# Theory for Quantization of Gravity

### Text-only Preview

May 17, 2012

Author: Patrick Linker

Contact information:

Rheinstrasse 13

61206 Woellstadt

Germany

Phone: +49 (0)6034 905296

E-Mail: [email protected]

Affiliation:

Technische Universitaet Darmstadt

Schlossgartenstrasse 7

64289 Darmstadt

Phone: +49 (0)176 70122964

1

Theory for Quantization of Gravity

Patrick Linker

2

Abstract

The unification of Einstein's field equations with Quantum field theory is a prob-

lem of theoretical physics. Many models for solving this problem were done, i.e.

the String Theory or Loop quantum gravity were introduced to describe grav-

ity with quantum theory. The main problem of this theories is that they are

mathematically very complicated. In this research text, there is given another

description of gravity unified with Quantum field theory. In this case, grav-

ity is described so that for weak gravitational fields the (semi)classical gravity

desciption is equivalent.

Introduction

Quantization of gravity is a difficult problem of theoretical physics, because

there is a difficulty in renormalization of a gravity quantum field theory [1]. One

of the problems in Quantum Gravity is the singularity, which appears at the

Big Bang or in Black Holes [2]. For solving this problem, there were developed

different theories. In [3] and [4] the String Theory was used for unify all four

well-known fundamental forces. The theory needs extra dimensions to describe

the universe with its fundamental forces. Another model for unify gravity with

all other fundamental forces is the Loop quantum gravity [5].

There is no

semiclassical limit shown, which exists in Loop quantum gravity. Furthermore,

this theories for quantum gravity are mathematically very complicated. The way

described in this research text is the derivation of a gravity field equation, which

is similar to special-relativistic field equations of quantum field theory. There

are existing quantization relations and the field equation yielding Einstein's

field equations of general relativity, if the gravity field strength is weak. The

equations have basically a minkowski metric (the non-gravitational case) and

if gravity exists, there are introduced paths, which have a projective character

on relativistic energies. This projection properties of the paths make massive

objects to attract each other, i.e. paths that passing through 4-dimensional

spacetime leading to the masses, which attracts or repulses a certain particle.

For each amount of energy it is existing a path that is oriented to the position

of the amount of energy and a co-path , that is oriented outwards from amount

of energy. For very dense matter, the outward path is dominating (repulsion

of matter), otherwise the attractional path has the dominance. So, if matter

is not very dense, gravity has the well-known properties like attractivity and the

independence of gravity force on materia positioned between the line of gravity

interaction (because there are all kinds of projective paths possible).

3

Physical theory

From special relativity, the following metric is well-known:

ds2 = -dt2 + dx2 + dy2 + dz2 = dsds.

(1)

. For a line element ds in 4-dimensional spacetime, the metric tensor is flat,

i.e. for Christoffel's symbols holds the relation = 0. May be , : [0, 1] X

open curves that passing a set of points X R4 in a topological space T (here:

Minkowski space). Then, can be projective (expanded to the projective

metric f), if the direction of projection is given by and . The projectivizion

mapping p : f is given by the relationship:

f = gh.

(2)

Here, the two path metrics g (describing path elements in pi) and h (de-

scribing path elements in omega) are introduced. From (2), the line element

has the form:

ds2 = dsfds = dsghds = dsds.

(3)

Equation (3) defines a scalar product in a projected minkowski space with the

mappings pds = -1 : ds ds. For = the mapping is the identity, i.e.

there is no gravitational interaction available. According to Einstein's General

Relativity, Gravitation is the effect of curved spacetime. In this theory, gravity

occurs, if the two path metrics are different, so that the closed path made from

joining and on their end points, i.e. := has an interior area A. So,

the projection line follows from the crossing path between the two paths.

May be h : x [0, 1] a homotopy that maps the curve to . Then the

following diagram is commutative:

i

x [0, 1]

/ .

(4)

h

h

/

i

The upper relation on (4) is the isomorphy i between the interval [0, 1] and ,

by taking the two end points of this interval on the two paths that build up .

Analogous, to the tensor product space a homotopy h can be declared.

