General Relativity by Kawaguchi geometry
Erico Tanaka1,2,3,a
1Department of Mathematics,Torino University, via Carlo Alberto 10, 10123 Torino, Italy
2Department of Algebra and Geometry, Faculty of Science, Palacky University, 17. listopadu 12, 771 46
Olomouc, Czech Republic
3Physics Department, Ochanomizu University, 2-1-1 Ootsuka Bunkyo, Tokyo, Japan
Abstract.We construct a parameterisation invariant Lagrange theory of fields up to
sec-ond order by using multivector bundles and Kawaguchi geometry. In this setup, the spacetime is an dynamical object which is a submanifold of the greater manifold, and the actual spacetime is the solution of Euler-Lagrange equations. Such theory is a reasonable mathematical foundation to describe an extended theory of Einstein’s general relativity, and is capable of being a stage for unification with other physical fields.
1 Introduction
The standard physical theory of fields is constructed on the fibre bundle structure, with the spacetime being the base manifold, and fields being its section. In such construction, the base manifold is chosen at the beginning, therefore becomes the "background", and naturally the target of study is mainly on the fields, not on its background spacetime. However, in such cases when one needs to consider spacetime itself as a dynamical object, it is apparent that this structure is insufficient. Such considerations arise especially in the past few decades research on the attempt to combine gravity with other fundamental forces or obtain its quantisation, where spacetime itself is the topic of discussion. In this letter, we will introduce a geometric foundation for considering such cases. The spacetime is introduced as a submanifoldΣof a bigger manifoldMthat is the total space of all dynamical variables to be considered, namely, physical fields and spacetime. This way of considering spacetime together with fields have been proposed in [1, 2], for a local coordinate system using non-linear forms. Here we will use a structure called multivector bundles on manifolds. In contrast to the standard fibre bundle approach, we will assume no fibration, however, there still exists a natural bundle structure called the multivector bundle overM, which is an extension of the tangent bundle. To consider physically meaningful theory, we endow this multivector bundle a structure called Kawaguchi functionK, and together with the total spaceM, it constructs a Kawaguchi manifold which is a natural extension of Finsler manifold.>From this Kawaguchi function, it is possible to construct a Kawaguchi differential form, which could be taken as a Lagrangian. This form integrated over ak-dimensional submanifold is the action of the whole system, and by the calculus of variation, we will obtain the Euler-Lagrange equations of the submanifold, which its extremal corresponds to the actual spacetime.
In the following, we will give the basic foundations, first for the case when the dimension of spacetime is 1, namely the mechanics, and then for the fields.
ae-mail: [email protected] C
2 First order mechanics
In this section, we briefly review the Lagrange formulations of first order mechanics on Finsler mani-fold. We begin with the definition of a Finsler manimani-fold.
Definition 2.1. Finsler manifold
LetMbe aC∞-differentiable manifold, (T M, τM,M) its tangent bundle,T0M:=T M\0 the slit tangent bundle excluding the zero section fromT M, and (U, ϕ), ϕ=(xµ, yµ), µ=1,· · ·,na chart onT M. The
n-dimensional Finsler manifoldis a pair (M,F) whereFis aC0function onT MandC∞function on
T0M, satisfying the following homogeneity conditions,
F(xµ, λyµ)=λF(xµ, yµ), λ >0. (2.1)
This condition is also equivalent to the condition of Euler’s homogeneous function theorem,
∂F ∂yµy
µ
=F. (2.2)
Function with such properties is called aFinsler function.
We will use this minimal definition as our Finsler manifold.
Given a Finsler manifold, we obtain an important geometrical structure called aHilbert form[3].
Definition 2.2. Hilbert form
TheHilbert formF is a 1-form onT0M, which in local coordinates are expressed by
F = ∂F
∂yµdx µ.
(2.3)
The Hilbert form is invariant with respect to the coordinate transformations byxµ→x˜µ=x˜µ(xν),
yµ→y˜µ= ∂x˜
µ
∂xνy ν
, and acts as a Lagrangian of mechanics, when integrated over a one dimensional submanifold, a curve. LetCbe this parameterisable curve onM, and suppose we have an immersion
σfrom an intervalI = [ti,tf] ∈ Rto this curveC, i.e.,C = σ(I). Then the Hilbert form defines a
Finsler length lCF[σ] ofCby
lCF[σ]=
Z
ˆ σ(I)
F =
Z tf
ti ∂F ∂yµ( ˆσ(t))
d(xµ(σ(t)) dt dt=
Z tf
ti
F( ˆσ(t))dt (2.4)
where ˆσis a tangent lift ofσdefined by ˆσ(t)= d(x
µ◦σ) dt t ∂ ∂xµ !
σ(t)
. The Finsler length is invariant
with respect to change of parameterisationρ=σ◦φ,φ∈Diff(R), which preserves orientation, and fixed at the boundary. We consider this as anaction of mechanics.
