Table Of ContentReference Frame in General Relativity
AleksKleyn
8 Abstract. Areferenceframeineventspaceisasmoothfieldoforthonormal
0
bases. Every referenceframeis equipped by anholonomic coordinates. Using
0
anholonomiccoordinatesallowstofindoutrelativespeedoftwoobserversand
2
appropriateLorentztransformation.
n Synchronization of a reference frame is an anholonomic time coordinate.
a Simplecalculationsshowhowsynchronizationinfluencestimemeasurementin
J thevicinityoftheEarth.
Measurement of Doppler shift from the star orbiting the black hole helps
7
to determine mass ofthe black hole. Accordingobservations ofSgr A, ifnon
2
orbiting observer estimates age of S2 about 10 Myr, this star is 0.297 Myr
younger.
3
v
7
2
0 1. Geometrical Object and Invariance Principle
5
0 Theinvarianceprinciplewhichwestudied[14]islimitedbyvectorspacesandwe
4 can use it only in frame of the special relativity. Our task is to describe structures
0 which allow to extend the invariance principle to general relativity.
/
c A measurement of a spatial interval and a time length is one of the major tasks
q of general relativity. This is a physical process that allows the study of geometry
-
r in a certain area of spacetime. From a geometrical point of view, the observer
g uses an orthonormal basis in a tangent plane as his measurement tool because an
:
v orthonormal basis leads to the simplest local geometry. When the observer moves
i from point to point he brings his measurement tool with him.
X
Notion of a geometrical object is closely related with physical values measured
r
a in space time. The invariance principle allows expressing physical laws indepen-
dently from the selection of a basis. On the other hand if we want to examine a
relationship obtained in the test, we need to select measurement tool. In our case
thisisbasis. Choosingthebasiswecandefinecoordinatesofthegeometricalobject
corresponding to studied physical value. Hence we can define the measured value.
Every reference frame is equipped by anholonomic coordinates. For instance,
synchronization of a reference frame is an anholonomic time coordinate. Simple
calculationsshowhow synchronizationinfluences time measurementin the vicinity
of the Earth. Measurement of Doppler shift from the star orbiting the black hole
helps to determine mass of the black hole.
Sections 9.1 and 10 show importance of calculations in orthogonalframe. Coor-
dinates that we use in event space are just labels and calculations that we make in
coordinatesmayappearnotreliable. Forinstanceinpapers[6,7]authorsdetermine
coordinate speed of light. This leads to not reliable answer and as consequence of
this to the difference of speed of light in different directions.
Aleks [email protected].
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Aleks Kleyn
Reference Frame in General Relativity
Few authors consider Einstein’s idea about variability of speed of light [13].
However Einstein proposed this idea when supposed to create theory of gravity
in Minkovsky space, therefore he assumed that scale of space and time does not
change. When Einstein learnedRiemann geometry he changedhis mind andnever
returnedtoideaaboutvariablespeedoflight. Scaleofspaceandtimeandspeedof
light are correlated in present theory and we cannot change one without changing
another.
There are few papers dedicated to a variable speed of light theory [8, 9]. Their
theoryisbasedonideathatmetrictensormaybescaleablerelativedilatation. This
ideaisnotnew. AssoonasEinsteinpublishedgeneralrelativityWeylintroducedhis
ideatomaketheoryinvariantrelativeconformaltransformation. HoweverEinstein
was firmly opposed to this idea because it broke dependence between distance and
proper time. You can find detailed analysis in [10].
We havestrong relationbetween the speed of light and units of length and time
in special and general relativity. When we develop new theory and discover that
speed of light is different we should ask ourselves about the reason. Did we make
accurate measurement? Do we have alternative way to exchange information and
synchronizereferenceframe? Didtransformationsbetweenreferenceframeschange
and do they create group?
Insomemodelsphotonmayhavesmallrestmass[11]. Inthiscasespeedoflight
is different from maximal speed and may depend on direction. Recent experiment
[12] put limitation on parameters of these models.
2. Reference Frame on Manifold
As is shown in section [14]-5 we canidentify frame manifold of vectorspace and
symmetry group of this space. Our interest was not details of structure of basis,
and stated theory can be generalized and extended on an arbitrary manifold. The
detailsofstructureofbasisdidnotinterestusandstatedtheorycanbegeneralized.
