ (77f)= 0 . (77g)Similarly, all the first order derivative terms that appear with ∂ ν vanish as well. So, all the first order derivative termsvanish in (76). Next, we consider all second order derivatives, both ∂ µ and ∂ ν , acting on vierbein fields. All the termswith both ∂ µ and ∂ ν are the following e λa e σb e τ c e τ ′ b ( ∂ µ e τ a ) ∂ ν e τ ′ c − e λa e σb e τ c e τ ′ b ( ∂ ν e τ a ) ( ∂ µ e τ ′ c ) − e λa e σb ( ∂ µ e τ b ) ∂ ν e τ a + e λa e σb ( ∂ ν e τ b ) ∂ µ e τ a = e λa e τ c ( ∂ µ e τ a ) ∂ ν e σc − e λa e τ c ( ∂ ν e τ a ) ∂ µ e σc − (cid:0) ∂ ν e λa (cid:1) (cid:0) ∂ µ e σb (cid:1) e τ b e τ a + (cid:0) ∂ µ e λa (cid:1) (cid:0) ∂ ν e σb (cid:1) e τ b e τ a (78a)= e λa e τ c [( ∂ µ e τ a ) ∂ ν e σc − ( ∂ ν e τ a ) ∂ µ e σc ] − ∂ ν e λa ∂ µ e σa + ∂ µ e λa ∂ ν e σa (78b)= − e λa e τ a ( ∂ µ e τ c ) ∂ ν e σc + e λa e τ a ( ∂ ν e τ c ) ∂ µ e σc − ∂ ν e λa ∂ µ e σa + ∂ µ e λa ∂ ν e σa (78c)= − δ λτ ( ∂ µ e τ c ) ∂ ν e σc + δ λτ ( ∂ ν e τ c ) ∂ µ e σc − ∂ ν e λa ∂ µ e σa + ∂ µ e λa ∂ ν e σa (78d)= − (cid:0) ∂ µ e λc (cid:1) ∂ ν e σc + (cid:0) ∂ ν e λc (cid:1) ∂ µ e σc − ∂ ν e λa ∂ µ e σa + ∂ µ e λa ∂ ν e σa (78e)= 0 . (78f)10 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 thefact ∂ µ (cid:0) e λa e τ a (cid:1) = 0, so as to swap the order of differentiation, ∂ µ (cid:0) e λa (cid:1) e τ a = − e λa ( ∂ µ e τ a ) . (79)Finally, as a second pass toward reducing (76) to its final form, we now consider all the remaining terms (no derivativesof the vierbein fields), and these lead to the curvature tensor expressed solely as a function of the affine connection: R λσµν = e λa e σb h e ρa e τ b (cid:0) ∂ µ Γ ρντ − ∂ ν Γ ρµτ (cid:1) + e ρa e τ ′ b (cid:0) Γ ρµτ Γ τντ ′ − Γ ρντ Γ τµτ ′ (cid:1)i (80a)= δ λρ δ τσ (cid:0) ∂ µ Γ ρντ − ∂ ν Γ ρµτ (cid:1) + δ λρ δ τ ′ σ (cid:0) Γ ρµτ Γ τντ ′ − Γ ρντ Γ τµτ ′ (cid:1) . (80b)Applying the Kronecker deltas, we arrive at the final re-sult R λσµν = ∂ µ Γ λνσ − ∂ ν Γ λµσ + Γ λµτ Γ τνσ − Γ λντ Γ τµσ , (81)which is identical to (72). If we had not already derivedthe curvature tensor, we could have written (81) downby inspection because of its similarity to (74), essentiallyreplacing the spin connection with the affine connection. V. MATHEMATICAL CONSTRUCTS
Here we assemble a number of preliminary identitiesthat we will use later to derive the Einstein equation. An identity we will need allows us to evaluate the traceof M − ∂ µ M where M is a 2-rank tensor T r [ M − ∂ µ M ] = ∂ µ ln | M | . (82)As an example of this identity, consider the following 2 × M = (cid:18) a bc d (cid:19) M − = 1 | M | (cid:18) d − b − c a (cid:19) . (83)A demonstration of the trace identity (82) for the sim-plest case of one spatial dimension isTr (cid:2) M − ∂ x M (cid:3) = Tr (cid:20) ad − bc (cid:18) d − b − c a (cid:19) (cid:18) ∂ x a ∂ x b∂ x c ∂ x d (cid:19)(cid:21) (84a)= Tr (cid:20) ad − bc (cid:18) ∂ x a d − b∂ x c ∂ x b d − b∂ x d − ∂ x a c + a∂ x c − ∂ x b c + a∂ x d (cid:19)(cid:21) (84b)= 1 ad − bc [ ∂ x ( ad ) − ∂ x ( bc )] (84c)= ∂ x ( ad − bc ) ad − bc (84d)= ∂ x ln | M | . (84e)This identity holds for matrices of arbitrary size. In ourcase, we shall need this identity for the case of 4 × g λµ g µν = δ λν , (85)so g µν → ( g µν ) − . (86)We also define the negative determinant of the metrictensor g ≡ − Det g µν . (87) A. Consequence of tetrad postulate
The tetrad postulate of Section III.C is that the vier-bein field is invariant under parallel transport ∇ µ e νa = 0 . (88)That the metric tensor is invariant under parallel trans-port then immediately follows ∇ µ g νλ = ∇ µ ( e νa e λc n ab ) (89a)= ( ∇ µ e νa ) e λb n ab + e νa (cid:0) ∇ µ e λb (cid:1) n ab (89b) ( ) = 0 . (89c)11 Affine connection in terms of the metric tensor VI GRAVITATIONAL ACTION
This is called metric compatibility.
