Gravitational fields and harmonic gauge in brane model of Universe

TL;DR

This paper proposes a brane universe model in higher-dimensional space, deriving equations linking gravitational fields, chiral currents, and gravitational waves, and showing how harmonic gauge conditions naturally emerge from symmetry considerations.

Contribution

It introduces a novel approach connecting brane symmetry with gravitational wave propagation and harmonic gauge conditions in a higher-dimensional universe model.

Findings

Derived Klein-Gordon equation from universe expansion symmetry.

Linked gravitationally induced chiral current to inertial and vortex gravitational fields.

Showed harmonic gauge conditions emerge from symmetry in weak gravitational fields.

Abstract

We suppose that our Universe is closed manifold in real embedding higher dimensional space. This model well describes expanding character of Universe where each point becomes more far from any other point with time. We have derived Klein-Gordon equation using the symmetry of Universe expansion. Comparing it with squared Dirac-Fock-Ivanenko equation we have derived expression for gravitationally induced chiral current. Starting from the expression for gravitationally induced chiral current obtained on the basis of brane symmetry we have shown that being placed into the Einstein's gravitational field, chiral current yields inertial gravitational field. This inertial gravitational field transforms into the vortex gravitational field and then back to the inertial gravitational field thus ensuring the propagation of such kind of gravitational excitation as constitutional part of gravitational wave. In the case of weak gravitation, these symmetry considerations and our consequent equations lead directly to harmonic gauge conditions that are necessary for supplementing Einstein equations.

Equations

$$\left\{g^{\mu \nu }{\nabla }_{\mu }{\nabla }_{\nu }+\frac{1}{4}R+{\left(\frac{mc}{\hslash }\right)}^2\right\}\Psi =0$$

$$i{\gamma }^{\mu }({\nabla }_{\mu }+{\Gamma }_{\mu })\psi =m\psi$$

$${\gamma }^{\mu }{\gamma }^{\nu }{D}_{\mu }{\Gamma }_{\nu }=-\frac{1}{4}R$$

$$j^{\alpha 5}{\epsilon }^{\mu \nu \tau \xi }{R}_{\mu \nu \tau \xi }=4i{\gamma }^{\alpha }{D}_{\mu }{\Gamma }^{\mu }$$

$${\gamma }^5\frac{1}{4!}{\epsilon }_{\zeta \lambda \chi \sigma }{\epsilon }^{\mu \nu \tau \xi }{R}_{\mu \nu \tau \xi }=\frac{1}{4!}4i{\epsilon }_{\zeta \lambda \chi \sigma }{\gamma }^{\alpha }{D}_{\mu }{\Gamma }^{\mu }$$

$$j^{\alpha 5}{\delta }_{\zeta \lambda \chi \sigma }^{\mu \nu \tau \xi }{R}_{\mu \xi \nu \tau }=\frac{i}{6}{\epsilon }_{\zeta \lambda \chi \sigma }{\gamma }^{\alpha }{D}_{\mu }{\Gamma }^{\mu },$$

$$j^{\alpha 5}{R}_{\zeta \lambda \chi \sigma }={\epsilon }_{\zeta \lambda \chi \sigma }{j}^{\alpha },$$

$$j^{\alpha }=\frac{i}{6}{\gamma }^{\alpha }{D}_{\mu }{\Gamma }^{\mu },$$

$$j^{\alpha 5}{R}_{\lambda \sigma }={\epsilon }_{\zeta \lambda \chi \sigma }{g}^{\zeta \chi }{j}^{\alpha },$$

$$R_{\lambda \sigma }-\frac{1}{2}{g}_{\lambda \sigma }R=\frac{4\pi G}{c^4}{T}_{\lambda \sigma },$$

$${\gamma }^{\mu }{\gamma }^{\nu }{D}_{\mu }{\Gamma }_{\nu }=-\frac{1}{4}R$$

$$D_{\mu }{\Gamma }^{\mu }\frac{1}{2}{\gamma }_{\mu }{\Gamma }_{\xi }(D_{\mu }{\Gamma }_{\xi }-D_{\xi }{\Gamma }_{\mu })=-\frac{1}{4}R$$

$$R_{\lambda \sigma }-\frac{1}{2}{g}_{\lambda \sigma }R=\frac{4\pi G}{c^4}{T}_{\lambda \sigma },$$

$$j^{\alpha 5}{R}_{\lambda \sigma }=\frac{i}{6}{\epsilon }_{\zeta \lambda \chi \sigma }{g}^{\zeta \chi }{\gamma }^{\alpha }{D}_{\mu }{\Gamma }^{\mu },$$

