Real Projective Geodesics Embedded In Complex Manifolds
aa r X i v : . [ m a t h . G M ] J u l REAL PROJECTIVE GEODESICS EMBEDDED IN COMPLEX MANIFOLDS
SWAGATAM SEN
Abstract.
Focus of this study is to explore some aspects of mathematical founda-tions for using complex manifolds as a model for space-time. More specifically,certain equations of motions have been derived as a Projective geodesic on a realmanifold embedded within a complex one. To that goal, first the geodesic on com-plex manifold has been computed using local complex and conjugate coordinates,and then its projection on the real sub-manifold has been studied. The projec-tive geodesic, thus obtained, is shown to have additional terms beyond the usualChristo ff el symbols, and hence expands the geodesic to capture e ff ects beyond themere gravitational ones. Introduction
Motivations behind studying Complex manifolds within the realms of gravita-tional physics, or specifically quantum gravity, is a topic that has drawn significantattention since the days of formulation of General Relativity [1][2][3]. Einsteinhimself had progressed to consider a complex hermitian metric to unify gravity andelectromagnetism, albeit defined on real 4-dimensional coordinates[4][5]. Muchbefore that similar proposals for a complex metric was put forward by Weyl[6][7]and Soh[8] among others. As an approach, it doesn’t provide adequately strongstructure to conceptually integrate the real and imaginary parts of the metric. Forexample, under a generic continuous transformation both parts of the metric be-have quite independently. That line of investigation has persisted over time[9][10][11][12], albeit in a much reduced capacity. We have a fundamental weakness inthese approaches which has restricted its successes, because of the metric beingconsidered as function of exclusively real coordinates, or In other words, the frame-work used in these studies are that of a real manifold with a complex vector bundlestructure around it. It doesn’t draw in the stronger structure of complex metric de-fined on a complex manifold.
Date : July 15, 2020.2010
Mathematics Subject Classification.
Primary 53Z05; Secondary 57N16, 83D05.
Key words and phrases.
Complex manifold, Geodesic, Gravity, Lorentz Field, Unified Field.
In a way, that same weakness can be attributed to the standard mathematical ap-proach developed later for Quantum Mechanics, wherein measurements are con-ceptualised as Operators on a complex Hilbert space, which in turn is defined onreal phase space coordinates for a quantum system[13]. Consequently the un-derlying mathematical structure for quantum mechanics developed through nu-merous groundbreaking work[14][15], while tremendously accurate, appears tobe incomplete in addressing the longstanding foundational challenges - that ofinterpretation[16][17][18], and that of measurement[19][20][21] among others.It was primarily through the development of Twistor theory by Penrose[22], thatthe idea of a complex underlying space to describe the reality branched out into anarea of sustained focus. Essentially within Twistor theory, ‘real’ physical fields arerepresented as complex objects defined on complex projective space, called Twistorspace[23]. Mathematically, that approach o ff ers a much richer structure to workwith using contour integral formalism[24][25], and using connections with Stringtheory[26][27].In the present work we adopt the view of a spacetime model that’s inherentlya complex manifold with a real Riemannian sub-manifold describing the measur-able outcome. Specifically we look into the behaviour of Geodesics on a complexmanifold and the corresponding trajectory it traces as a projection onto the realsub-manifold.Throughout the next section we would work with a complex n-manifold M andits projective real submanifold R ⊂ M . Our objective would be to derive the geo-desic on the complex manifold first using the usual variational technique. Then wewould study how that geodesic maps to a curve on the projective sub-manifold R .We would refer this as the projective real geodesic on R and analyse how that di ff ersfrom the true geodesic on R . Definition 1.1.
