The string-model analog of general relativity is shown to be unphysically dependent on an embedding gauge. Moreover, an explicit example confirms that it is inequivalent to Einstein's theory

Deser, S.

Pirani, F. A. E.

Robinson, D. C.

Caltech Authors - Main

PHYSICAL REVIE% D VOLUME 14, NUMBER 12
New embedding model of general relativity
15 DECEMBER 1976
S. Deser, ~ F. A. E. Pirani, and D. C. Robinson
Department ofMathematics, Eing 's College, Strand, London 8'.C 2, England
(Received 15 March 1976)
The string-model analog of general relativity is shown to be unphysically dependent on an embedding gauge.
Moreover, an explicit example confirms that it is inequivalent to Einstein s theory.
Hegge and Teitelboim have developed' a variation
of general relativity employing the string model
and have discussed certain difficulties associated
with it. In this note we shall show that their the-
ory, which we shall call the new embedding model
of general relativity, suffers from gauge depen-
dence in a way which appears to be physically un-
acceptable. We shall show further, by example,
that there are vacuum solutions of the Regge-
Teitelboim equations which are not vacuum solu-
tions of Einstein's equations, thereby confirming
their statement that the physical interpretations
of the two theories are different.
A typical string model is an intrinsically n-di-
mensional dynamical system S embedded (locally)
in a manifold M of dimension rn&n; the latter may
or may not be flat. The dynamical variables are
the "amplitudes" y "(x), 2 = I, . . . , m, which locate
the system in the embedding space M, while the
coordinates g", p, =1, . . . , n parametrize the em-
bedded system S. The action is of the form
but rather
0 = 6Z/6y "(x)
=- 2I.~""(y(x))y",„j„ (4)
Alternatively, one may take both the g„„and the
y" as field quantities and deduce Eqs. (2), as well
as (4), from the variational principle by introduc-
ing a Lagrange multiplier field I,"' and substituting
for Eq. (1) the action
I=- d X 8 g —~" gf v ~~ay, f y, v
With the help of the Noether identities
v 9""=0v
where V'„denotes the covariant derivative associ-
ated with g, one may rewrite the string-model, or
external-variables, field equations (4) in the mani-
festly S-covariant form
I= — d "x6t(g),
where S is any scalar density on S built up from
the metric g of S; this metric is induced from that
of M by the embedding map S-M. In the string
models discussed here M is flat, the y~ are
(pseudo-) Cartesian coordinates on M, and the
components of g are given explicitly by
g„„(x)=g„ay" „(x)ya „(x);
here and below subscript and superscript commas
denote partial differentiation and q» is the pseudo-
Cartesian metric on M. By "pseudo-Cartesian"
we mean that g is diagonal, with entries +1 and
—1. If +1 appears p times and -1 appears q
=m- p times, then M is denoted E~*'.
For a string model the external variables y",
not the g„„, are chosen as field quantities, the
density 6t(g) is written as a function $(g(y)) by
substituting for the g's from Eqs. (2), and the field
equations are not
0 = 6Z/6g„„(x) =- S"'(x),
Notice that for any given embedding the y" are
scalar functions on S. Equations (5) are invariant
under the action of the orthogonal group of signature
P, q, 0(P, q), onE ', whichproduces rigid rotations
or pseudorotations of the embedded manifold S in E '.
As Regge and Teitelboim point out, the identity
~~ay )t f v y
which follows from 7'„g „=0, prevents the infer-
ence from (5) that 9"'=0, since it shows that the
V„V„y~ are not linearly independent.
In the origina. l string model (geometrically, n
=2 is an exceptional case'), n=2 and (R = ( —g)' '.
In the new embedding model, n =4 and S is the
Einstein scalar curvature density. The (vacuum)
Euler-Lagrange equations for the new embedding
model are equations (5) with 9""=(-g)'~'G"", where
G"" is the Einstein tensor. If (R is replaced by 6'
=: 2X, where X is a constant, then the Euler-
I agrange equations are again (5), but now with
8"=(-g)' ' (G "+kg'"). As Regge and Teitelboim
point out, and as we confirm explicitly by an ex-
ample below, these new-embedding-model equa-