The extremal of the Finsler length is the solution curve of the Euler-Lagrange equations. The Euler-Lagrange equation of (2.4) is obtained by considering a flowαs,s ∈RonM, and comparing the value of (2.4) by every possible deformations. The variation of the action is then obtained by,
δlCF[σ] = lim s→0
1
s "Z
Tαs◦bσ◦idI−1(I)
F −
Z
b σ(I)
F
#
=
Z
b σ(I)
LXF.=
Z
I ˆ
σ∗LXF. (2.5)
Xis a vector field onT Mgenerated byTαs,X=
d(Tαs)
ds . In local coordinate expressions,
X=ξµ◦τM
∂ ∂xµ
!
+∂ξ
µ
∂xν ◦τM·y ν ∂
∂yµ !
whereξ=ξµ ∂
∂xµ is a vector field generated byαs,ξ= dαs
ds . The extremal condition is,
ˆ
σ∗ (
∂2F ∂xµ∂yρdx
ρ−d ∂F ∂yµ
!)
=0, σˆ∗
( ∂2F ∂yµ∂yρ
! dxρ
)
=0. (2.7)
The second equation becomes an identity by the homogeneity condition, and the first becomes the Euler-Lagrange equations, also by the homogeneity condition.
3 First order fields
Kawaguchi considered two directions of extending the Finsler geometry [4, 5]. The first in higher order derivatives and the second in the way to extend the parameter space, namely the spacetime. The latter is also called as areal metric geometry. For a higher order field theory, we need the combination. We will first define the geometric structure on the total space of a k-multivector bundle (ΛkT M,Λkτ
M,M). We will call this structure a first orderk-areal Kawaguchi function.
Definition 3.1. Kawaguchi manifold (First orderk-dimensional parameter space)
LetMbe an-dimensionalC∞-differentiable manifold, (U, ϕ), ϕ=(xµ, yµ1···µk),µ, µ
1,· · ·, µk=1,· · ·,n be a chart on ΛkT M, and K ∈ C∞(ΛkT M) withk 6 n that satisfies the following homogeneity condition,
K(xµ, λyµ1···µk)=λK(xµ, yµ1···µk), λ >0. (3.1)
We will call the function with such properties, afirst order k-areal Kawaguchi function, and the pair (M,K) an-dimensional k-areal Kawaguchi manifold, or simply aKawaguchi manifold.
The condition (3.1) is equivalent to the following,
1
k!
∂K ∂yµ1···µky
µ1···µk =K. (3.2)
We will call the manifoldMa total space, in the sense it contains both spacetime and the field. Given an-dimensionalk-areal Kawaguchi manifold (M,K), we can obtain a structure which we will call a
Kawaguchi k-form. Kawaguchik-form is constructed in accord with the condition (3.2), and gives the Lagrangian of a field theory when pulled back to the parameter space, namely the spacetime, by a certain parameterisation.
Definition 3.2. Kawaguchik-form (first order field theory)
TheKawaguchi k-formKis ak-form onΛkT M, which in local coordinates are expressed by
K= 1
k!
∂K ∂yµ1···µkdx
µ1∧ · · · ∧dxµk. (3.3)
With thisk-form,we can define a parameterisation invariantk-area of the submanifoldΣofM.
Definition 3.3. Kawaguchik-area
Consider the parameterisationσ: P→Σ⊂MwherePis a closed rectangleP=[ti1,t1f]×[t2i,t2f]× · · · ×[tk
i,t k f]⊂R
k. A Kawaguchik-area ofΣis defined by,
lΣK[σ] =
Z
ˆ σ(P)
K=
Z t1 f
t1 i
· · ·
Z tk f tk i 1 k! ∂K
∂yµ1···µk( ˆσ(t))y
µ1···µk( ˆσ(t))dt1∧ · · · ∧dtk
=
Z t1 f
t1 i
dt1· · ·
Z tk f
tk i
dtkK xµ(σ(t)),∂(x
[µ1(σ(t)))
∂t1 · · ·
∂(xµk](σ(t))) ∂tk
!
where ˆσis a multi-tangent lift ofσdefined by
ˆ
σ(t)= ∂(x
µ1◦σ)
∂t1 t· · ·
∂(xµk◦σ) ∂tk
t
∂
∂xµ1 ∧ · · · ∧ ∂ ∂xµk
!
σ(t)
, t∈P. (3.5)
Then we will have the following lemma,
Lemma 3.4.The arealΣK[σ] defined by (3.4) is invariant with respect to reparameterisationρ=σ◦φ,
φ∈Diff(Rk), whereρpreserves orientation, and the boundary ofPis fixed.
The Euler-Lagrange expressions could be obtained by similar considerations.
Theorem 3.5. Variational formula of Kawaguchik-area
The extremal conditions of (3.4) is given by,
ˆ
σLXˆK=0, (3.6)
where ˆXis a vector field generated by the multi-tangent flow,ΛkTα
s, induced by a flowαsonM.
In local coordinate expression, (3.6) becomes:
ˆ
σ∗ (
∂2K ∂xµ∂yρ1···ρkdx
ρ1−kd ∂K ∂yρ1···ρk
!!
∧dxρ2···ρk )
=0,
ˆ
σ∗ (
yνµ2···µk ∂ 2K
∂yµ1···µk∂yρ1···ρk !
dxρ1···ρk )
=0.