For a complex n-manifold M with a holomorphic atlas { ( z, U ) } , R ⊂ M is a projective real sub-manifold if ∀ holomorphic charts ( z, U ) on M , ∃ a projection π : M R and a continuous chart χ : π ( U ) R n such that χ ◦ π = Re ( z ) (cid:3) This would allow us to write down the local coordinates as z α = x α + it α and z β = x β − it β where x = χ ◦ π and t = I m ( z ).Next we would need to introduce a hermitian positive definite metric g on M with the usual distance 2-form in local coordinates, dτ = g αβ dz α dz β EAL PROJECTIVE GEODESICS EMBEDDED IN COMPLEX MANIFOLDS 3
Our end goal in the next section would be to derive a geodesic equation usingthe complex/conjugate coordinates which is summarised in Theorem 2.2. Then inTheorem 2.5, we would further prove that this geodesic allows a projective geodesicon the real coordinates which has the generic structural form as below, h µγ D x µ = Υ (1 , γαβ Dx α Dx β − Υ (1 , γαβ Dx β Dt α − Υ (0 , γαβ Dt α Dt β for some tensor h and symbols Υ (0 , , Υ (1 , , Υ (1 , all of which are functions of themetric.These theorems would then allow us to consider certain known special casesand illustrate how equations of motions for gravitational and Lorentz Fields cannaturally manifest from Theorem 2.5 under special configurations.2. Results
To achieve the initial goal of deriving the geodesic on the complex manifoldusing complex/conjugate coordinates, we would follow the usual approach of vari-ational calculus.Under the metric g , the arc length for a given path parametrised by σ ∈ [0 , τ = R L ( z , d z dσ ) dσ , where L = s g αβ dz α dσ dz β dσ is the Lagrangian for equation of motion. Note.
Since g is hermitian, L is real and we can think of minimising the arc lengthin the usual variational approach. (cid:3) Note. the path can be re-parametrized using τ instead of arbitrary σ and the thederivatives can be replaced as ddσ = L ddτ . Notationally henceforth we would use D = ddτ . (cid:3) Proposition 2.1.
Let Dz γ = dz γ dτ and Dz γ = dz γ dτ be path derivatives of the complex andconjugate coordinates. Also let ∂ α = ∂∂z α and ∂ β = ∂∂z β be the partial derivatives withrespect to complex and conjugate coordinates. Then, ddσ h ∂L∂ ( dz γ /dσ ) i = L h ∂ β g αγ Dz α Dz β + ∂ β g αγ Dz α Dz β + g αγ D z α i (cid:3) S. SEN
Proof.
From definition of L we know that, ∂L∂ ( dz γ /dσ ) = 12 L g αγ dz α dσ That, in turn, would allow us to write, ddσ ∂L∂ ( dz γ /dσ ) = ddσ L g αγ dz α dσ = L ddτ g αγ dz α dτ = L h dg αγ dτ dz α dτ + g αγ d z α dτ i = L h ∂ β g αγ dz α dτ dz β dτ + ∂ β g αγ dz α dτ dz β dτ + g αγ d z α dτ i (cid:3) We can now use this result to derive the geodesic equation in complex and con-jugate coordinates.
Theorem 2.2.
Let M be a complex manifold with a hermitian metric g . If τ is thecumulative length of a given path Ω on M , then Ω is a geodesic i ff g µγ D z µ = (cid:16) ∂ γ g αβ − ∂ β g αγ (cid:17) Dz α Dz β − ∂ α g βγ Dz α Dz β (cid:3) Proof.
We’d first note that, ∂L∂z γ = L ∂ γ g αβ Dz α Dz β Now the result follows from a direct application of the Euler-Lagrange Eqn onconjugate coordinates ∂L∂z γ = ddσ h ∂L∂ ( dz γ /dσ ) i and replacing the right hand side of the equation using Proposition 2.1. (cid:3) We can use Theorem 2.2, to understand the trace of the complex geodesic pro-jected on the real coordinates and how di ff erent that trajectory is from the geodesicon the real sub-manifold. But before that we’d need to introduce a set of objectsthat would serve the equivalent purpose of Christo ff el symbols in the projectivegeodesic. Definition 2.1.
For a hermitian, positive definite metric g on a Complex manifold M , we define Primary Field as the combination Φ = (cid:16) Φ γ ++ αβ , Φ γ + − αβ , Φ γ − + αβ , Φ γ −− αβ (cid:17) EAL PROJECTIVE GEODESICS EMBEDDED IN COMPLEX MANIFOLDS 5 where • Φ γ ++ αβ = h ∂ xγ g Rαβ − ∂ xα g Rβγ − ∂ xβ g Rαγ i • Φ γ + − αβ = h ∂ xγ g Iαβ − ∂ xα g Iβγ − ∂ xβ g Iαγ i • Φ γ − + αβ = h ∂ tγ g Rαβ − ∂ tα g Rβγ − ∂ tβ g Rαγ i • Φ γ −− αβ = h ∂ tγ g Iαβ − ∂ tα g Iβγ − ∂ tβ g Iαγ i are the ordinary Christo ff el symbols for real and imaginary parts of the metric inreal and imaginary coordinates respectively. (cid:3) Note. If g is real symmetric, then Φ γ + − αβ = Φ γ −− αβ = 0 Note. If g is independent of t , then Φ γ − + αβ = Φ γ −− αβ = 0 Note. Φ γ + − αα = Φ γ −− αα = 0 Proposition 2.3.