3301
3302 S. DESER, F. A. E. PIRANI, AND D. C. ROBINSON 14
tions are, on account of (6), weaker than, not
equivalent to, Einstein's vacuum field equations.
It appears to be the derivative character of the
change of field variables (2), rather than the non-
linearity, which is responsible for this weakening:
Observe that the (nonlinear) substitution g„,
=q ze„e~ embodying the introduction of a, vierbein
field e „actually increases the number of field
variables from 10 to 16, but the vierbein Euler-
Lagrange equations analogous to (5), namely
(—g)'/'G'"e„=0, are, in contrast, strictly equiv-
alent to Einstein's vacuum equations.
It is worth mentioning that although the substitu-
tion (2) expresses the g's in terms of the first
derivatives of the y's, wherefore one might expect
the field equations, which contain second deriva-
tives of the g's, to conta. in third derivatives when
written in terms of the y's, nevertheless, all the
third-derivative terms cancel out, so that thenew-
embedding- model field equations are in fact t~lineax
in the second derivatives of the external variables
y". This essential nonlinearity, which is of course
related to the geometrical interpretation in terms
of embedding, makes it difficult to formulate a
linearized theory representing weak excitations
of the y~. Heuristieally speaking, an assumption
of the form y = x+ &, where & is supposed weak,
yields, on substitution into Eqs. (2), a metric of
the form
g =flat metric
+ ga,uge terms removable by
coordinate transforma, tion
+terms quadratic in &,
so that the linearized weak-field approximation
g= q+h of Einstein's theory is inaccessible in
terms of the y's.
Clarke has shown recently' that global embedding
of any Lorentzian 4-m3,nifold S in a flat M is pos-
sible with m =91 (I =50 if S is compact). Local
embeddings are always possible with m =10,' al-
though the local minimal value of rn may vary over
S, and, as is well known and will be shown in the
examples below, local embeddings with rn &10 are
possible in particular cases, although a, nonflat
solution of the vacuum Einstein equations cannot
be embedded, even locally, with rn &6. Regge and
Teitelboim postulate m~ 10 so as to be able to in-
clude, at least locally, the generic case.
For any given manifold S, there is an infinity
of embeddings S-M (with M not necessarily al-
ways the same). We call the corresponding trans-
formations of the y's "changes of embedding gauge, "
and emphasize that
(a) if an embedding with a particula, r value of I
is given, then any nonsingular linear transforma-
tion of the y's, with constant coefficients, yields
another embedding with the same value of m, and
if, furthermore, both the original and the trans-
formed y's are pseudo-Cartesian, then the trans-
formation is an element of O(p, q);
(b) if an embedding with a particular value of m
is given, then there may exist a nonlinear trans-
formation of the y's which yields another pseudo-
Cartesian embedding with the same p, q, and rn;
this corresponds to a deformation of the embedded
manifold S (we shall give a particular example of
a deformation, but we have not worked out which
solutions of the vacuum Einstein equations admit
such deformations); and
(c) if an embedding with a particular value of m
is given, then an embedding with any larger value
of m may be constructed, more or less trivially,
by introducing supernumerary y's. For example,
one may adjoin y™+1(x)=y .~(x), with q„„„„
„=1. This leaves g„„, and hence all ob-
servable phenomena, unaltered, but y "may be an
arbitrary function on S and if 8""10, then equation
(5) will not in general be satisfied by y"" or y
even if y', . . . , y satisfy it. Thus the new embed-
ding model is unphysically dependent on the embed-
ding gauge.
We now give a concrete example which illus-
trates some of these points. We exhibit a class
C of spacetimes S with the properties:
(1) there is a local embedding S-Z" for each
S in C, satisfying the new-embedding-model vacu-
um equations (5);
(2) only those spacetimes in a subclass Co satisfy
Einstein's vacuum field equations 9""=0; and
