geometry and the nature of gravitation - galilei€¦ · geometry and the nature of gravitation...
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Geometry and the Nature ofGravitation
Waldyr A. Rodrigues Jr.
IMECC-UNICAMP
ICCA9-WEIMAR 2011
1
The Flat and the Curved Punctured SphereDo not Confuse Curvature with Bending
Levi Civita Connection D and Nunes Connection
1 2 1 1 2
1 1 2
2 2 1 1 1 2 1 2 212 21
1( , ) ( , ), 0 , 0 2 , , , , sin
sin
[ , ] , cot , sin .
x x dx x dxx x x
c c c x g dx dx x dx dx
e e
e e eki j ij k
2 2 121 12 22, cot , cos sin ,
, cot ,
- 0
cot .
, [ ] 0,
1: , :
2
D D
D
Levi Civita Connectio
D
D
D D
n
T
D
i
i j
kj ij k
k k ki j i j j i i j
k i j k kij k k
g
e
e e
e e
e ,e e ,e e e e ,e
e e
T
T
[ ]
10,
2
( , , ) ,
1.
D
T
D D D D D
j i i j i j
k i jij
a ak i j k
1 1 2 221 12 12 21
R e e e e e ,ee e ,e e
R R R R
0,
0,
[ ] [ ]
cot .
,
0Nunes Connec n
T
tio
T
i
j i
j
i j i j i j i j
2 221 12
e e
R
g
e ee ,e e e e ,e e ,e
2
The Gravitational Field in GR
1
1 1: ,
2 2
, sec are the ,
( , ) , ( ) , { } basis of , { } basis of ,
Einstein Equation
1 -
m
Ricci R R
Ricci R R T M Ricci for
TM
m fields
T M
g T T
g e e e e
a a a
a b a a bab b
a a aa b ab b b a
g
g g g
g g
( )
atrix with entries is (1, 1, 1, 1).diag ab
As mentioned in section 3.8, conservation laws have a grea
:
1. Which is the topology of ?
2. There are no conservation laws for energy-momentumand angular momentumi
Problems with the GR model
"
n .
M
GR
t predictive power.
It is a shame to lose the special relativistic total energy conservation law(Section 3.10.2) in general relativity.
Many of the attempts to resurrect it are quite interesting; man
(Sachs&Wu, General Relativity for Mathematicians, pp.
y are simply
97-98, Sprin
garbag
ger 1
e."
977)
:
i represent the gravitational field by a different geometrical model, e.g., e.g.,a , , , , ;
• (ii) represent the gravitational field as a fie
Possible soluti
ld
ons
in
teleparallel spacetime M g g
Faraday sense living in spacetime , , , , .Minkowski M D
models a gravitational field generated by a given energy-momentum tensor by a Lorentzian spacetime
, , , , , where is a noncompact (locally compact) 4-d Hausdorff manifold, is a Lore
GR T
M D M
gg g
4
ntzian metric field,
is the Levi-Civita connection of , sec is a orientation, is a time orientation.D T M gg
3
Alternative Representation for aReliable Gravitational Field
Suppose is , i.e.: four vector fields sec , 0,1, 2,3, and is a for .
is : Necessary and sufficient condition for a 4-dimensional Lo
parallelizable basM gl is
Geroch theorem
obal TM TM
Motivation
e ea aa
Geroch, R. Spinor Structure of Space-Times in General Relativity I,( .
rentzian manifold ,
to admit spinor fields is that the orthonormal frame bundle be trivial. (thus parallelizable)
J Ma
M g
. . , 1739-1744 (19 .)68)th Phys 9
4
Let , sec be the corresponding , ( ) . Suppose moreover
that not all are , i.e., 0, for at least some 0,1, 2,3.
sec defines a (positive) ieor
dual basis
closed
T M
d
T M
ea a a ab b
a a
0 1 2 3
a
g g g
g g
g g g g2
0
ntation for .
Define a in by . Define sec .
is a (according to ) and defines a for .
Lorentzian metric
global timelike vector field time orientation
M
M g T M
M
g e e
e g
a b abab a b
0
g g
1Since some of the 0, we have [ , ] and moreover .
2d c d c e e ea k a k a b
a b ab k abg g g g
• Next introduce metric compatible connections in , namely , the Levi-Civita
connection of , and , a connection.
, ,
M D
telepara
two different
llel
D D
e e
g
e ea a
k b b kb ab k akg g 0, 0.
