An Explicit Isochronal, Isometric Embedding

0.13 MB·English

by  Earnest Harrison

An Explicit Isochronal, Isometric Embedding
Earnest Harrison∗
February 7, 2008
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n Abstract
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Certain semi-RiemannianmetricsmaybedecomposedintoaRieman-
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nianpartandanisochronal part. WeusethisideaandanideaofKasner
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toconstructamanifoldin6+1Minkowskispacewithawellknownmetric.

The full embedding we display is isochronal which simplifies visualizing
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the properties of the manifold.
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It is well known that embeddings of manifolds are not unique, however an
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embeddingcanbeausefulrepresentationofthemanifoldthatfacilitatesunder-
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1 standing its properties. For example, an image of a manifold may be thought
0 of as a coordinate-free way of representing a given metric.
6 Wewillbeconcernedwithisometricembeddingsdefinedbyasetofequations
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inn+1Minkowskispacethatdefineaparticular3+1dimensionalsubspacewith
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c the desired metric tensor. These equations take the form [1]:
q
gr- gαβ =y,iαy,jβhij (1)
: α, β ∈{0, ...3} i, j ∈{0,...n} hij =diag(−1, 1...1)
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X Anembeddingisdefinedoncethe functions,yi,areidentified, whosederiva-
r tivescombinetoproducethedesiredmetrictensor. Inthefollowing,wewishto
a restrict our attention to embeddings that are isochronal. That is, embeddings
where the (single) time-like dimension is a linear function of time.
An Isochronal Embedding Example
The metric we shall consider is Schwarzschild’s vacuum metric expressed in
isotropic coordinates[2]:
2
2 2 2r−m 2 4 2 2 2
ds =−c dt +(1+m/2r) dr +r dΘ
2r+m (2)
(cid:18) (cid:19)
2 2 2 2 (cid:0) (cid:1)
dΘ =dθ +cos (θ)dφ ,
∗email: [e-mail]
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which we embed in 6+1 dimensional Minkowski space, Y, where y0 is the
isochronaldimension:
0
y =ct (timelike)
2
2r−m ct
1
y =2km 1− cos
s 2r+m 2km
(cid:18) (cid:19) (cid:18) (cid:19)
2
2r−m ct
2
y =2km 1− sin
s 2r+m 2km
(cid:18) (cid:19) (cid:18) (cid:19) (3)
3 2
y =(1+m/2r) rcos(φ)cos(θ)
4 2
y =(1+m/2r) rsin(φ)cos(θ)
5 2
y =(1+m/2r) rsin(θ)
m 4 m 2 2 (m−2r)2
y6 = 1+ − 1− −8k2m3 dr
s 2r 2r r (m+2r)4
Z (cid:16) (cid:17) (cid:18) (cid:16) (cid:17) (cid:19)
All of the functions remain real if r ≥0 and k2 ≤27.
Figure 1 shows slices of this manifold for the critical case of k2 = 27. The
Figure 1: Manifold slices y3 −y4 −y6, and y1 −y6 at t = φ = θ = 0. The
manifold rotates in the y1−y2 plane.
y1−y6 slice in this figure, appears as a line spinning in the y1−y2 plane at a
rate of ω = c . The tip of the curve corresponds to the event horizon, which
2km
is also at the waist of the y3−y4−y6 slice. This tip is spinning at the speed
of light. That is, the velocity vector becomes null at this point. The velocity of
eachpoint on the curve models the time dilation we see in a gravitationalfield,
which the Minkowski space, Y, takes care automatically.
The rotation of the manifold models acceleration. There are two compo-
nents of this acceleration: an in-manifold component and an out-of-manifold
component. Thein-manifoldcomponent,whichisinther direction,modelsthe
acceleration due to gravity.1
1Whileitmightseemthattheaccelerationmodeledbyarotationwouldbeawayfromthe
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Isochronal Embedding Methodology
Consider the line element for a static, spherically symmetric manifold:
2 2 2 2 2 2 2 2
ds =−c f (r) dt +g (r) dr +h (r) dΘ (4)
The first step is to define the isochronal, time-like function:
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y =k1ct,
where k1 is a free parameter. We then remove that term from (4) leaving a
Riemannian sub-metric, which we then embed.
WeuseanideaduetoKasner[4]ascarriedforwardbyFronsdal[5]2tocreate
the function, f, while compensating for the term introduced by y0:
ct
y1 =2k2m k12−f2 cos
2k2m
q (cid:18) (cid:19) (5)
ct
y2 =2k2m k12−f2 sin
2k2m
q (cid:18) (cid:19)
The term, m, is included in the circular function arguments to balance units.
We ensurethaty1 andy2 remainrealbydemanding k12 ≥f2. The term,k2,
is a second free parameter.
The spherical term can be captured in the following three functions:
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y =hcos(θ)cos(φ)
4
y =hcos(θ)sin(φ) (6)
5
y =hsin(θ)
So far, we have a metric tensor that defines a line segment as:
2
2 ′
ds2 =−c2f2dt2+h2dΘ2+ h′ + 2k2m k12−f2 dr2
 ! 
(cid:16) (cid:17) q
 