From this construction, the mapping i is an isomorphism, if h and h have the

same (projective) behavior. May be H the group of homotopies in R4, then

H -1HH H is the subgroup, which is equal to H, when H = H. So, the

quotient

(H , H) = H/H -1HH

(5)

is the identity for H = H. The following short sequence is exact:

i

q

0

/

/

/ (H , H)

/ 0 .

(6)

4

From (6) follows that (H , H)

= /i (), because i () ker(q). Hence,

can be partitioned into a kernel i () and a non-kernel part (H , H), i.e.

= i () (H , H) = h ( ) (H , H) = h( x [0, 1]).

(7)

The relation (7) is equivalent to (H , H) = h( x [0, 1])/h ( ), where for

h h it must follow (H , H) = id; hence

h( ) = h( x [0, 1])

= x [0, 1].

(8)

The conclusion in (8) follows from the uniqueness of the definition of h. The

commutation of the paths , , i.e. C := - vanishes , if they have different

starting and ending points. If the starting and ending points are equal, from

relation (8) there exists a homeomorphism between the closed path and a finite

interval and because closed paths leading to the same point for infinite winding

around that paths, the value of C is infinite. This result can be summarized

with:

(X - Y ) = (X)(Y )- (X)(Y ); [h(X), g(Y )]- = (X -Y ). (9)

Equation (9) is the quantization condition. For computation of volume elements

in projective minkowski space, the value A must be determined. May be

A : [0, ] a measurable function on . This measure is zero, if = =

h( x [0, 1]) = = h ( ) = h( ) (where the last conclusion comes from

(8)), i.e.:

(A = 0 h = h ) (A = 0 (H , H) = id).

(10)

This measure must be -additive and by the definition

A := |(H , H)|,

(11)

there exists the connexion that every equivalence class that is added contributes

to the from enclosed area. With = h( x [0, 1]) and = h ( ) =

h ( x [0, 1]) = follows

= h h -1( ) (H , H) = /

(12)

, that means that all paths between and , denoted by , projected in direction

of is the group (H , H). If lies in the enclosed area of , the homotopy h,

which leading from to can be chosen so, that is approximately isomorphic

to , whereas follows:

A = |(H , H)| 0.

(13)

In general relativity, the following Lagrangian can be used to derive Einstein's

field equations with classical metrix g [6]

L =

d4x -g(Rg - S(g)),

(14)

5

where is a coupling constant, g = det(g ), R is the Ricci tensor and S is

the Lagrangian density of all other fields. In projective formulation, the volume

element g can be approximated as follows (with g f and (13)):

1

-f =

-det(( + gh - )) 1 - tr((gh - ) ) =

2

adj

1

1

1 +

(gh - ) = -1 + gh.

2

2

(15)

Here, a linear approximation on the from enclosed area was done. By substi-

tuting (15) into (14) with g f the Lagrangian of the field theory is:

1

L = -

d4x(1 -

gh)(Rf - S(f )).

(16)

2

Variation on the fields g and h yields two coupled field equations. Conclu-

sions

The equation (16) tends to linearized Einstein's equations for the limit f

. Otherwise, the metrics g and h become different. So, the product gh

is finite, if both metrics grow at the same order. The field equation (16) can

be used as a model to describe gravity strength at the Big Bang [7]. Further,

this field theory is a model that describes the gravitational interactions of dark

matter.

6

References

[1] Edward Anderson (2010), "The Problem of Time in Quantum Gravity",

arXiv:1009.2157

[2] Robin Schumann (2006), "Quantengravitation", Shaker, Aachen , ISBN 978-

3-8322-5683-8

[3] J. Polchinski (1998), "String Theory", Cambridge University Press, Cam-

bridge, UK

[4] Hawking, Stephen (2010), "The Grand Design", Bantam Books

[5] Rovelli, Carlo (2011), "Zakopane lectures on loop gravity", arXiv:gr-qc/1102.3660

[6] U.E.Schroeder (2000), "Gravitation - Einfuehrung in die allgemeine Relativ-

itaetstheorie", Harri Deutsch, Frankfurt a.M., ISBN 978-3-8171-1874-8

[7] Martin Bojowald (2009), "Zurueck vor den Urknall", S. Fischer, Frankfurt

a.M. , ISBN 978-3-10-003910-1

7

Author: Patrick Linker

Contact information:

Rheinstrasse 13

61206 Woellstadt

Germany

Phone: +49 (0)6034 905296

E-Mail: [email protected]

Affiliation:

Technische Universitaet Darmstadt

Schlossgartenstrasse 7

64289 Darmstadt

Phone: +49 (0)176 70122964

1

Theory for Quantization of Gravity

Patrick Linker

2

Abstract

The unification of Einstein's field equations with Quantum field theory is a prob-

lem of theoretical physics. Many models for solving this problem were done, i.e.

the String Theory or Loop quantum gravity were introduced to describe grav-

ity with quantum theory. The main problem of this theories is that they are

mathematically very complicated. In this research text, there is given another

description of gravity unified with Quantum field theory. In this case, grav-

ity is described so that for weak gravitational fields the (semi)classical gravity

desciption is equivalent.

Introduction

Quantization of gravity is a difficult problem of theoretical physics, because

there is a difficulty in renormalization of a gravity quantum field theory [1]. One

of the problems in Quantum Gravity is the singularity, which appears at the

Big Bang or in Black Holes [2]. For solving this problem, there were developed

different theories. In [3] and [4] the String Theory was used for unify all four

well-known fundamental forces. The theory needs extra dimensions to describe

the universe with its fundamental forces. Another model for unify gravity with

all other fundamental forces is the Loop quantum gravity [5].

There is no

semiclassical limit shown, which exists in Loop quantum gravity. Furthermore,

this theories for quantum gravity are mathematically very complicated. The way

described in this research text is the derivation of a gravity field equation, which

is similar to special-relativistic field equations of quantum field theory. There

are existing quantization relations and the field equation yielding Einstein's

field equations of general relativity, if the gravity field strength is weak. The

equations have basically a minkowski metric (the non-gravitational case) and

if gravity exists, there are introduced paths, which have a projective character

on relativistic energies. This projection properties of the paths make massive

objects to attract each other, i.e. paths that passing through 4-dimensional

spacetime leading to the masses, which attracts or repulses a certain particle.

For each amount of energy it is existing a path that is oriented to the position

of the amount of energy and a co-path , that is oriented outwards from amount

of energy. For very dense matter, the outward path is dominating (repulsion

of matter), otherwise the attractional path has the dominance. So, if matter

is not very dense, gravity has the well-known properties like attractivity and the

independence of gravity force on materia positioned between the line of gravity

interaction (because there are all kinds of projective paths possible).

3

Physical theory

From special relativity, the following metric is well-known:

ds2 = -dt2 + dx2 + dy2 + dz2 = dsds.

(1)

. For a line element ds in 4-dimensional spacetime, the metric tensor is flat,

i.e. for Christoffel's symbols holds the relation = 0. May be , : [0, 1] X

open curves that passing a set of points X R4 in a topological space T (here:

Minkowski space). Then, can be projective (expanded to the projective

metric f), if the direction of projection is given by and . The projectivizion

mapping p : f is given by the relationship:

f = gh.

(2)

Here, the two path metrics g (describing path elements in pi) and h (de-

scribing path elements in omega) are introduced. From (2), the line element

has the form:

ds2 = dsfds = dsghds = dsds.

(3)

Equation (3) defines a scalar product in a projected minkowski space with the

mappings pds = -1 : ds ds. For = the mapping is the identity, i.e.

there is no gravitational interaction available. According to Einstein's General

Relativity, Gravitation is the effect of curved spacetime. In this theory, gravity

occurs, if the two path metrics are different, so that the closed path made from

joining and on their end points, i.e. := has an interior area A. So,

the projection line follows from the crossing path between the two paths.

May be h : x [0, 1] a homotopy that maps the curve to . Then the

following diagram is commutative:

i

x [0, 1]

/ .

(4)

h

h

/

i

The upper relation on (4) is the isomorphy i between the interval [0, 1] and ,

by taking the two end points of this interval on the two paths that build up .

Analogous, to the tensor product space a homotopy h can be declared.