• are the connection 1-forms of and 0 are the connection 1-forms of
in th
T
e basis {
hen we immediately have two possible structu e
.
:s
}
r
D
e ee
e
a a
bb
k k b k k ba ab a ab
a
g
g g
4
40
Now, before proceeding we suppose that:
( , ),
where ( , ) is the hom
rr
Clifford bundle of no
T M T M
n ogeneous differen
g
tg i
M
M
#
and we use the conventions about the
scalar product, left and right contractions and the Hodge star operator and the Hodge coderivative operator
as in th
W. A. Rodrigues Jr. a
e book:
nd
g g
al forms
E. Capelas de Oliveira, , .
http://www.ime.unicamp.br/~walrod/errata1101201err
, Springer 200
0.pd
7
fat t
.
a a :
The Many Faces of Maxwell Dirac and Einstein Equations A Clifford
Bundle Approach
2 2
, , , , , : 0, : 0;
where sec and sec are respectively the torsion and curvature 2-forms of .
, , , , , : , :
M D d d
T M T M D
M d d d
g
g
g
g
a a a b a a a cb b b c b
a ab
a a a b a a ab b b
g g
g g g
2 2
0;
where sec and sec are respectively the torsion and curvature 2-forms of .d T M T M
a cc b
a a abg
5
The Potentials { } as Representatives of the Gravitational Fieldag
:
is the gravitational Lagrangian
is the matter Lagrangian
:
1 1 1( ) ( ). ( )
2 2 4
The form of this Lagr
g m
g
m
gg g gg g
d
Lagran
d d
gian Densit
d
y
Postulate
a a a ba a a bg g g g g g g g
angian is notable, the first term is Yang-Mills like, the second one is a kind of
gauge fixing term and the third term is an autointeraction term describing the interaction of the
of the pvorticities otentials.
Before proceeding we observe that this Lagrangian is not invariant under arbitrary point
of the basic cotetrad fields. In fact, if , , wher
dependent
Lorentz rotations R R x M a a a b abg g g g
1,3 1,3
e
( ) , the homogeneous and orthochronous Lorentz group and ( ) we get
exact differential.
So, the field equations derive
g g
x L R x Spin
ab
d from the variational principle results invariant under a change of gauge.
6
The Field Equations
7
, (1 )g g g
d t d d d
1: ( ) ( )
2
1( ) ( ), (2)
2
g
g g g
g g g g g gg
d dd
d d
d
a ad a d d ad
a ad d a d a
d d
g g g g g gg
g g g g g g
g
: ( )
1( ) ( )
2
1 1 1( ) ( ) )
2 2 2
1 1( ) ( )] (
4 4
g g
gg g g
g gg g
g g g g g g gg g
g gg g
t dd
d d d d
d d d d d d
d d d
ad d dd d
a a ad a d
a a ad a d a d a
a c ca d c d
g g gg g
g g g g g g
g g g g g g g (g g
g g [g g g [g g g
(3)
)] ( ),g
d ac ag g
. (4)m
g gT
d ddg
Maxwell Like Form of the Field EquationsRecall that and define:
1: [( ) ( )], (5)
2
and a
g g g g
possible legitimate energy momentum tensor for the gravit
d
d d
ational field
h
d d
ad d a d a
g
g g g g g
a
. (6)
Then, recalling Eq.(1) and the definition of , we can write the Maxwell like
equations for the grav
g
t
hd d dt
itational field: (a) 0, (b) ( ). (7)g
d dd d dt
1 1 2
Compare Eqs.(7a) and (7b) with Maxwell Equations in the Structure , , , .
sec , sec , sec ,
0, . (8)
e
eg
M
A T M J T M F dA T M
dF F J
gg
8
acting on sec ( , ) in , , , , :
: = ; , . 9( )
rr
r r r rg gg g
A T M M g MDirac operator
Sq
D
D d A dA A A
The Many Faces of Maxwell and Einstein Equations
g
e
g
a
ag
2 2
1
( ( ; ( ) ,
( , ( ; ( , (
is called the Ricci operator and: ( sec ( , ),
( is the cova
r r r r rg g
g g g g
A A A A d d A
d d d d
R Tu re
Ma o
gf
M
a a a bbg g
2
(10)
riant D'Alembertian, ( ) is called Hodge Laplacian.
acting on sec ( , ) in , , , , :
: , ; ( ),
g g
rr
r r r r rgg g g
Dirac ope
d d
A T M M g M
A dA A
r
A
at r
A
o
g
e
g
a
a aag g
( ). (10)rg
A aag
: . (11)
: ( ) ( ). (12)
, , , ,
, , ,,
e
eg g
Maxwell Equation in M D
Maxwell Equation i
F J
M F J F Fn
g
g
g
g a aa a g g
2
?