Inthissectionwegeneralizethedefinitionofaframeandintroduceareferenceframe
onamanifold. Incaseofaneventspaceofgeneralrelativityitleadsustoanatural
definition of a reference frame and the Lorentz transformation.
When we study manifold V the geometry of tangent space is one of important
factors. In this section, we will make the following assumption.
• All tangent spaces have the same geometry.
• Tangent space is vector space V of finite dimension n.
• Symmetry group of tangent space is Lie group G.
Definition 2.1. Set e=<e ,i∈I > of vector fields e is called G-reference
(i) (i)
frame, if for any x∈V set e(x)=<e (x),i ∈I > is a G-basis1 in tangent space
(i)
T .2 We use notation e ∈e for vector fields which form G-reference frame e. (cid:3)
x (i)
Vector field a has expansion
(2.1) a=a(i)e
(i)
relative reference frame e.
Ifwedonotlimitdefinitionofareferenceframebysymmetrygroup,thenateach
pointofthemanifoldwecanselectreferenceframe∂ =<∂ >basedonvectorfields
i
1Accordingtosection[14]-5wecanidentifybasise(x)withanelementofgroupG.
2IneachparticularcaseweneedtoproveexistenceofG-referenceframeonmanifold.
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Aleks Kleyn
Reference Frame in General Relativity
tangent to lines xi =const. We call this field of bases the coordinate reference
frame. Vector field a has expansion
(2.2) a=ai∂
i
relative coordinate reference frame. Then standard coordinates of reference frame
e have form ek
(i)
(2.3) e =ek ∂
(i) (i) k
Becausevectorse arelinearlyindependentateachpointmatrixkek khasinverse
(i) (i)
matrix ke(i)k
k
(2.4) ∂ =e(i)e
k k (i)
Weusealsoamoreextensivedefinitionforreferenceframeonmanifold,presented
informe=(e ,e(k))whereweusethe setofvectorfieldse anddualformse(k)
(k) (k)
such that
(2.5) e(k)(e )=δ(k)
(l) (l)
at each point. Forms e(k) are defined uniquely from (2.5).
In a similar way, we can introduce a coordinate reference frame (∂ ,dsi). These
i
reference frames are linked by the relationship
(2.6) e =ei ∂
(k) (k) i
(2.7) e(k) =e(k)dxi
i
From equations (2.6), (2.7), (2.5) it follows
(2.8) e(k)ei =δ(k)
i (l) (l)
In particular we assume that we have GL(n)-reference frame (∂,dx) raised by
n differentiable vector fields ∂ and 1-forms dxi, that define field of bases ∂ and
i
cobases dx dual them.
If we have function ϕ on V than we define pfaffian derivative
dϕ=∂ ϕdxi
i
3. Reference Frame in Event Space
Startingfromthissection,weconsiderorthogonalreferenceframee=(e ,e(k))
(k)
inRiemannspacewithmetricg . Accordingtodefinition,ateachpointofRiemann
ij
space vector fields of orthogonalreference frame satisfy to equation
g ei ej =g
ij (k) (l) (k)(l)
whereg =0,if(k)6=(l),andg =1org =−1dependingonsignature
(k)(l) (k)(k) (k)(k)
of metric.
We can define the reference frame in event space V as O(3,1)-reference
frame. To enumeratevectors,weuse index k =0,...,3. Index k=0 correspondsto
time like vector field.
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Aleks Kleyn
Reference Frame in General Relativity
Remark 3.1. We can prove the existence of a reference frame using the orthogono-
lization procedure at every point of space time. From the same procedure we get
that coordinates of basis smoothly depend on the point.
A smooth field of time like vectors of each basis defines congruence of lines that
are tangent to this field. We say that each line is a world line of an observer or
a local reference frame. Therefore a reference frame is set of local reference
frames. (cid:3)
We define the Lorentz transformation as transformationof a reference frame
x′i =fi(x0,x1,x2,x3)
(3.1) e′i =aib(l)ej
(k) j (k) (l)
where
∂x′i
ai =
j ∂x′j
δ b(i)b(l) =δ
(i)(l) (j) (k) (j)(k)
We call the transformation ai the holonomic part and transformation b(l) the an-
j (k)
holonomic part.