For a hermitian metric g = g R + ig I with Primary Field Φ , we canwrite,(1) ∂ γ g αβ − ∂ β g αγ − ∂ α g βγ = Φ γ ++ αβ + i Φ γ + − αβ + i ∂ tγ g Rαβ (2) ∂ γ g αβ − ∂ β g αγ + ∂ α g βγ = ∂ xγ g Rαβ − Φ γ ; −− αβ + i Φ γ − + αβ Proof.
As we know,, ∂ γ g αβ − ∂ β g αγ = 12 h ∂ xγ g Rαβ − ∂ tγ g Iαβ − ∂ xβ g Rαγ + ∂ tβ g Iαγ i + i h ∂ xγ g Iαβ + ∂ tγ g Rαβ − ∂ xβ g Iαγ − ∂ tβ g Rαγ i we can write, ∂ γ g αβ − ∂ β g αγ − ∂ α g βγ = 12 h ∂ xγ g Rαβ − ∂ xβ g Rαγ − ∂ xα g Rβγ i + i h ∂ xγ g Iαβ − ∂ xβ g Iαγ − ∂ xα g Iβγ i + i ∂ tγ g Rαβ = Φ γ ++ αβ + i Φ γ + − αβ + i ∂ tγ g Rαβ
Similarly, ∂ γ g αβ − ∂ β g αγ + ∂ α g βγ = 12 ∂ xγ g Rαβ − h ∂ tγ g Iαβ − ∂ tβ g Iαγ − ∂ tα g Iβγ i + i h ∂ tγ g Rαβ − ∂ tβ g Rαγ − ∂ tα g Rβγ i = − Φ γ −− αβ + i Φ γ − + αβ + 12 ∂ xγ g Rαβ (cid:3)
S. SEN
Definition 2.2.
For a hermitian, positive definite metric g on a Complex manifold M , we define Secondary Field as F = (cid:16) F xαγβ , F tαγβ (cid:17) where • F xαγβ = ∂ xγ g Iαβ − ∂ xβ g Iαγ • F tαγβ = ∂ tγ g Iαβ − ∂ tβ g Iαγ (cid:3)
Note. Secondary Field symbols are anti-symmetric in β, γ
Note. If g is real symmetric, then F xαγβ = F tαγβ = 0 Note. Secondary Field symbols are covariant 2-tensors in β, γ indices under holo-morphic coordinate changes.
Proposition 2.4.
For a hermitian metric g = g R + ig I with Secondary field F , we canwrite, ∂ γ g Iαβ + i∂ xβ g αγ + ∂ tα g βγ = F xαγβ + ∂ tα g Iβγ − iF tβγα + i∂ xβ g Rαγ
Proof.
Clearly,2 ∂ γ g Iαβ + i∂ xβ g αγ + ∂ tα g βγ = ∂ xγ g Iαβ + i∂ tγ g Iαβ + i∂ xβ g Rαβ − ∂ xβ g Iαγ + ∂ tα g Rβγ + i∂ tα g Iβγ = ( ∂ xγ g Iαβ − ∂ xβ g Iαγ ) + ∂ tα g Iβγ − i ( ∂ tγ g Iαβ − ∂ tα g Iβγ ) + i∂ xβ g Rαγ = F xαγβ + ∂ tα g Iβγ − iF tβγα + i∂ xβ g Rαγ (cid:3)
Definition 2.3.
For a hermitian metric g on a Complex manifold M with g = g R + ig I , we define Link tensor as ǫ ηγ = g Iνγ g R ; νη (cid:3) Note. If g is real symmetric then ǫ ηγ = 0 Theorem 2.5.