, , , , : (13)
1( ( ( ( )
2
, , :, ,
Gravitational Equation in M D
Gravitational Eq
R
uation in M
g
g
g
g
T T T
d d d d d d
d d d d d d d d d d d d
d d
t g t
g t g t g t + g
( ) ( ). (14)g g
d a d a da a t g g
in , , , :
0, eg
dF F
ME
J
M
gg
:
0, ( ),
.
i , , ,n
gd
GE M
d
gg
dd d d
d d
t
g
9
The Lagrangian density (Eq.( )) for the gravitational field ,
1 1 1( ) ( ),
2 2 4
1differs from the Einstein Hilbert Lagrangian density by an exact di
2
gg g g g gg g
EH g
d
d d d d
R
a a
a a a ba a a b
F
g
g g g g g g g g
fferential, i.e.,
( ).
This can be seen with s ...ome algebra (details in ) once we recall that
1 1
2 2
EH gg
EH g
The Many Fac
d d
e of
R
s
aa
g g
1( )
2
and that the connection 1-form fields of (the Levi-Civita connection of ) can be written as:
1[ ( ) ]
2
g g
g g g g
d
D
d d d
g
c d c c a dcd d a d c
cd d c c d c d aa
g g g g
g g g g g g g g
2
.
This warrants the equivalence of the equations:
, (1) (14)
1( ( ( ( )
2
and we have the interesting equatio
g g gd t
R
T T T
d d d d d dd d d
d d d d d d d d d d d
t g t
g t g t g + g
n for the energy-momentum 1-forms of the gravitational field
1
The important lesson we learn from this exercise is tha
( . (15)
E n
2
t i s
R d d dt g + g
:
, (1
tein equations can be written in the structure simlpy as
where no connection, no curvature, no tor
) 0, ( ) ,
s on
,
i
g g g gd t d d d d d
d d d d d d t g
, is involved. 10
From the Eq.(7b) ( ( )) it follows trivially that
( ) 0, (16)
may be interpreted as a
g
g
legitimate energ
d d d
d d
t
t
in a teleparallel structure , , , ,
or in the particular teleparallel structure , , , , , the Minkowski spacetime stru
-
cture.
In any of those structures we can
y momentum conservation Mla
M D
w
gg
in an obvious way identify all tangent spaces of .
Indeed, if ( ) sec and ( ) sec we define the equivalence
relation ( ) in by if and only if ( ) ( ).
We defin
x x y yx y
x y
M
v v x T M v v y T M
TM v v v x v y
e ea aa a
a a
e [ ]. The set of all obtained from all the sec defines a 4- real vector space .
We can take as a basis for the ordered set { , , , } with [ ].
Thus using Eq.(7) and Stokes t
x x x
x
v v T M d
V
V E E E E E ea a
v v
heorem we can define the total energy-momentum of the gravitational plus
matter fields by:
1 1: , : ( ) . (17)
8 8B Bg g
vector
P P P
Ed d d d dd t
Eq.(1) [ ], permit us to define an alternative conserved energy-momentum law by:
1 1: , : ( ) . (18)
8 8
g g g
B Bg g
d t
P P P t
E
d d d
d d d d dd
Energy-Momentum Conservation
11
Hamiltonian FormalismIf we define as usual the canonical momenta associated to the potentials by /
and suppose that this equation can be solved for the as function of the we can introd
,
uce a
gg
Legen
d
d
a aa a
aa
g p g
g p
with respect to the fields d by
: ( ) ( ) ( ( )) (19)
Put ( ): ( ( )). Observe that defining
(
, , ,
, ,
,
g
g g
g
d d
d
dre transformation
a
a a a a aa a a a
a a aa a
a
L L
g
g p g p g p g g p
L g p g g p
L g
, , , (20)
we can obtai de
) / / ( ) / /
( ) ( ) / ( )
tails inn ( )
,
...