(2) for the spacetimes in a subclass C„disjoint
from C„ there is a local embedding S-F."not
satisfying the new-embedding- model vacuum equa-
tions (5).
The class C consists of the plane-fronted wave
solutions of the Einstein-Mmrwell equations'. Let
ds' = 2 du dv + 2H(x, y, u) du' + dx' + dy',
with (for convenience) H&0 in the spacetime do-
main under consideration. The Einstein tensor is
given by
G „=u „u„~H,
where H=H „„+H„.A straightforward calcula-
tion gives
G""V Vy = Hy„„.
An embedding in E" is given' by
1 —(2H)1/2 y2 —2-1/2( )
y'=2 '/'(u+v), y'=x, y'=y,
y' = (2H)'/' cosu, y' = (2H)' ' sinu.
NEW EMBEDDING MODEL OF GENERAL RELATIVITY 3303
It is evident from (7) that 9""V„V„y"=0
(A=I, . . . , 7), while G""=0 if and only if H=O.
This last condition defines the subclass C, . The
rest of the spacetimes in the class C are there-
fore interpreted in two different ways in the two
theories. As solutions of the Einstein-Maxwell
equations they represent plane-fronted electro-
magnetic and gravitational waves propagating in
the same direction without dispersion or diffusion,
while as solutions of the new-embedding-model
equations they represent purely "gravitational"
plane- fronted waves.
However, for some other spacetimes in C there
are embeddings, still in E", which do not satisfy
the new- embedding- model vacuum equations. For
example, if H(x, y, u) =-,'x', then such ar. embedding
is given by
y' = o.'(v), y' = 2 'i'(u —v) + P(v),
y'=2 ' '(&+v)+P(v) y'=r(v),
3J X COSQ~ $ X SlIlQp
where dPldv = 2 ' 't (do.jd v)' —(dy/dv)'], and at
least one of the otherwise arbitrary functions n,
P, and y is nonlinear. This is one example of the
class C„and others may easily be constructed.
It may be inferred from our remark (c) above
that only trivial modifications of these examples
are needed to increase rn to 10. The dependence
on the embedding gauge therefore makes it difficult
to regard the present new embedding model as an
interesting theory of gravitation. Admittedly we
have found no manifest conflict with observation
such as a spherically symmetric static solution
yielding the wrong perihelion motion, but we con-
tend that the gauge dependence is nevertheless un-
acceptable.
We have not found any satisfactory way out of
this difficulty. Regge and Teitelboim suggest the
ad hoc addition of further field equations which
make the new embedding model equivalent to Ein-
stein's theory. A more satisfactory alternative
would be to find another action functional l(y) whose
Euler-I agrange equations (4) were equivalent to
Einstein's equations, but neither they nor we have
been able to do this. If it were possible to find an
action which removed the excessive gauge freedom,
even though the field equations were not equivalent
to Einstein's equations, this too might resolve the
difficulty. However, one final example suggests
that the linearization instability described earlier
may impede any easy resolution: Einstein's equa-
tions with cosmological constant
G~v+ gg~v 0
are not satisfied by the flat spacetime metric g„„
= q „. Yet the corresponding new-embedding-mod-
el equations
(G""+Xg"")V,V„y"= 0
are trivially satisfied by
and by a variety of supernumerary y's.
*On leave from Brandeis University, Waltham, Massa-
chusetts, U.S.A.; work supported in part by the Na-
tional Science Foundation, the Science Research Coun-
cil, and NATO.
~T. Regge and C. Teitelboim, in Proceedings of the
Marcel Grossman meeting, Trieste, 1975 (unpub-
lished).
2C. Aragone and S. Deser, Nucl. Phys. 896, 327 (1975).
3C. J. S. Clarke, Proc. R. Soc. London A314, 417 (1970).
4M. Janet, Ann. Soc. Polon. Math 5, 38 (1926); E. Car-
tan, ibid. 6, 1 (1927).
5J. Ehlers and W. Kundt, in Gravitation: an introduc-
tion to current research, edited by Witten (Wiley,
New York, 1962).
6J. Rosen, Rev. Mod. Phys. 37, 204 (1965). It has been
shown by R. Penrose, Bev. Mod. Phys. 37, 215 (1965),
that noglobal embedding in &~' with q (orP) =1 is
possible.


New embedding model of general relativity

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New embedding model of general relativity

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