Let R be a real projective manifold embedded within a complex manifold M . Let g be a hermitian, positive definite metric on M with g = g R + ig I . Additionallylet Φ , F be the primary and secondary fields respectively and let ǫ be the link symbol.Then R is endowed with a real projective geodesic described by (cid:16) g Rµγ + ǫ ηγ g Iµη (cid:17) D x µ = Υ (1 , γαβ Dx α Dx β − Υ (1 , γαβ Dx β Dt α − Υ (0 , γαβ Dt α Dt β where EAL PROJECTIVE GEODESICS EMBEDDED IN COMPLEX MANIFOLDS 7 • Υ (1 , γαβ = Φ γ ++ αβ + ǫ ηγ Φ γ − + αβ + ǫ ηγ ∂ tη g Rαβ • Υ (1 , γαβ = F xαγβ − ǫ ηγ F tβηα + ∂ tα g Rβγ + ǫ ηγ ∂ xβ g Rαη • Υ (0 , γαβ = Φ γ −− αβ − ∂ xγ g Rαβ − ǫ ηγ Φ γ − + γαβ (cid:3) Proof.
First of all we can write, h ∂ γ g αβ − ∂ β g αγ i Dz α Dz β = h ∂ γ g αβ − ∂ β g αγ ih Dx α Dx β + Dt α Dt β + i (cid:16) Dt α Dx β − Dt β Dx α (cid:17)i = h ∂ γ g αβ − ∂ β g αγ ih Dx α Dx β + Dt α Dt β i + i h ∂ γ g Iαβ − ∂ β g αγ + ∂ α g βγ i Dt α Dx β Also, ∂ β g αγ Dz α Dz β = ∂ β g αγ h Dx α Dx β − Dt α Dt β + i (cid:16) Dt α Dx β + Dt β Dx α (cid:17)i = ∂ α g βγ h Dx α Dx β − Dt α Dt β i + i h ∂ β g αγ + ∂ α g βγ i Dt α Dx β That implies, h ∂ γ g αβ − ∂ β g αγ i Dz α Dz β − ∂ β g αγ Dz α Dz β = h ∂ γ g αβ − ∂ β g αγ − ∂ α g βγ i Dx α Dx β + h ∂ γ g αβ − ∂ β g αγ + ∂ α g βγ i Dt α Dt β − h ∂ γ g Iαβ + i∂ xβ g αγ + ∂ tα g βγ i Dt α Dx β Now using this last equation alongside Theorem 2.2, Propositions 2.3 and 2.4,we get, g µγ D z µ = h ∂ γ g αβ − ∂ β g αγ i Dz α Dz β − ∂ β g αγ Dz α Dz β = ( Φ γ ++ αβ + i Φ γ + − αβ + i ∂ tγ g Rαβ ) Dx α Dx β + ( 12 ∂ xγ g Rαβ − Φ γ ; −− αβ + i Φ γ − + αβ ) Dt α Dt β − ( F xαγβ + ∂ tα g Iβγ − iF tβγα + i∂ xβ g Rαγ ) Dt α Dx β But we’d note that, g µγ D z µ = (cid:16) g Rµγ D x µ − g Iµγ D t µ (cid:17) + i (cid:16) g Rµγ D t µ + g Iµγ D x µ (cid:17) Equating real and imaginary parts we get the dual identity, g Rµγ D x µ − g Iµγ D t µ = Φ γ ++ αβ Dx α Dx β + ( 12 ∂ xγ g Rαβ − Φ γ ; −− αβ ) Dt α Dt β − ( F xαγβ + ∂ tα g Iβγ ) Dt α Dx β and S. SEN g Rµη D t µ + g Iµη D x µ = ( Φ η + − αβ + 12 ∂ tη g Rαβ ) Dx α Dx β + Φ η − + αβ Dt α Dt β − ( ∂ xβ g Rαη − F tβηα ) Dt α Dx β Multiplying the second identity by ǫ ηγ and adding the identities, we arrive at thedesired result. (cid:3) Next we’d investigate some of the specific ramifications and special cases of The-orem 2.5 through a set of corollaries.
Corollary.
Let M be a 4-dimensional complex manifold with a real symmetric metric g and let R be a projective real sub-manifold. Additionally let’s assume that ∀ α, β , ∂ x g αβ = ∂ t g αβ = ∂ t g αβ = ∂ t g αβ = 0 Then the projective geodesic is identical to the geodesic on R . (cid:3) Proof.