( , ) ( , ) . (21)/
To de
g
g g g
Gravitation as a Pl
d
d
asti
d
c
a a a a aa a a a
a a a a a a a a aa a
L Lp g p g L g p p g p
g g g g L g p g L g p p p
fine the Hamiltonian form we need something to act the role of time for our manifold, and we choose this "time" to
be given by the fow of an arbitrary timelike vector field sec , such that ( , ) 1TM gZ Z Z1
. Moreover we define
( , ) sec is generated by the Lie derivative .
Cartan's magical formula and
( , ). With this choise the variatio
some trick algebra (details in
nZ T
Gravitatio
M g
n as a Plas
M g ZZ # L
, ( ) ( ),
and using Eq.(2
)
( ) ( ) ( ) = / /
(
1) we have
( )./
...
)
g g g g g g
g g
g g
g
d Z Z d d
d Z
tic
a a a aa aZ Z aZ Z
Z Z
a
a a aa
L g p L L L g p g L g p L p
g p L g g
L L L L
L L (22)
, (23)( ) : .
From Eqs.( 22) and (23)we have when the field
Hamiltonian
equations are satisfied ( / ) that
3-for
0
:
0
m
gg
g
Z
d
Za a
a a
a
g p g p L
g
L
24.
.is a conserved Noether current12
Meaning
/ . (26)
. Consider an arbitrary spacelike surface .
Write: (25)
Some algebra shows that:
of the boundary t
,
erm
g
Z dB
B Z
B
Quasi - local Energy
aa
a aa a
g p
( )
0, i.e., the field equations are satisfied we are left with th
1: (27)
8
When /
the : (28)
If { }is
e
1.
8
a basis fo such thatr
-
g
quasi local
T
ener
M
Z dB
Bgy
e
aa
a
a
H
E
g
g
0
( ) and if we choose
1that: ,we get recalling that (29)
which we recognize as the same quantity given (when 0) b
8
y
g g
Z
e ea a
a
b
a
b 0
E
d
p
Eq.(18), i.e
1 1:
.,
( ) .8 8B Bg g
P t
d d d d
13
2 1
Instead of choosing an arbitrary unit timelike vector field start with a global timelike vector field
sec such that ( ,) sec ( , ), where iTM n N dt T M M g N
Relation with the ADM Energy Concept
g
Z
n n #
0
s the .
Then, 0 and Frobenius theorem says that induces a foliation of where is a spacelike
hypersurface with normal and x , with {x x } coordinates in EPL
t t
t
n dn n
lapse func i
M
t
t on
a an,
1
gauge, i.e., ( x , x ) .
For sec ( , ) write , with and the tangent and normal components
to . We have: ( ), , . (30)
Put : ( ). C
tg
g
d d
A T M M g A A A A A
A n dt A A dt A A n A
dA dt n dA
#
artan's magical formula gives: ( ) . (31)
First fundamental form on : . (32) : . (33)
Writting ( , , ,
,
( , ) ,, ) ( )g
tg
g
dA dt A d
nA A
N
n
A d A
d dt dn d
mm g n n n ni
i
i i i ia an
i in
ii
g g
g g g g g g p g
n L
L L , (34)
Some (trick) algebra gives , ( )= , , (35)
1When 0, we get exactly the energy : , (36)
8
Indeed, ta
/ /
e
/
k
t
g
i
g A
g
D
g
M
nn dB B N d
ADM N d
m
m
m
i
i i i ii
ii
i
i
E =
p
g g g
g g g
a 2-sphere at infinity. Then,
1, ( / x / x ) and ( / x / x ) (37)
8
x , 0, 1,
t
t
ADMd h
h d h N
h h h h
m m m
ji ij ij i
i ij k j k
j
i ij ik ijj
ik k
kE
g
g g g g
14
If we choose it may happen that 0, and thus it does not determine
a spacelike hypersuperface . However, all algebraic calculations up to Eq.(35) above
are va
ADM
t
n d
0 0 0
E E for Isolated Systems
g g g
lid (and of course ). So, if we take a spacelike hypersurface such that at
spatial infinity the ( ( ) ) are tangent to , and / is orthogonal to ,
then we have since in this
t
i i
i ii j j 0
E E
g g
ge e e
0
0
case recalling Eq.(2) for 0, i.e.,
1( ) ( )
2
we see that
( ) (38)
g g g
g
d d
N d d
a a0 a 0 a
i ii i
d
g g
g g g g g g
g g gm
is the asymptotic value of (taking into account that at spatial infinity 0).g
d 0 0g
15
Conclusions• We recalled that a gravitational field generated by a given energy-
momentum tensor can be represented by distinct geometricalstructures and if we prefer, we can even dispense all those geometricalstructures and simply represent the gravitational field as a field in theFaraday's sense living in Minkowski spacetime. The explicit Lagrangiandensity for this theory has been given and the equations of motionpresented in a Maxwell like form and shown to be equivalent toEinstein's equations in a precise mathematical sense. We hope thatour study clarifies the real difference between mathematical modelsand physical reality and leads people to think about the real physicalnature of the gravitational field (and also of the electromagnetic fieldas suggested, e.g., by the works of Laughlin and Volikov. We discussedalso an Hamiltonian formalism for our theory and the concepts ofenergy defined by Eq.(29) and the one given by the ADM formalism.