B. Affine connection in terms of the metric tensor
Now we can make use of (89) to compute the affineconnection. Permuting indices, we can write ∇ ρ g µν = ∂ ρ g µν − Γ λρµ g λν − Γ λρν g µλ = 0 (90a) ∇ µ g νρ = ∂ µ g νρ − Γ λµν g λρ − Γ λµρ g νλ = 0 (90b) ∇ ν g ρµ = ∂ ν g ρµ − Γ λνρ g λµ − Γ λνµ g ρλ = 0 . (90c)Now, we take (90a) − (90b) − (90c): ∂ ρ g µν − ∂ µ g νρ − ∂ ν g ρµ + 2Γ λµν g λρ = 0 . (91)Multiplying through by g σρ allows us to solve for theaffine connectionΓ σµν = 12 g σρ ( ∂ µ g νρ + ∂ ν g ρµ − ∂ ρ g µν ) . (92)Then, contracting the σ and µ indices, we haveΓ µµν = 12 g µρ ( ∂ µ g νρ + ∂ ν g ρµ − ∂ ρ g µν ) (93a)= 12 g µρ ( ∂ ν g ρµ + ∂ µ g ρν − ∂ ρ g µν ) (93b)= 12 g µρ ∂ ν g ρµ + 12 g µρ { ∂ µ g ρν − ∂ ρ g µν } . (93c)Since the metric tensor is symmetric and the last term inbrackets is anti-symmetric in the µ ρ indices, the productmust vanish. Thus Γ µµν = 12 g µρ ∂ ν g ρµ . (94)Furthermore, rewriting (94) as the trace of thesimilarity transformation of the covariant derivativeTr[ M − ∂ ν M ] = ∂ ν ln Det M that we demonstrated in(84), we haveΓ µµν = 12 Tr (cid:0) g λρ ∂ ν g ρµ (cid:1) (95a)= 12 ∂ ν ln Det g ρµ (95b)= 12 ∂ ν ln( − g ) , g ≡ − Det g ρµ (95c)= ∂ ν ln √− g (95d)= 1 √− g ∂ ν √− g. (95e)Equating (94) to (95e), we have ∂ ν √− g = 12 √− g g µρ ∂ ν g ρµ . (96)For generality, we write this corollary to (95e) as follows: δ √− g = 12 √− g g µν δ g µν . (97) C. Invariant volume element
Now, we consider the transport of a general 4-vector ∇ ν V µ = ∂ ν V µ + Γ µνλ V λ . (98)Therefore, the 4-divergence of V µ is ∇ µ V µ = ∂ µ V µ + Γ µµλ V λ (99a) ( ) = ∂ µ V µ + 1 √− g (cid:0) ∂ λ √− g (cid:1) V λ (99b)= 1 √− g ∂ µ (cid:0) √− gV µ (cid:1) . (99c)If V µ vanishes at infinity, then integrating over all spaceyields Z d x √− g ∇ µ V µ = Z d x ∂ µ (cid:0) √− gV µ (cid:1) = 0 , (100)which is a covariant form of Gauss’s theorem where theinvariant volume element is dV = √− g d x. (101) D. Ricci tensor
The Ricci tensor is the second rank tensor formed fromthe Riemann curvature tensor as follows: R σν ≡ R λσλν = g λµ R µσλν , (102)where the Riemann tensor is R ρσµν = ∂ µ Γ ρνσ − ∂ ν Γ ρµσ + Γ ρµλ Γ λνσ − Γ ρνλ Γ λµσ . (103)Therefore, the Ricci tensor can be written as R σν = ∂ ρ Γ ρνσ − ∂ ν Γ ρρσ + Γ ρρλ Γ λνσ − Γ ρνλ Γ λρσ (104a)= ∂ ρ Γ ρνσ − Γ λνρ Γ ρλσ − ∂ ν Γ ρσρ + Γ λνσ Γ ρλρ . (104b)Now, using the correction for a covariant vector ∇ ρ A ν = ∂ ρ A ν − Γ λρν A λ , (105)we can write (104b) as R σν = ∇ ρ Γ ρνσ − ∇ ν Γ ρσρ . (106)(106) is known as the Palatini identity. The scalar cur-vature is the following contraction of the Ricci tensor R ≡ g µν R µν . (107) VI. GRAVITATIONAL ACTIONA. Free field gravitational action
The action for the source free gravitational field is I G = 116 πG Z d x √− g R ( x ) . (108)12 Free field gravitational action VI GRAVITATIONAL ACTION
The equation of motion for the metric tensor can be de-termined by varying (108) with respect to the metric ten-sor field. The variation is carried out in several stages.The variation of the Lagrangian density is δ (cid:0) √− gR (cid:1) = √− gR µν δg µν + R δ √− g + √− g g µν δR µν . (109)From the Palatini identity (106), the change in the Riccitensor can be written as δR µν = −∇ ν δ Γ λµλ + ∇ λ δ Γ λµν , (110) where we commute the variational change with the co-variant derivative. Now, we can expand the third termon the R.H.S. of (109) as follows: √− g g µν δR µν ( ) = −√− g (cid:2) g µν ∇ ν δ Γ λµλ − g µν ∇ λ δ Γ λµν (cid:3) (111a) ( ) = −√− g ∇ ν (cid:0) g µν δ Γ λµλ (cid:1)| {z } like V ν −∇ λ (cid:0) g µν δ Γ λµν (cid:1)| {z } like V µ , (111b)since ∇ µ g νρ = 0. Now from (99c) we know ∇ ν V ν = √− g ∂ ν ( √− gV ν ), so for the variation of the third term we have √− g g µν δR µν = − ∂ ν (cid:0) √− g g µν δ Γ λµλ (cid:1) + ∂ λ (cid:0) √− g g µν δ Γ λµν (cid:1) . (111c)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) δ √− g = 12 √− g g µν δg µν into the second term on the R.H.S. of (109) we can write the variation of the gravitational action (108) entirely interms of the variation of the metric tensor field δI G = 116 πG Z d x √− g (cid:20) R µν δg µν + 12 g µν R δg µν (cid:21) . (112)The variation of identity vanishes, δ [ δ µλ ] = δ [ g µτ g τλ ] = ( δg µτ ) g τλ + g µτ δg τλ = 0 , (113)from which we find the following useful identity δg µτ g νλ g τλ + g νλ g µτ δg τλ = 0 (114)or δg µν = − g νλ g µτ δg τλ . (115)With this identity, we can write the variation of the free gravitational action as δI G = − πG Z d x √− g (cid:20) R µν g µτ g νλ δg τλ − g µν Rδg µν (cid:21) (116a)= − πG Z d x √− g (cid:20) R µν − g µν R (cid:21) δg µν . (116b)Since the variation of the metric does not vanish in general, for the gravitational action to vanish the quantity insquare brackets must vanish. This quantity is call the Einstein tensor G µν ≡ R µν − g µν R. (117)So, the equation of motion for the free gravitation field is simply G µν = 0 . (118)13 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 g µν ( x ) = e µa ( x ) e ν b ( x ) η ab , (119)so its variation can be directly written in terms of the variation of the vierbein δg µν = δe µa e νb η ab + e µa δe νb η ab (120a)= δe µa e νa + e µa δe νa (120b) ( ) = − e µa δe νa − δe µa e νa . (120c)Now, we can look upon δe µa as a field quantity whose indices we can raise or lower with the appropriate use of themetric tensor. Thus, we can write the variation of the metric as δg µν = − g νλ e µa δe λa − g µλ δe λa e νa (120d)= − ( g µλ e νa + g νλ e µa ) δe λa . (120e)Therefore, inserting this into (116b), the variation of the source-free gravitational action with respect to the vierbeinfield is δI G = 116 πG Z d x √− g (cid:18) R µν − g µν R (cid:19)(cid:18) g µλ e νa + g νλ e µa (cid:19) δe λa (121a)= 116 πG Z d x √− g (cid:18) R λν e νa − δ νλ R e νa + R µλ e µa − δ µλ R e µa (cid:19) δe λa (121b)= 18 πG Z d x √− g (cid:20)(cid:18) R µλ − δ µλ R (cid:19) e µa (cid:21) δe λa . (121c)Since the variation of the vierbein field does not vanish ingeneral, for the gravitational action to vanish the quan-tity in square brackets must vanish. Multiplying this by g λν , the equation of motion is G µν e µa = 0 , (122)which leads to (118) since e µa = 0. Yet (122) is a moregeneral equation of motion since it allows cancelationacross components instead of the simplest case whereeach component of G µν vanishes separately. C. Action for a gravitational source
The fundamental principle in general relativity is thatthe presence of matter warps the spacetime manifold inthe vicinity of the source. The vierbein field allows us toquantify this principle in a rather direct way. The varia-tion of the action for the matter source to lowest order islinearly proportional to the variation of the vierbein field δI M = Z d x √− g u λa δe λa , (123)where the components u λa are constants of proportion-ality. However, the usual definition of the matter action is as a functional derivative with respect to the metric δI M δg µν ≡ Z d x √− g T µν . (124)So, in consideration of (123) and (124), we should write u λa δe λa = 12 T µν δg µν (125a) ( ) = − T µν ( g µλ e νa + g νλ e µa ) δe λa , (125b)which we can solve for T µν . Dividing out δe λa and thenmultiplying through by g λβ we get u βa = − T µν (cid:0) δ βµ e νa + δ βν e µa (cid:1) (126a)= − (cid:0) T βν e νa + T µβ e µa (cid:1) (126b)= − T βν e νa , (126c)since the energy-momentum tensor is symmetric. Thus,we have T µν = − u µa e νa . (127)14 Full gravitational action VII EINSTEIN’S ACTION
Alternatively, again in consideration of (123) and (124),we could also write u λa δe λa = 12 T µν δg µν (128a) ( ) = 12 T µν ( δe µa e νa + e µa δe νa ) (128b)= 12 T µν ( δe µa e νa + e µa δe νa ) (128c)= 12 (cid:0) T λν δe λa e νa + T µλ e µa δe λa (cid:1) (128d)= 12 ( T λν e νa + T µλ e µa ) δe λa (128e)= T λν e νa δe λa . (128f)This implies that the energy-stress tensor is proportionalto the vierbein field T µν = e µa u νa . (129)Consequently, the variation of the energy-stress tensor isthen δT µν = δe µa u νa = g µλ δe λa u νa . (130)This can also be written as T λν = e λa u νa → δT λν = δe λa u νa . (131)Inserting this back into the action for a graviationalsource (123) we have I M = Z d x √− g T λλ (132a)= Z d x √− g g µν T µν . (132b) D. Full gravitational action
The variation of the full gravitational action is the sumof variations of the source-free action and gravitationalaction for matter δI = δI G + δI M . (133)Inserting (121c) and (132b) into (133) then gives δI G = Z d x √− g (cid:20) πG (cid:18) R µλ − δ µλ R (cid:19) e µa + u λa (cid:21) δe λa . (134)Therefore, with the requirement that δI G = 0, we obtainthe equation of motion of the vierbein field (cid:18) R µλ − δ µλ R (cid:19) e µa = − πGu λa . (135)Multiplying through by e νa (cid:18) R µλ − δ µλ R (cid:19) e µa e νa = − πGe νa u λa (136)gives the well known Einstein equation R νλ − g νλ R = − πG T νλ . (137) VII. EINSTEIN’S ACTION