Proof follows by considering g I = 0 and setting the specific partial deriva-tives to 0, in which case Theorem 2.5 simplifies to D x µ = g µγ Φ γ ++ αβ Dx α Dx β − g µγ ∂ t g βγ Dt Dx β − g µγ ∂ xγ g ( Dt ) where α, β, µ > D y µ = − Γ µab Dy a Dy b where a, b > , µ > y = ( t , x , x , x ). Γ denotes the usual Christo ff el symbolsand the equation precisely signifies the classical geodesic path. (cid:3) We’d refer the quantity G µ = − Γ µab Dy a Dy b as the Gravitation field.It’s interesting to note that Theorem 2.5 doesn’t prescribe a single equation for D x µ . Rather it describes a family of geodesics parametrised by the extrinsic pa-rameters Dt α . One particular choice of such parameters and consequentially, geodesics,would be the Root Geodesics, where Dt α = 1 , ∀ α Corollary.
Let M be a 4-dimensional complex manifold with a hermitian metric g with ǫ ηγ → . Also let R be a projective real sub-manifold. Additionally let’s assume that ∀ α, β , ∂ x g αβ = ∂ t g αβ = ∂ t g αβ = ∂ t g αβ = 0 Then the projective root geodesic is given by, D x µ = G µ + L µ EAL PROJECTIVE GEODESICS EMBEDDED IN COMPLEX MANIFOLDS 9 where L µ = − g R ; µγ F x γβ Dx β − g R ; µγ Φ γ −− (cid:3) Proof.
Proof follows by admitting the limit ǫ ηγ → Dt = 1 in Theo-rem 2.5 g Rµγ D x µ = Φ γ ++ αβ Dx α Dx β − ∂ t g βγ Dx β − g µγ ∂ xγ g − F x γβ Dx β − Φ γ −− = g Rµγ G γ − F x γβ Dx β − Φ γ −− (cid:3) L µ would be referred as the Lorentz field. Indeed when g Rµγ = δ µγ is Euclidean,we have a familiar form of a Lorenz field, L µ = − F x γβ Dx β − Φ γ −− where F x γβ represents anti-symmetric Magnetic tensor and Φ γ −− represents Elec-tric field. In fact, since, F x γβ = ∂ xγ g I β − ∂ xβ g I γ naturally we can characterise A γ = g I γ as the magnetic potential. If we allow φ such that ∂ xγ φ = Φ γ −− then we can write ( φ, A , A , A ) as the 4-potential for thefield.In the most general case when ǫ ηγ isn’t insignificant but still k ǫ k <
1, we canrewrite Theorem 2.5 for D x µ as an infinite sum of perturbations involving dimin-ishing powers of ǫ . 3. Discussion and Conclusion
In the preceding section we have been able to derive the most generic form forthe
Projective real geodesics. While Theorem 2.2 describes the said geodesic overcomplex and conjugate coordinates, in Theorem 2.5 we’ve been able to arrive ata much more useful expression for the geodesic in terms of real and imaginarycoordinates. Through the results, we have seen that in general the derived projec-tive geodesic deviates from the true geodesic on the real coordinates, and indeedcontains additional contributions to the ‘Force field’. Careful examination of theseadditional terms reveal that they vanish when the underlying metric is real sym-metric and the manifold itself can be locally mapped to the Minkowski spacetime.For these
Gravitational systems it is shown that the Projective geodesic is identicalto the true geodesic prescribed by General Relativity.
However, in presence of an anti-symmetric imaginary component of the metric,we have seen that Theorem 2.5 naturally yields additional terms. When the imagi-nary component of the metric is relatively small in magnitude compared to the realpart, these additional terms are shown to manifest as familiar Lorentz field with 4-potential being a function of the metric itself. This is a remarkable result becauseunlike the existing relativistic approaches to Electromagnetism, it doesn’t merelyshow that Lorenz Field is compatible with GR, but rather it is a manifestation ofthe spacetime itself, as is gravity.For the case when both real and imaginary parts of the metric are of comparablescale, the Projective geodesic incrementally covers an infinite series of perturbativecorrections involving diminishing contributions from the higher powers of the linktensor. This general case and perturbative terms would need to be investigated inmore details within a separate follow up study.
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