(3.7)
The second equation becomes an identity by the homogeneity condition, and the first becomes the Euler-Lagrange equations, also by the homogeneity condition.
4 Second order fields
Here we will briefly describe the geometrical structures for higher order field theory. We will consider this by referring to the second order mechanics. The structure used for second order mechanics is the second order tangent bundle (T2M, π,M), which is not a vector bundle. It is constructed in the following way. Let (T T M, τT M,T M) be an iterated tangent bundle. Then, (T2M, τT M|T2M,T M) is a
sub-bundle of (T T M, τT M,T M), defined byT2M ={∀v ∈ T T M|TτM(v) =τT M(v)}, whereτM is a tangent projectionτM : T M → M, andτT M is a tangent projectionτT M : T T M → T M. Then the composed projectionπ=τM◦τT M|T2Mgives the second order tangent bundle. We similarly consider
the second order multivector space by this construction.
Definition 4.1. The second orderk-multivector bundle The structure ((ΛkT)2M, π,M) with the fol-lowing properties is called thesecond order k-multivector bundle.
1.(ΛkT
)2M={∀v∈ΛkT
ΛkT M|ΛkT
ΛkτM(v)= ΛkτΛkT M(v)} (4.1)
2. π= ΛkτM◦ΛkτΛkT M|(ΛkT)2M is a surjective submersion. (4.2)
The second orderk-areal Kawaguchi manifold is defined as follows.
Definition 4.2. Second orderk-areal Kawaguchi manifold
Let (M,K) be a pair of n-dimensional C∞-differentiable manifold M and a function K ∈
µ, µ1,· · ·, µk, ν2,· · ·, νk = 1,· · ·,n, and multi index notation: Ij := µij1· · ·µijk, on (ΛkT)2M, satis-fies the followingsecond order homogeneity condition,
K(xµ, λyµ1···µk,(λ)2zI1ν2,···νk +λν2···νkyI1,(λ)2zI1I2ν3,···νk+λν3···νkyI1yI2,· · ·,(λ)2zI1I2···Ik+λ0yI1yI2· · ·yIk)
=λK(xµ, yµ1···µk,zI1ν2,···νk,zI1I2ν2,···νk,· · ·,zI1I2···Ik), (4.3)
forλ >0, andλν2···νk, λν3···νk,· · ·, λνk, λ0being arbitrary constants. We will call the function with such
properties, asecond order k-areal Kawaguchi function, and the pair (M,K) an-dimensional second order k-areal Kawaguchi manifold.
Using the homogeneity condition as a guide, we can construct a geometric structure.
Definition 4.3. The second order Kawaguchi k-form K is a k-form on (ΛkT)2M
, which in local coordinates are expressed by
K= 1
k!
∂K ∂yµ1···µkdx
µ1···µk + 2
(k−1)!
∂K ∂zI1ν2···νkdy
I1∧dxν2···νk+ 2
(k−2)!
∂K ∂zI1I2ν3···νkdy
I1∧dyI2∧dxν3···νk
+· · ·+2 ∂K
∂zI1I2···Ikdy
I1∧ · · · ∧dyIk. (4.4)
We used the abbreviation such asdxµ1···µk :=dxµ1∧· · ·∧dxµk,dyI1∧dxν2···νk :=dyI1∧dxν2∧· · ·∧dxνk.
Now we can define thek-area;
Definition 4.4. Thesecond order Kawaguchi k-areais defined by,
lΣK[σ]=
Z
σ2(P)
K=
Z t1 f
t1 i
dt1 Z t2
f
t2 i
dt2· · ·
Z tk f
tk i
dtkKσ2(t), t∈P, (4.5)
whereσ2is a second order lift of parameterisationσ→ Σ, defined by the iterated tangent lift with
conditionΛkTΛkτ
M(σ(t))= ΛkτΛkT M(σ(t)), t∈P.
Lemma 4.5.The arealΣK[σ] defined by (4.5) is invariant with respect to reparameterisationρ=σ◦φ,
φ∈Diff(Rk), whereρpreserves orientation, and the boundary ofPis fixed.
This is the action of second order field theory. To obtain the Euler-Lagrange equations, we con-sider a flowαsonM, and it induces a flow (ΛkT)2αson (ΛkT)2M. Then we will have the following:
Theorem 4.6. Variational formula of second order Kawaguchik-area
The extremal conditions of (4.5) are given by,
σ2LX2K =0, (4.6)
whereX2is a vector field generated by the flow, (ΛkT)2αs.
To find the solution of Euler-Lagrange equations is equivalent to find the extremal of the action, which is a submanifold that corresponds to our spacetime embedded into a greater space together with other dynamical variables. General relativity could be regarded as an embedded theory, when a semi-Riemannian metric is induced on this submanifold.
References
[1] T. Ootsuka, arXiv:1206.6040v1 (2012)
[2] R. Yahagi, T. Ootsuka, E. Tanaka, Soryuushiron Kenkyu13(2012)
[3] W.H.C. S. S. Chern, K.S. Lam,Lectures on Differential Geometry(World Scientific, Singapore, 2000)
[4] A. Kawaguchi, Periodica Mathematica Hungarica7, 291 (1976)