In this section, we review the derivation of the equa-tion of motion of the metric field in the weak field ap-proximation. We start with a form of the Lagrangiandensity presented in (Einstein, 1928a) for the vierbeinfield theory. Einstein’s intention was the unification ofelectromagnetism with gravity.With h denoting the determinant of | e µa | ( i.e. h ≡√− g ), the useful identity δ √− g = 12 √− g g µν δ g µν can be rewritten strictly in terms of the vierbein field asfollows δh = 12 h g µν δg µν (138a)= 12 h g µν δ ( e µa e νb ) η ab (138b)= 12 h g µν δe µa e νb η ab + 12 h g µν e µa δe νb η ab (138c)= 12 h δe µa e µa + 12 h e νb δe νb (138d)= h δe µa e µa . (138e) A. Lagrangian density form 1
With the following definitionΛ ναβ ≡ e νa ( ∂ β e αa − ∂ α e βa ) , (139)the first Lagrangian density that we consider is the fol-lowing: L = h g µν Λ µαβ Λ νβα , (140a)= h g µν e αa e βb ( ∂ β e µa − ∂ µ e βa )( ∂ α e νb − ∂ ν e αb ) . (140b)For a weak field, we have the following first-order expan-sion e µa = δ µa − k µa · · · . (141)15 Lagrangian density form 2 VII EINSTEIN’S ACTION
The lowest-order change (2nd order in δh ) is δ L = h η µν δ αa δ βb ( ∂ β k µa − ∂ µ k βa )( ∂ α k νb − ∂ ν k αb )(142a)= h η µν ( ∂ β k µα − ∂ µ k βα )( ∂ α k νβ − ∂ ν k αβ ) (142b)= h η µν ( ∂ β k µα ∂ α k νβ − ∂ β k µα ∂ ν k αβ (142c) − ∂ µ k βα ∂ α k ν β + ∂ µ k βα ∂ ν k αβ )= h (cid:16) η µα ∂ β k µν ∂ ν k αβ − η µν ∂ β k µα ∂ ν k αβ (142d) − η µα ∂ µ k βν ∂ ν k αβ + η µν ∂ µ k βα ∂ ν k αβ (cid:17) = h − η µα ∂ β ∂ ν k µν + η µν ∂ β ∂ ν k µα (142e)+ η µα ∂ µ ∂ ν k βν − η µν ∂ µ ∂ ν k β α ) k αβ . (142f)So δ R d x L = 0 implies the equation of motion − ∂ β ∂ ν k αν + ∂ β ∂ ν k να + ∂ α ∂ ν k βν − ∂ k βα = 0 (143a)or ∂ k βα − ∂ µ ∂ β k µα + ∂ β ∂ µ k αµ − ∂ µ ∂ α k βµ = 0 . (143b)The above equation of motion (143b) is identical toEq. (5) in Einstein’s second paper. B. Lagrangian density form 2
Now the second Lagrangian density we consider is thefollowing: L = h g µν g ασ g βτ Λ µαβ Λ νστ (144a)= h g µν g ασ g βτ e µa e νb ( ∂ β e αa − ∂ α e βa ) ( ∂ τ e σb − ∂ σ e τb ) . (144b)We will see this leads to the same equation of motion thatwe got from the first form of the Lagrangian density. Thelowest-order change is δ L = h η µν η ασ η βτ δ µa δ νb ( ∂ β k αa − ∂ α k βa ) ( ∂ τ e σb − ∂ σ e τb )(145a)= h η ασ η βτ ( ∂ β k αν − ∂ α k βν ) ( ∂ τ k σν − ∂ σ k τ ν ) (145b)= h ∂ β k αν − ∂ α k βν ) (cid:0) ∂ β k αν − ∂ α k βν (cid:1) (145c)= h (cid:0) ∂ β k αν ∂ β k αν − ∂ β k αν ∂ α k βν − ∂ α k βν ∂ β k αν + ∂ α k βν ∂ α k βν (cid:1) (145d)= h (cid:0) ∂ β k αν ∂ β k αν − ∂ β k αν ∂ α k βν − ∂ β k αν ∂ α k βν + ∂ β k αν ∂ β k αν (cid:1) (145e)= h (cid:0) − ∂ k αν + ∂ β ∂ α k βν (cid:1) k αν . (145f) This implies the following: ∂ k αν − ∂ β ∂ α k βν = 0 (146a)or ∂ k βα − ∂ µ ∂ β k µα = 0 . (146b)These are the first two terms in Einstein’s Eq. (5).Notice that we started with a Lagrangian density withthe usual quadratic form in the field strength of the form(145c), which is L = h F αβν F αβν , (147)where the field strength is F αβν = ∂ α k βν − ∂ β k αν . (148)If we had varied the action with respect to k βν , then wewould have obtained the same equation of motion (146b). C. First-order fluctuation in the metric tensor
The metric tensor expressed in terms of the vierbeinfield is g αβ = e αa e βa = ( δ aα + k αa ) ( δ βa + k βa ) . (149)So the first order fluctuation of the metric tensor field isthe symmetric tensor g αβ ≡ g αβ − δ αβ = k αβ + k βα · · · . (150)We define the electromagnetic four-vector by contractingthe field strength tensor ϕ µ ≡ Λ αµα = 12 e αa ( ∂ α e µa − ∂ µ e αa ) . (151)This implies ϕ µ = 12 δ αa ( ∂ α k µa − ∂ µ k αa ) , (152)so we arrive at 2 ϕ µ = ∂ α k µα − ∂ µ k αα . (153) D. Field equation in the weak field limit
The equation of motion for the fluctuation of the met-ric tensor from the first form of the Lagrangian densityis obtained by adding (143b) to itself but with α and β exchanged: ∂ k βα − ∂ µ ∂ β k µα + ∂ β ∂ µ k αµ − ∂ µ ∂ α k βµ + ∂ k αβ − ∂ µ ∂ α k µβ + ∂ α ∂ µ k βµ − ∂ µ ∂ β k αµ = 0 , (154)16 Invariance in flat space VIII RELATIVISTIC CHIRAL MATTER IN CURVED SPACE which has a cancellation of four terms leaving ∂ g αβ − ∂ µ ∂ α k µβ − ∂ µ ∂ β k µα = 0 . (155)Similarly, we arrive at the same result starting with theequation of motion for the fluctuation of the metric ten-sor obtained from the second form of the Lagrangian den-sity, again by adding (146a) to itself but with α and β exchanged: ∂ k βα − ∂ µ ∂ β k µα + ∂ k αβ − ∂ µ ∂ α k µβ = 0 . (156)In this case, the sum is exactly the same as what we justobtained in (155) but with no cancellation of terms ∂ g αβ − ∂ µ ∂ α k µβ − ∂ µ ∂ β k µα = 0 . (157)Using (153) above, just with relabeled indices,2 ϕ α = ∂ µ k αµ − ∂ α k µµ . (158)Taking derivatives of (158) we have ancillary equationsof motion: − ∂ µ ∂ β k αµ + ∂ α ∂ β k µµ = − ∂ β ϕ α (159a)and − ∂ µ ∂ α k βµ + ∂ α ∂ β k µµ = − ∂ α ϕ β . (159b)Adding the ancilla (159) to our equation of motion (157)gives − ∂ g αβ + ∂ µ ∂ α ( k µβ + k βµ ) + ∂ µ ∂ β ( k µα + k αµ ) − ∂ α ∂ β k µµ = 2( ∂ β ϕ α + ∂ α ϕ β ) . (160)Then making use of (150) this can be written in terms ofthe symmetric first-order fluctuation of the metric tensor field12 (cid:18) − ∂ g αβ + ∂ µ ∂ α g µβ + ∂ µ ∂ β g µα − ∂ α ∂ β g µµ (cid:17) = ∂ β ϕ α + ∂ α ϕ β . (161)This result is the same as Eq. (7) in Einstein’s secondpaper. In the case of the vanishing of φ α , (161) agrees tofirst order with the equation of General Relativity R αβ = 0 . (162)Thus, Einstein’s action expressed explicitly in terms ofthe vierbein field reproduces the law of the pure gravita-tional field in weak field limit. VIII. RELATIVISTIC CHIRAL MATTER IN CURVEDSPACEA. Invariance in flat space
The external Lorentz transformations, Λ that act on4-vectors, commute with the internal Lorentz transfor-mations, U (Λ) that act on spinor wave functions, i.e. [Λ µν , U (Λ)] = 0 . (163)Note that we keep the indices on U (Λ) suppressed, justas we keep the indices of the Dirac matrices and the com-ponent indices of ψ suppressed as is conventional whenwriting matrix multiplication. Only the exterior space-time indices are explicitly written out. With this con-vention, the Lorentz transformation of a Dirac gammamatrix is expressed as follows: U (Λ) − γ µ U (Λ) = Λ µσ γ σ . (164)The invariance of the Dirac equation in flat spaceunder a Lorentz transformation is well known(Peskin and Schroeder, 1995):[ iγ µ ∂ µ − m ] ψ ( x ) LLT −→ h iγ µ (cid:0) Λ − (cid:1) νµ ∂ ν − m i U (Λ) ψ (cid:0) Λ − x (cid:1) (165a)= U (Λ) U (Λ) − h iγ µ (cid:0) Λ − (cid:1) ν µ ∂ ν − m i U (Λ) ψ (cid:0) Λ − x (cid:1) (165b) ( ) = U (Λ) h i U (Λ) − γ µ U (Λ) (cid:0) Λ − (cid:1) ν µ ∂ ν − m i ψ (cid:0) Λ − x (cid:1) (165c) ( ) = U (Λ) h i Λ µσ γ σ (cid:0) Λ − (cid:1) νµ ∂ ν − m i ψ (cid:0) Λ − x (cid:1) (165d)= U (Λ) h i Λ µσ (cid:0) Λ − (cid:1) ν µ γ σ ∂ ν − m i ψ (cid:0) Λ − x (cid:1) (165e)= U (Λ) [ i δ νσ γ σ ∂ ν − m ] ψ (cid:0) Λ − x (cid:1) (165f)= U (Λ) [ i γ ν ∂ ν − m ] ψ (cid:0) Λ − x (cid:1) . (165g) B. Invariance in curved space
Switching to a compact notation for the interiorLorentz transformation, Λ ≡ U (Λ), (164) isΛ − γ µ Λ = Λ µσ γ σ , (166) where I put a minus on the subscript to indicate the in-verse transformation, i.e. Λ − ≡ U (Λ) − . Of course,17 Invariance in curved space VIII RELATIVISTIC CHIRAL MATTER IN CURVED SPACE exterior Lorentz transformations can be used as a simi-larity transformation on the Dirac matricesΛ µσ γ σ (cid:0) Λ − (cid:1) νµ = γ ν . (167)Below we will need the following identity:Λ µλ e λa γ a (Λ − ) νµ Λ ( ) = Λ Λ − (cid:0) Λ µλ e λa γ a (cid:1) Λ (Λ − ) νµ (168a) ( ) = Λ Λ µσ (cid:0) Λ σλ e λa γ a (cid:1) (Λ − ) νµ (168b) ( ) = Λ Λ νλ e λa γ a . (168c)We require the Dirac equation in curved space be invari-ant under Lorentz transformation when the curvature ofspace causes a correction Γ µ . That is, we require e µa γ a ( ∂ µ + Γ µ ) ψ ( x ) LLT −→ Λ µλ e λa γ a (Λ − ) νµ ( ∂ ν + Γ ′ ν ) Λ ψ (Λ − x ) (169a)= Λ µλ e λa γ a (Λ − ) νµ Λ (cid:16) ∂ ν + Λ − Γ ′ ν Λ (cid:17) ψ (Λ − x )+ Λ µλ e λa γ a (cid:0) Λ − (cid:1) νµ (cid:16) ∂ ν Λ (cid:17) ψ (Λ − x ) (169b) ( ) = Λ Λ νλ e λa γ a h(cid:16) ∂ ν + Λ − Γ ′ ν Λ (cid:17) ψ (Λ − x )+ Λ − (cid:16) ∂ ν Λ (cid:17) ψ (Λ − x ) i (169c)= Λ Λ νλ e λa γ a [( ∂ ν + Γ ν ) − Γ ν + Λ − Γ ′ ν Λ + Λ − ∂ ν (cid:16) Λ (cid:17)| {z } =0 (cid:3) ψ (Λ − x ) . (169d)In the last line we added and subtracted Γ ν . To achieveinvariance, the last three terms in the square bracketsmust vanish. Thus we find the form of the local “gauge”transformation requires the correction field to transformas follows: − Γ ν + Λ − Γ ′ ν Λ + Λ − ∂ ν (cid:16) Λ (cid:17) = 0 (170a)or Γ ′ ν = Λ Γ ν Λ − − ∂ ν (cid:16) Λ (cid:17) Λ − . (170b)Therefore, the Dirac equation in curved space iγ a e µa ( x ) D µ ψ − m ψ = 0 (171)is invariant under a Lorentz transformation provided thegeneralized derivative that we use is D µ = ∂ µ + Γ µ , (172)where Γ µ transforms according to (170b). This is anal-ogous to a gauge correction; however, in this case Γ µ isnot a vector potential field. C. Covariant derivative of a spinor field