From this construction, the mapping i is an isomorphism, if h and h have the

same (projective) behavior. May be H the group of homotopies in R4, then

H -1HH H is the subgroup, which is equal to H, when H = H. So, the

quotient

(H , H) = H/H -1HH

(5)

is the identity for H = H. The following short sequence is exact:

i

q

0

/

/

/ (H , H)

/ 0 .

(6)

4

From (6) follows that (H , H)

= /i (), because i () ker(q). Hence,

can be partitioned into a kernel i () and a non-kernel part (H , H), i.e.

= i () (H , H) = h ( ) (H , H) = h( x [0, 1]).

(7)

The relation (7) is equivalent to (H , H) = h( x [0, 1])/h ( ), where for

h h it must follow (H , H) = id; hence

h( ) = h( x [0, 1])

= x [0, 1].

(8)

The conclusion in (8) follows from the uniqueness of the definition of h. The

commutation of the paths , , i.e. C := - vanishes , if they have different

starting and ending points. If the starting and ending points are equal, from

relation (8) there exists a homeomorphism between the closed path and a finite

interval and because closed paths leading to the same point for infinite winding

around that paths, the value of C is infinite. This result can be summarized

with:

(X - Y ) = (X)(Y )- (X)(Y ); [h(X), g(Y )]- = (X -Y ). (9)

Equation (9) is the quantization condition. For computation of volume elements

in projective minkowski space, the value A must be determined. May be

A : [0, ] a measurable function on . This measure is zero, if = =

h( x [0, 1]) = = h ( ) = h( ) (where the last conclusion comes from

(8)), i.e.:

(A = 0 h = h ) (A = 0 (H , H) = id).

(10)

This measure must be -additive and by the definition

A := |(H , H)|,

(11)

there exists the connexion that every equivalence class that is added contributes

to the from enclosed area. With = h( x [0, 1]) and = h ( ) =

h ( x [0, 1]) = follows

= h h -1( ) (H , H) = /

(12)

, that means that all paths between and , denoted by , projected in direction

of is the group (H , H). If lies in the enclosed area of , the homotopy h,

which leading from to can be chosen so, that is approximately isomorphic

to , whereas follows:

A = |(H , H)| 0.

(13)

In general relativity, the following Lagrangian can be used to derive Einstein's

field equations with classical metrix g [6]

L =

d4x -g(Rg - S(g)),

(14)

5

where is a coupling constant, g = det(g ), R is the Ricci tensor and S is

the Lagrangian density of all other fields. In projective formulation, the volume

element g can be approximated as follows (with g f and (13)):

1

-f =

-det(( + gh - )) 1 - tr((gh - ) ) =

2

adj

1

1

1 +

(gh - ) = -1 + gh.

2

2

(15)

Here, a linear approximation on the from enclosed area was done. By substi-

tuting (15) into (14) with g f the Lagrangian of the field theory is:

1

L = -

d4x(1 -

gh)(Rf - S(f )).

(16)

2

Variation on the fields g and h yields two coupled field equations. Conclu-

sions

The equation (16) tends to linearized Einstein's equations for the limit f

. Otherwise, the metrics g and h become different. So, the product gh

is finite, if both metrics grow at the same order. The field equation (16) can

be used as a model to describe gravity strength at the Big Bang [7]. Further,

this field theory is a model that describes the gravitational interactions of dark

matter.

6

References

[1] Edward Anderson (2010), "The Problem of Time in Quantum Gravity",

arXiv:1009.2157

[2] Robin Schumann (2006), "Quantengravitation", Shaker, Aachen , ISBN 978-

3-8322-5683-8

[3] J. Polchinski (1998), "String Theory", Cambridge University Press, Cam-

bridge, UK

[4] Hawking, Stephen (2010), "The Grand Design", Bantam Books

[5] Rovelli, Carlo (2011), "Zakopane lectures on loop gravity", arXiv:gr-qc/1102.3660

[6] U.E.Schroeder (2000), "Gravitation - Einfuehrung in die allgemeine Relativ-

itaetstheorie", Harri Deutsch, Frankfurt a.M., ISBN 978-3-8171-1874-8

[7] Martin Bojowald (2009), "Zurueck vor den Urknall", S. Fischer, Frankfurt

a.M. , ISBN 978-3-10-003910-1

7