Where the prime denotes differentiation with respect to r. We will use y6 to
correct the coefficient of dr2:
′ 2
y6 = g2−(h′)2− 2k2m k12−f2 dr (7)
s
Z (cid:18) q (cid:19)
The requirementthat eachof these functions remainreal,places constraintson
the two free parameters, k1 and k2.
mass,itisnot. TheaccelerationistowardtheeventhorizonascanbeseeninFigure1.
2The circular functions have the nice property of allowing us to include both r and t in
these functions without creating unwanted off-diagonal terms in the metric tensor. Kasner
usedthispropertytorepresenttheSchwarzschildmetricinsixdimensionsusingtwotime-like
dimensions. Fronsdal used hyperbolic functions and required only one time-like dimension.
Neitheroftheseembeddingswereisochronal,however.
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Note that the manifold is dynamic, even though the metric tensor is static.
This is a general result: any isochronal manifold that admits a metric that has
nonvanishing gravitational field must be dynamic. If it were not, there would
be no accelerationdue to gravitysince the space-timewe areembedding within
is flat.
As demonstrated above, the IsochronalEmbedding Method is applicable to
metrics that can be placed in diagonal form with bounded, real functions, f,
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2 ′
g and h, and with g2− h′ − 2k2m k12−f2 everywhere non-negative
for some fixed value of(cid:16)k1(cid:17)and (cid:18)k2. Whpile these c(cid:19)onstraints exclude a global
embedding of the Schwarzschild metric, we can find a piece of a manifold that
has the Schwarzschildmetric (the piece outside the event horizon):
2m r
2 2 2 2 2 2
ds =−c 1− dt + dr +r dΘ
r r−2m
(cid:18) (cid:19)
0
y =ct
2m ct
1
y =2km cos
r 2km
r (cid:18) (cid:19)
2m ct
2
y =2km sin (8)
r 2km
r (cid:18) (cid:19)
3
y =rcos(φ)cos(θ)
4
y =rsin(φ)cos(θ)
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y =rsin(θ)
r m3
y6 = −1−2k2 dr
r−2m r3
Z r
We canseethatthe twomanifolds,(3)and(8), areidenticalovertheir rangeof
mutual validity – just use the transformation to isotropic coordinates[2]:
m 2
r= 1+ R
2R
(cid:16) (cid:17)
WecanalsoseethisqualitativelyinFigure2. Wecanseeclearlythatembedding
(8) fails at the event horizon, while embedding (3), does not. This offers a way
to analytically continue a manifold in a natural way.3
This example alsoshows that the manifold itself is coordinate-free. That is,
the manifold image (as seen in Figures 1 and 2, in this case) is independent of
the coordinate system used to express the metric (over the range of validity of
the embedding).
3While isotropic coordinate transformation “folds” the functions y1 - y5 about the event
horizon,itdoes notfoldy6,sincey6 isdefinedbyanintegralwithanon-negativeintegrand.
Thuswehaveananalyticcontinuation, notafoldingofthemanifold.
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Figure 2: The manifold defined using the Schwarzschild metric (8) is identical
to the one defined using (3) over the range of mutual validity. This embedding
method fails at the event horizon because of the coordinate singularity in (8).
Summary
Wehavedemonstratedamethodfordecomposingcertainsemi-Riemannianmet-
ric tensors into a Riemannian tensor and an isochronal tensor. We have shown
that we can embed the Riemannian portion in Euclidean space, Rn, and then
add a single time-like isochronal dimension to that space to create a n + 1
Minkowski space. The resulting manifold, embedded in the Minkowski space,
admitstothegivenmetrictensor. Theresultingmanifoldprovidesaparticularly
simple way to visualize the force and time dilation caused by gravity.
For this class of metric, the various embedding theorems that apply to Rie-
mannian manifolds can be extended to apply to semi-Riemannian manifolds.
Andsincethereisconsiderableexperienceinthemathematicalcommunitywith
Riemannian manifolds, it is hoped that progress may be made applying other
tools to metrics that admit to at least a local, isochronalembedding.
References
[1] Dirac,P.A.M.,GeneralTheoryofRelativity,(JohnWiley&Sons,NewYork:
1975), p 11.
[2] Stephani,H.,General Relativity, AnIntroduction totheTheory ofthe Grav-
itationalField,(CambridgeUniversityPress,NewYork: 1990),pp113-114.
[3] Dirac, P.A.M., op cit, pp 13, 14.
[4] Kasner, Edward, Am. J. Math. No. 43, p 130 (1921)
[5] Fronsdal, C., Phys Rev No. 116, p 778 (1959)
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