The Lorentz transformation for a spinor field isΛ = 1 + 12 λ ab S ab , (173)where the generator of the transformation is anti-symmetric S ab = − S ba . The generator satisfies the fol-lowing commutator[ S hk , S ij ] = η hj S ki + η ki S hj − η hi S kj − η kj S hi . (174)Thus, the local Lorentz transformations (LLT) of aLorentz 4-vector, x a say, and a Dirac 4-spinor, ψ say,are respectively:LLT: x a → x ′ a = Λ ab x b (175)and LLT: ψ → ψ ′ = Λ ψ. (176)The covariant derivative of a 4-vector is ∇ γ X α = ∂ γ X α + Γ αβγ X β , (177)and the 4-vector at the nearby location is changed by thecurvature of the manifold. So we write it in terms of theoriginal 4-vector with a correction X α k ( x + δx α ) = X α k ( x ) − Γ α k βγ ( x ) X β ( x ) δx γ , (178)as depicted in Fig. 3. ❇❇❇❇▼ ❇❇❇❇▼ ✂✂✂✂✍✛ x α x α + δx α X α k ( x ) X α k ( x + δx )Γ α k βγ ( x ) X β ( x ) δx γ rr FIG. 3
Depiction of the case of an otherwise constant field dis-torted by curved space. The field value X α ( x ) is parallel trans-ported along the curved manifold (blue curve) by the distance δx α going from point x α to x α + δx α . Likewise, the correction to the vierbein field due thecurvature of space is e µ k k ( x + δx α ) = e µ k k ( x ) − Γ µ k βα ( x ) e β k ( x ) δx α . (179)The Lorentz transformation of a 2-rank tensor field isΛ aa ′ Λ bb ′ η ab = η a ′ b ′ . (180)Moreover, the Lorentz transformation is invertibleΛ ia Λ ja = δ ij = Λ aj Λ ai , (181)where the inverse is obtained by exchanging index labels,changing covariant indices to contravariant indices and18 Covariant derivative of a spinor field VIII RELATIVISTIC CHIRAL MATTER IN CURVED SPACE contravariant to covariant. In the case of infinitesimaltransformations we haveΛ ij ( x ) = δ ij + λ ij ( x ) , (182)where 0 = λ ij + λ ji = λ ij + λ ji . (183)Lorentz and inverse Lorentz transformations of thevierbein fields are¯ e µh ′ ( x ) = Λ h ′ a ( x ) e µa ( x ) (184)and e µh ( x ) = Λ a ′ h ( x )¯ e µa ′ ( x ) , (185)where temporarily I am putting a bar over the trans-formed vierbein field as a visual aid. Since the vierbeinfield is invertible, we can express the Lorentz tranforma-tion directly in terms of the vierbeins themselves¯ e a ′ µ ( x ) e µh ( x ) = Λ a ′ h ( x ) . (186)Now, we transport the Lorentz transformation tensor it-self. The L.H.S. of (186) has two upper indices, the Latinindex a ′ and the Greek index µ , and we choose to use theupper indices to connect the Lorentz transformation ten-sor between neighboring points. These indices are treateddifferently: a Taylor expansion can be used to connect aquantity in its Latin non-coordinate index at one pointto a neighboring point, but the affine connection must beused for the Greek coordinate index. Thus, we haveΛ h ′ k ( x + δx α ) = ¯ e h ′ µ k ( x + δx α ) e µ k k ( x + δx α ) (187a)= ¯ e hµ ( x ) + ∂ ¯ e hµ ∂x α δx α ! e µ k k ( x + δx α )(187b) ( ) = ¯ e hµ ( x ) + ∂ ¯ e hµ ∂x α δx α ! (187c) × (cid:16) e µk − Γ µβα ( x ) e βk ( x ) δx α (cid:17) = δ hk + ∂ ¯ e hµ ∂x α δx α e µk − Γ µβα e βk δx α ¯ e hµ (187d)= δ hk + ∂ ¯ e hµ ∂x α δ µβ − Γ µβα ¯ e hµ ! e βk δx α (187e)= δ hk + (cid:16) e µk ∂ α ¯ e hµ − Γ µβα ¯ e hµ e βk (cid:17) δx α (187f)= δ hk − ω αhk δx α , (187g)where the spin connection ω αhk = − e µk ∂ α ¯ e hµ + Γ µβα ¯ e hµ e βk (188) is seen to have the physical interpretation of generalizingthe infinitesimal transformation (182) to the case of in-finitesimal transport in curved space. Relabeling indices,we have ω µab = − e νb ∂ µ e νa + Γ σµν e σa e νb (189a)= − e νb (cid:0) ∂ µ e νa − Γ σµν e σa (cid:1) (189b)= − e νb ∇ µ e νa , (189c)where here the covariant derivative of the vierbien 4-vector is not zero. Writing the Lorentz transformationin the usual infinitesimal formΛ hk = δ hk + λ hk (190)implies λ hk = − ω αhk δx α (191a)= e νk (cid:0) ∇ α e νh (cid:1) δx α (191b)or λ hk = e βk ( ∇ α e βh ) δx α . (191c)Using (173), the Lorentz transformation of the spinorfield is Λ ψ = (cid:18) λ hk S hk (cid:19) ψ = ψ + δψ. (192)This implies the change of the spinor is δψ = 12 e βk ( ∇ α e βh ) δx α S hk ψ (193a)= Γ α ψ δx α , (193b)where the correction to the spinor field is found to beΓ α = 12 e βk ( ∇ α e βh ) S hk (194a)= 12 e βk (cid:0) ∂ α e βh − Γ σµβ e σh (cid:1) S hk (194b)= 12 e βk ( ∂ α e βh ) S hk −
12 Γ σµβ ( e σh e βk ) S hk (194c)= 12 e βk ( ∂ α e βh ) S hk , (194d)where the last term in (194c) vanishes because e σh e βk issymmetric whereas S hk is anti-symmetric in the indices h and k . Thus, we have derived the form of the covariantderivative of the spinor wave function D µ ψ = ∂ µ ψ + Γ µ ψ (195a)= (cid:18) ∂ µ + 12 e βk ∇ µ e βh S hk (cid:19) ψ (195b)= (cid:18) ∂ µ + 12 e βk ∂ µ e βh S hk (cid:19) ψ. (195c) Remember the Tetrad postulate that we previously derived ∇ µ e νa = ∂ µ e νa − e σa Γ σµν + ω µab e νb = 0is exactly (189b). ACKNOWLEDGEMENTS
This is the generalized derivative that is needed to cor-rectly differentiate a Dirac 4-spinor field in curved space.