4. Anholonomic Coordinates
Let E(V,G,π) be the principal bundle, where V is the differential manifold of
dimension n and class not less than 2. We also assume that G is symmetry group
of tangent plain.
We define connection form on principal bundle
(4.1) ωL =λLdaN +ΓLdxi ω =λ daN +Γdx
N i N
We call functions Γ connection components.
i
If fiber is group GL(n), than connection has form
(4.2) ωa =Γa dxc
b bc
ΓA =Γa
i bi
A vector field a has two types of coordinates: holonomic coordinates ai rel-
ative coordinate reference frame and anholonomic coordinates a(i) relative ref-
erence frame. These two forms of coordinates also hold the relation
(4.3) ai(x)=ei (x)a(i)(x)
(i)
at any point x.
We can study parallel transfer of vector fields using any form of coordinates.
Because (3.1) is a linear transformationwe expect that parallel transfer in anholo-
nomiccoordinateshas the samerepresentationasinholonomiccoordinates. Hence
we write
dak =−Γkaidxj
ij
da(k) =−Γ(k) a(i)dx(j)
(i)(j)
It is required to establish link between holonomic coordinate of connection
Γk and anholonomic coordinates of connection Γ(k)
ij (i)(j)
(4.4) ai(x+dx)=ai(x)+dai =ci(x)−Γi ck(x)dxp
kp
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Aleks Kleyn
Reference Frame in General Relativity
(4.5) a(i)(x+dx)=a(i)(x)+da(i) =c(i)(x)−Γ(i) c(k)(x)dx(p)
(k)(p)
Considering (4.4), (4.5), and (4.3) we get
ai(x)−Γi ak(x)dxp
kp
(4.6)
=ei (x+dx) a(i)(x)−Γ(i) e(k)(x)ai(x)e(p)(x)dxp
(i) (k)(p) i p
(cid:16) (cid:17)
It follows from (4.6) that
Γ(i) e(k)(x)e(p)(x)ai(x)dxp =a(i)(x)−e(i)(x+dx) ai(x)−Γi ak(x)dxp
(k)(p) i p i kp
=ai(x)e(i)(x)−e(i)(x+(cid:0)dx) ai(x)−Γi ak(x(cid:1))dxp
i i kp
=ai(x) e(i)(x)−e(i)(x+dx(cid:0)) +e(i)(x)Γj ai(x)dx(cid:1)p
i i j ip
(cid:16) ∂e(cid:17)(i)(x)
=e(i)(x)Γj ai(x)dxp−ai(x) i dxp
j ip ∂xp
∂e(i)(x)
= e(i)(x)Γj − i ai(x)dxp
j ip ∂xp
!
Because ai(x) and dxp are arbitrary we get
∂e(i)(x)
Γ(i) e(k)(x)e(p)(x)=e(i)(x)Γj − i
(k)(p) i p j ip ∂xp
(i)
∂e
(4.7) Γ(i) =ei ep e(i)Γj −ei ep i
(k)(p) (k) (p) j ip (k) (p) ∂xp
We introduce symbolic operator
∂ ∂
(4.8) =ep
∂x(p) (p)∂xp
From (2.8) it follows
∂e(k) ∂ei
(4.9) ei i +e(k) (l) =0
(l) ∂xp i ∂xp
Substitude (4.8) and (4.9) into (4.7)
∂ei
(4.10) Γ(i) =ei ep e(i)Γj −e(i) (k)
(k)(p) (k) (p) j ip i ∂x(p)
Equation (4.10) shows some similarity between holonomic and anholonomic co-
ordinates. We introduce symbol ∂ for the derivative along vector field e
(k) (k)
∂ =ei ∂
(k) (k) i
Then (4.10) takes the form
Γ(k) =ei er e(k)Γj −ei ∂ e(k)
(l)(p) (l) (p) j ir (l) (p) i
Therefore when we move from holonomic coordinates to anholonomic, the con-
nectiontransformsthe waysimilarlyto whenwe movefrom one coordinatesystem
to another. This leads us to the model of anholonomic coordinates.