16
References
•Arnowitt, B., Deser, S., and Misner, C. W., The Dynamics of General Relativity, in Witten, L. (ed.), Gravitation, an Introduction toCurrent Research, pp. 227-265, J. Willey &Sons, New York, 1962. [http://arXiv.org/abs/gr-qc/0405109v1]•Bohzkov, Y., and Rodrigues, W. A. Jr., Mass and Energy in General Relativity, Gen. Rel. and Grav. 27, 813- 819 (1995).•Choquet-Bruhat, Y., DeWitt-Morette, C. and Dillard-Bleick, M., Analysis, Manifolds and Physics (revised edition), North HollandPubl. Co., Amsterdam, 1982.Cartan, E., On the Generalization of the Notion of the Curvature of Riemann and spaces with Torsion Comptes Rendus Acad. Sci.vol. 174, 593-595 (1922).• Clarke, C. J. S., On the Global Isometric Embedding of Pseudo-Riemannian Manifolds, Proc. Roy. Soc. A 314, 417-428 (1970).•de Andrade, V. C., Arcos, H. I., and Pereira, J. G., Torsion as an Alternative to Curvature in the Description of Gravitation, PoS WC,028 922040, [arXiv:gr-qc/0412034].• Fernández, V. V., and Rodrigues, W. A. Jr., Gravitation as a Plastic Distortion of the Lorentz Vacuum, Fundamental Theories ofPhysics 168, Springer, Heidelberg, 2010.•Geroch, R. Spinor Structure of Space-Times in General Relativity I, J. Math. Phys. 9, 1739-1744 (1968).•Hessenberg, G., Vektorielle Begründung der Differentialgeometrie”, Mathematische Annalen 78, 187–217 (1917).•Laughlin, R.B., A Different Universe: Reinventing Physics from the Bottom Down, Basic Books, New York, 2005.•Meng, F.-F., Quasilocal Center-of-Mass Moment in General Relativity, MSc. thesis, National Central University, Chungli,2001.[http://thesis.lib.ncu.edu.tw/ETD-db/ETD-search/view_etd?URN=89222030]•Misner, C. M., Thorne, K. S. and Wheeler,J. A., Gravitation, W.H. Freeman and Co. San Francesco, 1973.•Notte-Cuello, E. A. and Rodrigues, W. A. Jr., Freud's Identity of Differential Geometry, the Einstein-Hilbert Equations and theVexatious Problem of the Energy-Momentum Conservation in GR, Adv. Appl. Clifford Algebras 19, 113-145 (2009)•Notte-Cuello, E. A., da Rocha, R., and Rodrigues, W. A. Jr., Some Thoughts on Geometries and on the Nature of the GravitationalField, J. Phys. Math. 2, 20-40 (2010).• Rodrigues, W.A. Jr. and Capelas de Oliveira, E., The Many Faces of Maxwell, Dirac and Einstein Equations. A Clifford BundleApproach. Lecture Notes in Physics 722, Springer, Heidelberg, 2007•Sachs, R. K., and Wu, H., General Relativity for Mathematicians, Springer-Verlag, New York 1977.• Volovik, G. E., The Universe in a Helium Droplet, Clarendon Press, Oxford (2003).•Szabados, L. B., Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article, Living Reviews in Relativity,[http://www.livingreviews.org/lrr-2004-4]•Thirring, W. and Wallner, R., The Use of Exterior Forms in Einstein's Gravitational Theory, Brazilian J. Phys. 8, 686-723 (1978).•Wallner, R. P., Asthekar's Variables Reexamined, Phys. Rev. D. 46, 4263-4285 (1992). 17
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