IX. CONCLUSION
A detailed derivation of the Einstein equation from theleast action principle and a derivation of the relativisticDirac equation in curved space from considerations ofinvariance with respect to Lorentz transformations havebeen presented. The field theory approach that was pre-sented herein relied on a factored decomposition of themetric tensor field in terms of a product of vierbein fieldsthat Einstein introduced in 1928. In this sense, the vier-bein field is considered the square root of the metric ten-sor. The motivation for this decomposition follows natu-rally from the anti-commutator { e µa ( x ) γ a , e νb ( x ) γ b } = 2 g µν ( x ) , (196)where γ a are the Dirac matrices. Dirac originally dis-covered an aspect of this important identity when hesuccessfully attempted to write down a linear quantumwave equation that when squared gives the well knownKlein-Gordon equation. Thus, dealing with relativisticquantum mechanics in flat space, Dirac wrote this iden-tity as { γ a , γ b } = 2 η ab , (197a)where η = diag(1 , − , − , − e µa ( x ) e ν b ( x ) η ab = g µν ( x ) . (197b)Combining (197a) and (197b) into (196) is essential tocorrectly develop a relativistic quantum field theory incurved space. However, (197b) in its own right is a suf-ficient point of departure if one seeks to simply derivethe Einstein equation capturing the dynamical behaviorof spacetime. X. ACKNOWLEDGEMENTS
I would like to thank Carl Carlson for checking thederivations presented above. I would like to thank HansC. von Baeyer for his help searching for past Englishtranslations of the 1928 Einstein manuscripts (of whichnone were found) and his consequent willingness to trans-late the German text into English.
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Phys. Rev. , 96(1):191–195. ppendices The following two manuscripts, translated here in English by H.C. von Baeyer and into L A TEX by the author, originallyappeared in German in Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-MathematischeKlasse in the summer of 1928.
Einstein’s 1928 manuscript on distant parallelism iemann geometry with preservation of the concept of distant parallelism A. EinsteinJune 7, 1928Riemann geometry led in general relativity to a phys-ical description of the gravitational field, but does notyield any concepts that can be applied to the electromag-netic field. For this reason the aim of theoreticians is tofind natural generalizations or extensions of Riemann ge-ometry that are richer in content, in hopes of reaching alogical structure that combines all physical field conceptsfrom a single point of view. Such efforts have led me toa theory which I wish to describe without any attemptat physical interpretation because the naturalness of itsconcepts lends it a certain interest in its own right.Riemannian geometry is characterized by the facts thatthe infinitesimal neighborhood of every point P has aEuclidian metric, and that the magnitudes of two line el-ements that belong to the infinitesimal neighborhoods oftwo finitely distant points P and Q are comparable. How-ever, the concept of parallelism of these two line elementsis missing; for finite regions the concept of direction doesnot exist. The theory put forward in the following is char-acterized by the introduction, in addition to the Riemannmetric, of a “direction,” or of equality of direction, or of“parallelism” for finite distances. Correspondingly, newinvariants and tensors will appear in addition to those ofRiemann geometry. I. n -BEIN AND METRIC At the arbitrary point P of the n -dimensional con-tinuum erect an orthogonal n -Bein from n unit vectorsrepresenting an orthogonal coordinate system. Let A a bethe components of a line element, or of any other vector,w.r.t. this local system ( n -Bein). For the description ofa finite region introduce furthermore the Gaussian coor-dinate system x ν . Let A ν be the ν -components of thevector ( A ) w.r.t. the latter, furthermore let h aν be the ν components of the unit vectors that form the n -Bein.Then A ν = h νa A a · · · . (1)By inverting (1) and calling h νa the normalized sub-determinants (cofactor) of h aν we obtain A a = h µa A µ · · · . (1a)The magnitude A of the vector ( A ), on account of theEuclidian property of the infinitesimal neighborhoods, is We use Greek letters for the coordinate indices, Latin letters forBein indices. given by A = X A a = h µa h νa A µ A ν · · · . (2)The metric tensor components g µν are given by the for-mula g µν = h µa h νa , · · · (3)where, of course, the index a is summed over. With fixed a , the h aν are the components of a contravariant vector.The following relations also hold: h µa h νa = δ νµ · · · (4) h µa h µb = δ ab , · · · (5)where δ = 1 or δ = 0 depending on whether the two in-dices are equal or different. The correctness of (4) and(5) follows from the above definition of h νa as normal-ized subdeterminants of h νa . The vector character of h νa follows most easily from the fact that the left hand side,and hence also the right hand side, of (1a) is invariant un-der arbitrary coordinate transformations for any choiceof the vector ( A ).The n -Bein field is determined by n functions h νa ,while the Riemann metric is determined by merely n ( n +1)2 quantities g µν . According to (3) the metric is given bythe n -Bein field, but not vice versa. II. DISTANT PARALLELISM AND ROTATIONALINVARIANCE
By positing the n -Bein field, the existence of the Rie-mann metric and of distant parallelism are expressed si-multaneously. If ( A ) and ( B ) are two vectors at thepoints P and Q respectively, which w.r.t. the local n -Beins have equal local coordinates ( i.e. A a = B a ) theyare to be regarded as equal (on account of (2)) and as“parallel.”If we consider only the essential, i.e. the objectivelymeaningful, properties to be the metric and distant par-allelism, we recognize that the n -Bein field is not yet com-pletely determined by these demands. Both the metricand distant parallelism remain intact if one replaces the n -Beins of all points of the continuum by others whichresult from the original ones by a common rotation. Wecall this replaceability of the n -Bein field rotational in-variance and assume: Only rotationally invariant math-ematical relationships can have real meaning.eeping a fixed coordinate system, and given a metricas well as a distant parallelism relationship, the h aµ arenot yet fully determined; a substitution of the h aν is stillpossible which corresponds to rotational invariance, i.e. the equation A ∗ a = d am A m · · · (6)where the d am is chosen to be orthogonal and indepen-dent of the coordinates. ( A a ) is an arbitrary vector w.r.t.the local coordinate system; ( A ∗ a ) is the same one in termsof the rotated local