The vector field e generates lines defined by the differential equations
(k)
∂t
ej =δ(k)
(l)∂xj (l)
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Aleks Kleyn
Reference Frame in General Relativity
or the symbolic system
∂t
(4.11) =δ(k)
∂x(l) (l)
Keepingin mind the symbolic system(4.11) we denote the functionalt as x(k) and
call it the anholonomic coordinate. We call the regular coordinate holonomic.
From here we can find derivatives and get
∂x(i)
(4.12) =e(i)
∂xk k
The necessary and sufficient conditions of complete integrability of system (4.12)
are
c(i) =0
(k)(l)
where we introduced anholonomity object
∂e(i) ∂e(i)
(4.13) c(i) =ek el k − l
(k)(l) (k) (l) ∂xl ∂xk
!
Therefore each reference frame has n vector fields
∂
∂ = =ei ∂
(k) ∂x(k) (k) i
which have commutator
[∂ ,∂ ]= ek ∂ el −ek ∂ el e(m)∂ =
(i) (j) (i) k (j) (j) k (i) l (m)
(cid:16) (cid:17)
ek el −∂ e(m)+∂e(m) ∂ =c(m) ∂
(i) (i) k l l l (m) (k)(l) (m)
For the same reason we in(cid:16)troduce forms (cid:17)
dx(k) =e(k) =e(k)dxl
l
and an exterior differential of this form is
d2x(k) =d e(k)dxi
i
(4.14) = ∂(cid:16)e(k)−(cid:17)∂ e(k) dxi∧dxj
j i i j
=−(cid:16)c(m) dx(k)∧d(cid:17)x(l)
(k)(l)
Therefore when c(i) 6= 0, the differential dx(k) is not an exact differential
(k)(l)
and the system (4.12) in general cannot be integrated. However we can create
a meaningful object that models the solution. We can study how function x(i)
changes along different lines. We call such cordinates anholonomic coordinates
on manifold.
Remark 4.1. Function x(i) is a natural parameter along a flow line of vector field
e . Westudyaninstanceofsuchfunctioninsection8. Thepropertimeisdefined
(i)
alongworldlineoflocalreferenceframe. Asweseeinremark3.1worldlinesoflocal
referenceframescoverspacetime. To makepropertime oflocalreferenceframesas
time of reference frame we expect that proper time smoothly changes from point
to point. To synchronizeclocksof localreference frames we use classicalprocedure
of exchange light signals.
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Aleks Kleyn
Reference Frame in General Relativity
From mathematical point of view this is problem to integrate differential form.
However,a change of function along a loop is
∆x(i) = dx(i)
I
(4.15) = c(i) dx(k)∧dx(l)
(k)(l)
Z Z
= c(i) e(k)e(l)dxk∧dxl
(k)(l) k l
Z Z
Somebody may have impression that we cannot synchronize clock, however this
conflicts with our observation. We accept that synchronization is possible until
we introduce time along non closed lines. Synchronization breaks when we try
synchronize clocks along closed line.
This means ambiguity in definition of anholonomic coordinates. (cid:3)
Fromnowonwe willnotmakeadifferencebetweenholonomicandanholonomic
coordinates. Also,wewilldenoteb(l) asa−1(l) intheLorentztransformation(3.1).
(k) (k)
Even form dx(k) is not exact differential, we can see that form d2x(k) is exterior
differential of form dx(k). Therefore
(4.16) d3x(k) =0
We can represent exterior differential of form, written in anholonomic coordi-
nates, as
d(a dx(i1)∧...∧dx(in))
(i1)...(in)
=a dxp∧dx(i1)∧...∧dx(in)
(i1)...(in),p
−a ddx(i1)∧...∧dx(in)−...−(−1)n−1a dx(i1)∧...∧ddx(in)
(i1)...(in) (i1)...(in)
=a e(p)ep dx(r)∧dx(i1)∧...∧dx(in)
(i1)...(in),(p) p (r)
−a c(i1) dx(p)∧dx(r)∧...∧dx(in)−...
