Einstein equations in terms of internal and external curvatures by Vladimir L. Kalashnikov, [email protected]

Abstract: Notion of an external curvature in the theory of General Relativity is considered. The sections of space-time by space-like hyperplanes allow to formulate the Hamiltonian formalism for integration of dynamic equations.
Application Areas/Subjects: Science, Physics, Applied Examples, General Relativity, Tensor Analysis
Keywords: Curvature tensor, Einstein equations, Hamilton equations, spherical universe

Here we will consider some computation aspects of the theory of General Relativity concerning so-called dynamics of geometry. The last will be understudied as the splitting of 4-dimensional metric manifold on 3-dimensional space sections, which evaluates in the course of time. The main problem here is the absence of uniform course of time in the GR-theory, that produces the certain arbitrariness due to choice of coordinates, i.e. choice of initial conditions.

In the beginning let consider the notion of external curvature in 4-dimensional space-time containing space-like 3-dimensional section (hyperplane).

> restart:
with(tensor):
with(plots):
with(plottools):
with(plots):
with(linalg):

q := plot3d(x^2+y^2,x=-1..1,y= -1..1):
l1 := line([0,0,10], [0,0,0], color=red, linestyle=3):
l2 := line([0,0.5,10], [0,0.5,0], color=green, linestyle=3):
display([q,translate(q,0,0,10)],orientation=[200,270]):
display(%,l1,l2);

We select two space-like sections (see figure) corresponding to the different times: t and t+dt . The relative position of these sections is defined by the "rigid" lines connecting the hyperplanes in the different space points (in the Fig. only two lines are shown by red and green colors). These lines are defined by the 4-vector connecting the coordinates on the lower and upper hyperplanes: = - (t, )dt .

This allows to define the 4-dimensional metric from 3-dimensional metric and . Then the line element in 4-dimensional space-time is:

> ds^2 = expand( g[-i, -j]*(dx[i]+N[i]*dt)*(dx[j]+N[j]*dt)-(N*dt)^2 );# metric in 4-space-time from 3-metric and translating vector N

Here the indexed components of translating vector correspond to its projection on hyperplane, the time component is signed as N.

For the 3-hyperplane in 4-dimensional space-time we can define so-called internal and external curvatures. The internal curvature is defined by usual Riemann tensor on 3-metric . The external curvature is defined by following tensor (see C. W. Misner, K. S. Thorn, J. A. Wheeler, Gravitation, W.H.Freeman and Co., San Francisco 1973 and external):

= [ + - ] ,

where ".. i" depicts the covariant derivative on hyperplane (we use also the symbol [-.. i ]). The next procedure Cur_ext calculates the external curvature from the given 3-metric g in basis coord for the translating vector with N_space - space and N_time - time components (we use the unit tensor for representation of curvature):

> Cur_ext := proc( g, coord, N_space, N_time )#Tensor of external curvature
d1g := d1metric( g, coord ):
ginv := invert( g, 'detg' ):
Cf1 := Christoffel1 ( d1g ):
Cf2 := Christoffel2 ( ginv, Cf1 ):
cd_N := cov_diff( N_space, coord, Cf2 ):
cd_N_1 := get_compts( cd_N ):
cd_N_2 := transpose( cd_N_1 ):
D_g := map( diff,get_compts(g),t ):
evalm( (cd_N_1+cd_N_2-D_g)*get_compts(ginv)/2/N_time ):
create( [-1,-1], % );
end:

Let consider a very simple example with 3-hyperplane, which has the geometry of the sphere, but the radius of this sphere depends on the time:

> animate3d(x,theta=0..2*Pi,phi=0..Pi,x=1..10,coords=spherical);# oscillating sphere

The corresponding 3-metric is:

> coord := [chi, theta, phi]:# spherical 3-coordinates
g_compts := array(symmetric ,sparse ,1..3,1..3):# simple 3-metric
g_compts[1,1] := a(t)^2:
g_compts[2,2] := a(t)^2*sin(chi)^2:
g_compts[3,3] := a(t)^2*sin(chi)^2*sin(theta)^2:
g := create([-1,-1], eval(g_compts));

We define the translating vector as:

> N_space := create([-1], vector([0, 0, 0]));# space part
N_time := N;# time part

Then the external curvature is:

> Cur_ext( g, coord, N_space, N_time );

The components of this tensor have a very simple sense: they are the relative increase of sphere's size per unit of intrinsic time interval d = - Ndt . For the comparison we show the Ricci scalar for internal curvature of sphere:

> Curv_inn := proc(g, coord)# Ricci tensor for internal curvature
ginv := invert( g, 'detg' ):
D1g := d1metric ( g, coord ):
D2g := d2metric ( D1g, coord ):
Cf1 := Christoffel1 ( D1g ):
RMN := Riemann( ginv, D2g, Cf1 ):
RICCI := Ricci( ginv, RMN ):
end:

> Ricciscalar( invert( g, 'detg' ), Curv_inn(g, coord) ):
RS := simplify(get_compts(%));# Ricci scalar for sphere

The meaning of external curvature is defined by its connection with curvature of 4-dimensional space-time:

= + ( K K - K K )/( n*n ) ,

where is the internal curvature, and the denominator contains the scalar product of the time-like normal to hyperplane. The usual representation for this normal is n = - Ndt+0+0+0 . Hence we can describe the Einstein equation through internal and external curvatures:

G = - ( -Sp( )/( n*n ), (1)

G = - ( K - (Sp K )[ -..m ])/( n*n ) (2)

where Sp is the spur, n is the component corresponding to normal to hyperplane. Eq. (1) is equal to 8 ( is the mass-energy density) and Eq. (2) is equal to 8 *covariant i -component of the momentum density.

Now calculate the spur of K and .

> M := get_compts(Cur_ext( g, coord, N_space, N_time )):
tr_1 := trace( M );
multiply( M, M ):
tr_2 := trace( % );

As result we have the Einstein equation (compare with univ_gravity):

> 16*Pi*rho(t) = -RS+subs( N=1, (tr_1^2-tr_2)/N^2 );# first Einstein equation from four equations for initial conditions in general case

This equation describes the cyclic evolution of universe with spherical symmetry (see univ_gravity) and is resulted, in fact, from the equation for the initial conditions in model (four equations in general case).

In general case, we have to add the equations allowing finding the metric on the each step of the space dynamics. These equations have the form of the canonical dynamic equations in Hamiltonian form (we will consider this formalism in other worksheet). The role of the coordinate is played by metric tensor. The canonical momentum is defined as:

= ( Sp K - ).

Then the first canonical equation is

= ( - Sp ) + + (3)

and the second is:

= -N ( - R )+ (Sp - (Sp )^2)-2 ( - Sp )+

( N [..ij]- N [..m -..m])+( )[-..m] - [-..m] - [-..m] (4)

Unlike our elementary example with one degree of freedom, the process of modeling in general case is following:

1) We select the initial geometry, i.e. g and on initial hyperplane.

2) Eqs. (1, 2) produce the translating vector N .

3) Then Eqs. (3, 4) result in the geometry of space-section of manifold at next time moment ( t is rather parameter, not some absolute time!)

...

etc.

...

In future we plan to investigate this procedure in detail.

So, we considered the important in practice form of Einstein equations based on the notion of space-like section of spice-time manifold and internal and external curvatures of the space-like hyperplanes in this manifold. As result, there is the procedure for the simulation of geometry-dynamics, which is based on the Hamiltonian formalism of canonical first-order equations. As trivial example, the equation for the evolution of spherical universe was obtained.

2000©Kalashnikov