system. According to (1a), equation(6) yields h ∗ µa A µ = d am h µm A µ or h ∗ µa = d am h µm , · · · (6a)where d am d bm = d ma d mb = δ ab , · · · (6b) ∂d am ∂x ν = 0 . · · · (6c)The assumption of rotational invariance then requiresthat equations containing h are to be regarded as mean-ingful only if they retain their form when they are ex-pressed in terms of h ∗ according to (6). Or: n -Bein fieldsrelated by local uniform rotations are equivalent. Thelaw of infinitesimal parallel transport of a vector in goingfrom a point ( x ν ) to a neighboring point ( x ν + dx ν ) isevidently characterized by the equation dA a = 0 · · · (7)which is to say the equation0 = d ( h µa A ν ) = ∂h µa ∂x τ A µ dx τ + h µa dA µ = 0 . Multiplying by h νa and using (5), this equation becomeswhere dA ν = − ∆ νµσ A µ dx τ ∆ νµσ = h νa ∂h µa ∂x σ . (7a)This parallel transport law is rotationally invariant and isunsymmetrical with respect to the lower indices of ∆ νµσ .If the vector ( A ) is moved along a closed path accord-ing to this law, it returns to itself; this means that theRiemann tensor R , defined in terms of the transport co-efficients ∆ νµσ , R ik,lm = − ∂ ∆ ikl ∂x m + ∂ ∆ ikm ∂x l + ∆ iαl ∆ αkm − ∆ iαm ∆ αkl will vanish identically because of (7a)—as can be verifiedeasily. Besides this parallel transport law there is another(nonintegrable) symmetrical law of transport that be-longs to the Riemann metric according to (2) and (3).It is given by the well-known equations dA ν = − Γ νµσ A µ dx τ Γ νµσ = g νa (cid:16) ∂g µα ∂x τ + ∂g τα ∂x µ − ∂g µσ ∂x α (cid:17) . (8)The Γ νµσ symbols are given in terms of the n -Bein field h according to (3). It should be noted that g µν = h µa h νa . · · · (9)Equations (4) and (5) imply g µλ g νλ = δ µν which defines g µν in terms of g µν . This law of transportbased on the metric is of course also rotationally invariantin the sense defined above. III. INVARIANTS AND COVARIANTS
In the manifold we have been studying, there exist, inaddition to the tensors and invariants of Riemann geom-etry, which contain the quantities h only in the combi-nations given by (3), further tensors and invariants, ofwhich we want to consider only the simplest.Starting from a vector ( A ν ) at the point ( x ν ), the twotransports d and ¯ d to the neighboring point ( x ν + dx ν )result in the two vectors A ν + dA ν and A ν + dA ν . The difference dA ν − dA ν = (Γ ναβ − ∆ ναβ ) A α dx β is also a vector. HenceΓ ναβ − ∆ ναβ is a tensor, and so is its antisymmetric part12 (∆ ναβ − ∆ νβα ) = Λ ναβ . · · · The fundamental meaning of this tensor in the theoryhere developed emerges from the following: If this tensorvanishes, the continuum is Euclidian. For if0 = 2Λ ναβ = h a (cid:18) ∂h αa ∂x β + ∂h βa ∂x α (cid:19) , (10)then multiplication by h νb yields0 = ∂h αb ∂x β + ∂h βb ∂x α .
2e can therefore put h αb = ∂ Ψ b ∂x α . The field is therefore derivable from n scalars Ψ b . Wechoose the coordinates according to the equationΨ b = x b . (11)Then, according to (7a) all ∆ ναβ vanish, and the h µa aswell as the g µν are constant.Since the tensor Λ ναβ is evidently also formally the sim-plest one allowed by our theory, the simplest character-ization of the continuum will be tied to Λ ναβ , not to themore complicated Riemann curvature tensor. The sim-plest forms that can come into play here are the vectorΛ αµα as well as the invariants g µν Λ αµβ Λ βνα and g µν g ασ g βτ Λ µαβ Λ νστ . From one of the latter (or from linear combinations) aninvariant integral J can be constructed by multiplication with the invariant volume element h dτ, where h is the determinant of | h µa | , and dτ is the product dx . . . dx n . The assumption δJ = 0yields 16 differential equations for the 16 values of h µa .Whether one can get physically meaningful laws in thisway will be investigated later.It is helpful to compare Weyl’s modification of Rie-mann’s theory with the theory developed here:WEYL: Comparison neither of distant vectormagnitudes nor of directions;RIEMANN: Comparison of distant vectormagnitudes, but not of distant directions;THIS THEORY: Comparison of distant vec-tor magnitude and directions.3 instein’s 1928 manuscript on unification of gravity and electromagnetism ew possibility for a unified field theory of gravity and electricity A. EinsteinJune 14, 1928A few days ago I explained in a short paper in theseReports how it is possible to use an n-Bein-Field to for-mulate a geometric theory based on the fundamentalconcepts of the Riemann metric and distant parallelism.At the time I left open the question whether this the-ory could serve to represent physical relationships. Sincethen I have discovered that this theory—at least in firstapproximation—yields the field equations of gravity andelectromagnetism very simply and naturally. It is there-fore conceivable that this theory will replace the originalversion of the theory of relativity.The introduction of distant parallelism implies that ac-cording to this theory there is something like a straightline, i.e. a line whose elements are all parallel to eachother; of course such a line is in no way identical to ageodesic. Furthermore, in contrast to the usual generaltheory of relativity, there is the concept of relative rest oftwo mass points (parallelism of two line elements whichbelong to two different worldlines.)In order for the general theory to be useful immediatelyas field theory one must assume the following:1. The number of dimensions is 4 ( n = 4).2. The fourth local component A a ( a = 4) of a vectoris pure imaginary, and hence so are the componentsof the four legs of the Vier-Bein, the quantities h µ and h µ . The coefficients g µν (= h µα h να ) of course all become real.Accordingly, we choose the square of the magnitude of atimelike vector to be negative. I. THE UNDERLYING FIELD EQUATION
Let the variation of a Hamiltonian integral vanish forvariations of the field potentials h µα (or h µα ) that vanishon the boundary of a domain: δ (cid:26)Z H dτ (cid:27) = 0 . · · · (1) H = h g µν Λ µαβ Λ νβα , · · · (1a) Instead one could also define the square of the magnitude of thelocal vector A to be A + A + A − A and introduce Lorentztransformations instead of rotations of the local n-Bein. In thatcase all the h ’s would be real, but the immediate connection withthe general theory would be lost. where the quantities h (= det h µα ), g µν , and Λ αµν aredefined in (9) and (10) of the previous paper.Let the h field describe the electrical and the gravita-tional field simultaneously. A “purely gravitational field”results when equation (1) is fulfilled and, in addition, φ µ = Λ µαα · · · (2)vanish, which represents a covariant and rotationally in-variant subsidiary condition. II. THE FIELD EQUATION IN THE FIRSTAPPROXIMATION