(i1)...(in) (p)(r)
−(−1)n−1a dx(i1)∧...∧c(in) dx(p)∧dx(r)
(i1)...(in) (p)(r)
=(a −a c(r) −...−a c(r) )dx(p)∧dx(i1)∧...∧dx(in)
(i1)...(in),(p) (r)...(in) (p)(i1) (i1)...(r) (p)(in)
In case of form d3x(k) we get equation
d(c(k) dx(i)∧dx(j))
(4.17) (i)(j)
=(c(k) −c(k) c(r) −c(k) c(r) )dx(p)∧dx(i)∧dx(j)
(i)(j),(p) (r)(j) (p)(i) (i)(r) (p)(j)
From equations (4.16) and (4.17) it follows
(4.18) (c(k) −c(k) c(r) −c(k) c(r) )dx(p)∧dx(i)∧dx(j) =0
(i)(j),(p) (r)(j) (p)(i) (i)(r) (p)(j)
It is easy to see that
(−c(k) c(r) −c(k) c(r) )dx(i)∧dx(j)
(r)(j) (p)(i) (i)(r) (p)(j)
(4.19) =(−c(k) c(r) +c(k) c(r) )dx(i)∧dx(j)
(r)(j) (p)(i) (r)(i) (p)(j)
=−2c(k) c(r) dx(i)∧dx(j)
(r)(j) (p)(i)
Substituting from (4.19) into (4.18) gives
(4.20) (c(k) −2c(k) c(r) )dx(p)∧dx(i)∧dx(j) =0
(i)(j),(p) (r)(j) (p)(i)
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Aleks Kleyn
Reference Frame in General Relativity
From (4.20), it follows
c(k) +c(k) +c(k)
(4.21) (i)(j),(p) (j)(p),(i) (p)(i),(j)
=2c(k) c(r) +2c(k) c(r) +2c(k) c(r)
(r)(j) (p)(i) (r)(p) (i)(j) (r)(i) (j)(p)
We define the curvature form for connection (4.1)
Ω=dω+[ω,ω]
ΩD =dωD+CD ωA∧ωB =RDdxi∧dxj
AB ij
where we defined a curvature object
RD =∂ ΓD−∂ ΓD+CD ΓAΓB+ΓDck
ij i j j i AB i j k ij
The curvature form for the connection (4.2) is
(4.22) Ωa =dωa+ωa∧ωb
c c b c
where we defined a curvature object
(4.23) RD =Ra =∂ Γa −∂ Γa +ΓaΓc −Γa Γc +Γa ck
ij bij i bj j bi ci bj cj bi bk ij
We introduce Ricci tensor
R =Ra =∂ Γa −∂ Γa +Γa Γc −Γa Γc +Γa ck
bj baj a bj j ba ca bj cj ba bk aj
5. Metric-affine Manifold
For connection (4.2) we defined the torsion form
(5.1) Ta =d2xa+ωa∧dxb
b
From (4.2) it follows
(5.2) ωa∧dxb =(Γa −Γa )dxc∧dxb
b bc cb
Putting (5.2) and (4.14) into (5.1) we get
(5.3) Ta =Tadxc∧dxb =−ca dxc∧dxb+(Γa −Γa )dxc∧dxb
cb cb bc cb
where we defined torsion tensor
(5.4) Ta =Γa −Γa −ca
cb bc cb cb
Commutator of second derivatives has form
(5.5) uα −uα =Rα uβ −Tpuα
;kl ;lk βlk lk ;p
From (5.5) it follows that
(5.6) ξa −ξa =Ra ξd−Tpξa
;cb ;bc dbc bc ;p
In Rieman space we have metric tensor g and connection Γk. One of the
ij ij
featuresoftheRiemanspaceissymetricityofconnectionandcovariantderivativeof
metricis0. Thiscreatescloserelationbetweenmetricandconnection. Howeverthe
connection is not necessarily symmetric and the covariant derivative of the metric
tensor may be different from 0. In latter case we introduce the nonmetricity
(5.7) Qij =gij =gij +Γi gpj +Γj gip
k ;k ,k pk pk
Due to the fact that derivative of the metric tensor is not 0, we cannot raise or
lowerindexofatensorunderderivativeaswedoitinregularRiemannspace. Now
this operation changes to next
ai =gija +gija
;k j;k ;k j
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Aleks Kleyn
Reference Frame in General Relativity
This equation for the metric tensor gets the following form
gab =−gaigbjg
;k ij;k
Definition 5.1. Wecallamanifoldwithatorsionandanonmetricitythemetric-
affine manifold [3]. (cid:3)
If we study a submanifold V of a manifold V , we see that the immersion
n n+m
creates the connection Γα that relates to the connection in manifold as
βγ
∂el
Γα el =Γl emek + β
βγ α mk β γ ∂uγ
Therefore there is no smooth immersion of a space with torsion into the Riemann
space.