$$D_{\mu }{\Gamma }^{\mu }\frac{1}{2}{\gamma }_{\mu }{\Gamma }_{\xi }(D_{\mu }{\Gamma }_{\xi }-D_{\xi }{\Gamma }_{\mu })=-\frac{1}{4}R$$

$${\Gamma }_{\kappa \nu }^{\xi }=\frac{1}{2}{g}^{\xi \lambda }\left(\frac{g_{\lambda \kappa }}{\partial x_{\mu }}+\frac{g_{\lambda \mu }}{\partial x_{\kappa }}-\frac{g_{\kappa \mu }}{\partial x_{\lambda }}\right),$$

$${\Gamma }^{\kappa }=\frac{1}{2}{g}^{\alpha \beta }{\Gamma }_{\alpha \beta }^{\kappa },$$

$$g_{\mu \nu }(x)={\eta }_{\mu \nu }+h_{\mu \nu }.$$

$${\gamma }^{\mu }{\gamma }^{\nu }\frac{\partial }{\partial x_{\mu }}\left(\frac{\partial h_{\nu }^{\alpha }}{\partial x_{\alpha }}-\frac{1}{2}\frac{\partial h}{\partial x_{\nu }}\right)=0.$$

$${\gamma }^{\mu }{\gamma }^{\nu }\frac{\partial V_{\nu }}{\partial x_{\mu }}=0$$

$$V_{\nu }=\frac{\partial h_{\nu }^{\alpha }}{\partial x_{\alpha }}-\frac{1}{2}\frac{\partial h}{\partial x_{\nu }}=0,$$

Full text

††journal: Eur. Phys. J. C

1 Introduction

Brane can be regarded as multidimensional spatial manifold supporting the existence of particle in Universe. Zero dimen- sional brane is a point particle, one dimensional brane is a string. Brane with p dimensions is called p-brane. Open string describing the particle living on brane can be connected by its ends to brane and the brane, in this case, is called D-brane. Initially, dimensions of brane were considered as small scale ones but, in 1983, V.A. Rubakov and M.E. Shapo- shnikov had introduced the model of brane in large additional dimensions Rubakov . Branes like strings Dai , Horava , Polchinski , Horava2 can be closed manifolds. Already Alexander Friedmann considered the whole Universe in particular variant of his models as closed surface but this surface was regarded as hypothetical one Friedmann . Eventually, Merab Gogberashvilly had proposed the closed brane model of Universe as real spherical expanding shell Gogberashvili . He had introduced the model where our Universe is considered as an expanding four-dimensional bubble in five dimensions with a center of Big Bang in the fifth dimension. This model well describes expanding character of Universe where each point becomes more far from any other point with time in the course of galaxies runaway. In addition, this model describes well the isotropic character of relict background radiation in our Universe. Besides, this model can explain why Lorenz invariance is realized on 4d shell while it is lost in the fifth dimension as it was known earlier in extra-dimensional compactification theories Rizzo , Garcia . As an alternative to compactification theories, Randal and Sun- drum had proposed semi-phenomenological brane model Randall where it was shown that four-dimensional Newton and Ein- stein gravity live in five dimensional space. Cosmological extensions of this model were given soon in papers Mohapatra , Cline , Mukohyama , Flanagan , Cline2 . We have derived Klein-Gordon equation using the symmetry of Universe expansion Andrianov , Andrianov2 . Com- paring it with squared Dirac-Fock-Ivanenko equation, we have derived expression for gravitationally induced chiral current Andrianov3 . Here, we show that being placed, in its turn, into the Einstein’s gravitational field, chiral current yields inertial gravitational field. This inertial gravitational field transforms into the vortex gravitational field and then back to the inertial gravitational field thus ensuring the propagation of these fields along with gravitational wave. In addition, we analyze our equations in the limit of weak gravitation and show that harmonic gauge often using with Einstein’s equations is a consequence of brane rotational symmetry.

2 Gravitational fields equations

Symmetry of Universe in brane model regarding its expansion yields Klein Gordon equation for a particle with mass m and wave function moving on brane Andrianov , Andrianov2 :

[TABLE]

where $g^{\mu\nu}$ is metric tensor, $\nabla_{\mu},\nabla_{\nu}$ are Riemann connection covariant derivatives and $R$ is scalar curvature, Greek indices run from 0 to 3. Comparison of this Klein-Gordon equation with squared Dirac-Fock-Ivanenko equation

[TABLE]

where $\gamma^{\mu}$ is Dirac matrix and $\Gamma_{\mu}$ is spin connection, yields the following relation:

[TABLE]

where $D_{\mu}=\nabla_{\mu}+\Gamma_{\mu}$ is covariant derivative generalized on spin connection. In its turn, relation (3) leads to the following expression for determination of gravitationally induced chiral current $j^{\alpha 5}$:

[TABLE]

where $\varepsilon^{\mu\nu\tau\xi}$ is Levi-Civita tensor, $R_{\mu\nu\tau\xi}$ is Riemann curvature tensor, $\gamma^{\alpha}$ is Dirac matrix, $\Gamma^{\mu}=g^{\alpha\beta}\Gamma^{\mu}_{\alpha\beta}$ is spin connection, $g^{\alpha\beta}$ is metric tensor, $\Gamma^{\mu}_{\alpha\beta}$ is Christoffel symbol. By multi- plication of expression (4) both parts on Levi-Chevita tensor

[TABLE]

we come to the expression

[TABLE]

and then to

[TABLE]

where

[TABLE]

or alternatively to

[TABLE]

by using relations $R_{\lambda\sigma}=g^{\zeta\chi}R_{\zeta\lambda\chi\sigma}$ and $R=g^{\lambda\sigma}R_{\lambda\sigma}$. What is for $R_{\lambda\sigma}$ it can be found from Einstein’s equation

[TABLE]

Starting from the expression

[TABLE]

we get using relation

[TABLE]

the following equation

[TABLE]

Finally, we have the following set of equations

[TABLE]

[TABLE]

[TABLE]

The left side of equation (17) can be expressed solely through metric tensor and thus can serve in finding it via known energy-momentum tensor. In other words, energy dis- turbance produces Einstein’s metric gravitational field. It con- sists of the part connected with Ricci tensor curvature $R_{\lambda\sigma}$ and the part connected with scalar curvature $R$. In its turn, the tensor curvature part of field produces via the left side of equation (18) gravity inertial field $D_{\mu}\Gamma^{\mu})$ in its right part. Gravity inertial field is expressed through spin connection $\Gamma^{\mu}$ and is related to scalar curvature by equation (19). This gravity inertial field turns into gravity vortex field and vice versa ensuring propagation of this type gravitational fields according to this equation. It can be accomplished through creation and annihilation of virtual particles. This propagation is absorption less if the scalar curvature is constant. Thus, this set of equations completely describes the propagation of gravitational fields where gravitational wave is described by shown complex structure of gravitational fields supplementing Einstein equations up to full set.

3 Harmonic gauge.

Let’s consider equation (19) in the case of weak gravitation. It can be rewritten as

[TABLE]

Keeping in mind that

[TABLE]

and

[TABLE]

we introduce linear approximation

[TABLE]

where $\eta_{\mu\nu}$ is pseudo-Euclidian metrics. Assuming that not only $h_{\mu\nu}$ is small but, also, its derivatives we have at vacuum conditions $(R=0)$ from (25):

[TABLE]

or

[TABLE]

The only solution of equation (30) is trivial one

[TABLE]

that is harmonic gauge. Thus, we had shown that harmonic gauge is a consequence of brane symmetry.

4 Conclusion.

Thus, we have derived Klein-Gordon Ta-Pei , Weise on equation for a quantum particle starting from the symmetry properties of the brane. This had confirmed the validity of discussed above spherical model of Universe. Comparison of this Klein-Gordon equation with the squared Dirac-Fock-Ivanenko equa- tion yielded expression for chiral current that had given rise to chiral symmetry breaking at early stages of universe evo- lution and creation of particles baryonic mass constituting 99% of visible matter in Universe. It is shown that this chiral current serves as a source for special new type of gravitational fields that propagate as constitutional part of gravitational wave described by Einstein’s equation. These type gravita- tional fields propagate by transforming of gravitational inertial field into vortex field and vice versa through creation and annihilation of virtual particles. We had shown that obtained equations describe in the case of weak gravitation harmonic gauge relations that are usually supplement to Einstein equa- tions making them a closed set. Thus, harmonic gauge is the consequence of brane rotational symmetry in additional dimension space. This rotation symmetry provides invariance of our four-dimensional space located on four dimensional brane immersed in five dimensional universal space. This five dimensional space has preferable frame with the center of coordinates in the point of Big Bang serving as a center of above said rotation. Recent observations of gravitational waves Abbott confirm validity of harmonic gauge because it is the gauge leading to the wave like form of propagating gravitational excitations. Therefore, they confirm also the existence of Universal brane as a spherically symmetric shell in a higher dimensional space. Some recent theoretical papers consider a number of such symmetry consequences Belayev , Bobev , Troisi , Banerjee . In particular, the upgrade of this model was presented where the effect of mass was considered as a localized brane deformation imparted by the endpoint of the string stretched in additional dimension Banerjee .