6. Geometrical Meaning of Torsion
Suppose that a and b are non collinear vectors in a point A (see figure 6.1).
WedrawthegeodesicLathroughthe WedrawthegeodesicLbthroughthe
point A using the vector a as a tan- point A using the vector b as a tan-
gentvectortoLa inthepointA. Let gentvectortoLb inthe pointA. Let
τ be the canonical parameter on La ϕ be the canonical parameter on Lb
and and
dxk dxk
=ak =bk
dτ dϕ
We transfer the vector b along the We transfer the vector a along the
geodesic La from the point A into a geodesic Lb from the point A into a
point B that defined by any value of point D that defined by any value of
the parameter τ = ρ > 0. We mark the parameter ϕ = ρ > 0. We mark
the result as b′. the result as a′.
We draw the geodesic Lb′ through We draw the geodesic La′ through
the point B using the vector b′ as a the point D using the vector a′ as a
tangentvector to Lb′ in the point B. tangentvectorto La′ inthe pointD.
Letϕ′ bethecanonicalparameteron Letτ′ bethe canonicalparameteron
Lb′ and La′ and
dxk dxk
=b′k =a′k
dϕ′ dτ′
We define a point C on the geodesic We define a point E on the geodesic
Lb′ by parameter value ϕ′ =ρ La′ by parameter value τ′ =ρ
FormallylinesABandDEaswellaslinesADandBC areparallellines. Lengths
of AB and DE are the same as well as lengths of AD and BC are the same. We
call this figure a parallelogram based on vectors a and b with the origin in the
point A.
Theorem 6.1. Suppose CBADE is a parallelogram with a origin in the point
A; then the resulting figure will not be closed [4]. The value of the difference of
coordinates of points C and E is equal to surface integral of the torsion over this
parallelogram3
∆ xk = Tk dxm∧dxn
CE mn
ZZ
3Proofofthisstatement Ifoundin[5]
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Aleks Kleyn
Reference Frame in General Relativity
C
Q
B (cid:8)(cid:8)b′(cid:8)* Q E
6
a′
(cid:3)(cid:23)
a (cid:3)
(cid:3)
(cid:3) b(cid:17)3
(cid:17)
(cid:3)(cid:17) D
A
Figure 6.1. Meaning of Torsion
Proof. We can find an increase of coordinate xk along any geodesic as
dxk 1d2xk
∆xk = τ + τ2+O(τ2)=
dτ 2 dτ2
dxk 1 dxmdxn
= τ − Γk τ2+O(τ2)
dτ 2 mn dτ dτ
where τ is canonical parameter and we take values of derivatives and components
Γk in the initial point. In particular we have
mn
1
∆ xk =akρ− Γk (A)amanρ2+O(ρ2)
AB 2 mn
along the geodesic L and
a
1
(6.1) ∆ xk =b′kρ− Γk (B)b′mb′nρ2+O(ρ2)
BC 2 mn
along the geodesic Lb′. Here
(6.2) b′k =bk−Γk (A)bmdxn+O(dx)
mn
is the result of parallel transfer of bk from A to B and
(6.3) dxk =∆ xk =akρ
AB
with precision of small value of first level. Putting (6.3) into (6.2) and (6.2) into
(6.1) we will receive
1
∆ xk =bkρ−Γk (A)bmanρ2− Γk (B)bmbnρ2+O(ρ2)
BC mn 2 mn
Common increase of coordinate xK along the way ABC has form
∆ xk =∆ xk+∆ xk =
ABC AB BC
(6.4) =(ak+bk)ρ−Γk (A)bmanρ2−
mn
1 1
− Γk (B)bmbnρ2− Γk (A)amanρ2+O(ρ2)
2 mn 2 mn
Similar way common increase of coordinate xK along the way ADE has form
∆ xk =∆ xk+∆ xk =
ADE AD DE
10
