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The mean curvature of gravitational fields
Physica 114A (1982) 143-145 North-Holland Publishing Co.
THE MEAN CURVATURE
OF GRAVITATIONAL
FIELDS
Kishore B. MARATHE Department of Mathematics, Brooklyn College, Bedford Avenue and Aoenue H, Brooklyn, NY 11210, USA The mean curvature of a gravitational field is defined as a generalization of the average curvature in a given direction by using the gravitational sectional curvature function on nondegenerate tangent 2-planes to the space-time manifold. We find that the mean curvature of a gravitational field is independent of direction as determined by a unit vector. The converse of this result provides a new characterization of spaces admitting gravitational fields. We also define the average curvature (or bending) of a non-degenerate k-plane in a gravitational field and show that it is independent of the choice of non-degenerate k-plane.
We define a gravitational field F as a triple (M, g, T) which satisfies the following conditions: (1) M is a Qdimensional manifold with symmetric, fundamental tensor g of type (2,0) and signature (-, -,-, +) and with the unique torsion-free LeviCivita connection r. (2) T is a symmetric divergence-free tensor of type (2,O) on M. (3) g satisfies Einstein’s field equations with source T. Using a local coordinate chart and the induced basis of the tensor algebra of M, we can write conditions 2 and 3 in the familiar form Tii = Tji,
ViTij = 0;
(1)
and R 0 _ ;Rg ii = _ T ii,
(2)
where Vi is the covariant derivative along a/ax’ and the units are chosen so that the coefficient of T” in eq. (2) is - 1. When all the above conditions are satisfied we say that M is the carrier of the gravitational field F, or that the space M admits the gravitational field F. In what follows we assume that the triple (M, g, T) satisfies conditions 1 and 2 of the above definition unless otherwise stated. We say that a tensor S of the type (4,0) is a tensor of curvature type if it satisfies (point-wise) the algebraic properties of a Riemann curvature tensor. It is well known that if S is of curvature type then S,, (x E M), can be regarded as a symmetric, linear transformation of the space of second-order differential forms AgM). Now for each x E A4 we can regard g, and TX as endomorphisms of A;(M), the space of first-order differential forms. Their exterior product gJT, is then a symmetric linear transformation of AZ(M). The corresponding tensor of curvature type is denoted by gAT. 0378-4371/82/ooo(Mooo
BO2.75 @ 1982 North-Holland
144
KISHORE
We can now associate defined by
B. MARATHE
with triple (M, g, T) a tensor
W=R+gAT,
W of curvature
type
(3)
where R is the Riemann curvature tensor on M. The tensor W has been used by the author’.*) to obtain various characterizations of gravitational field equations. The algebraic and differential properties of W have been studied by ,Modugno3) which lead in particular, to new proofs of some classical results of Lichnerowicz. The gravitational sectional curvature function is defined on the space of non-degenerate cotangent 2-planes to the space-time manifold M and is denoted by f. If P is a non-degenerate 2-plane with orthonormal basis (U, V) then f is defined by f(P) = f(U, V) = s(U)s(V)G(W(U, where s(U) induced by orthonormal The mean
V), U, VI,
(4)
= g( U, U) is the sign of U and G is the inner product on AgM) g. It can be shown that f does not depend on the choice of basis of the plane P. curvature m(U) along a unit covector (l-form) is defined by 1
i=4
m(U) = j F, fW’, Vi), where U = U’, U*, U3, U4 is an orthonormal basis of A:(M). The average curvature or bending m(K) of a non-degenerate k-plane (k 5 2) is defined by 1-9 i f(U’, Vi), m(K) = k(4- k) i=l j=k+r
(6)
where the first k covectors form an orthonormal basis for K. This definition is a generalization of the definition of mean curvature given by Tachibana4) for the case of Riemannian manifolds. The calculation of m(K) is simplified by the observation that the plane K is completely determined by a unit decomposable k-form U’ A.. . A Uk. Now if (M, g, T) is a gravitational field, the field equations can be expressed by the vanishing of the trace-free part of Ric (W), the Ricci tensor of W. Using this result in (5) and (6) we get: Theorem 1. Let U be a unit covector and K the 3-plane which is the orthogonal complement of U. Then m(U) = m(K) = -Tr(Ric( W))/12, where Tr denotes the trace operator.
(7)
THE MEAN CURVATURE
OF GRAVITATIONAL
FIELDS
145
The converse of this theorem is also true and provides a new characterization of spaces admitting gravitational fields. We state it as Theorem 2. If eq. (7) holds for any unit covector or non-degenerate triple (M, g, T) defines a gravitational field. For 2-planes the sectional curvature function characterization of gravitational fields.
3-plane, the
itself provides the following
Theorem 3. (M, g, T) defines a gravitational field if and only if f(P) = f(P’) holds for all non-degenerate 2-planes P and their orthogonal complements P’.
References
THE MEAN CURVATURE
OF GRAVITATIONAL
FIELDS
Kishore B. MARATHE Department of Mathematics, Brooklyn College, Bedford Avenue and Aoenue H, Brooklyn, NY 11210, USA The mean curvature of a gravitational field is defined as a generalization of the average curvature in a given direction by using the gravitational sectional curvature function on nondegenerate tangent 2-planes to the space-time manifold. We find that the mean curvature of a gravitational field is independent of direction as determined by a unit vector. The converse of this result provides a new characterization of spaces admitting gravitational fields. We also define the average curvature (or bending) of a non-degenerate k-plane in a gravitational field and show that it is independent of the choice of non-degenerate k-plane.
We define a gravitational field F as a triple (M, g, T) which satisfies the following conditions: (1) M is a Qdimensional manifold with symmetric, fundamental tensor g of type (2,0) and signature (-, -,-, +) and with the unique torsion-free LeviCivita connection r. (2) T is a symmetric divergence-free tensor of type (2,O) on M. (3) g satisfies Einstein’s field equations with source T. Using a local coordinate chart and the induced basis of the tensor algebra of M, we can write conditions 2 and 3 in the familiar form Tii = Tji,
ViTij = 0;
(1)
and R 0 _ ;Rg ii = _ T ii,
(2)
where Vi is the covariant derivative along a/ax’ and the units are chosen so that the coefficient of T” in eq. (2) is - 1. When all the above conditions are satisfied we say that M is the carrier of the gravitational field F, or that the space M admits the gravitational field F. In what follows we assume that the triple (M, g, T) satisfies conditions 1 and 2 of the above definition unless otherwise stated. We say that a tensor S of the type (4,0) is a tensor of curvature type if it satisfies (point-wise) the algebraic properties of a Riemann curvature tensor. It is well known that if S is of curvature type then S,, (x E M), can be regarded as a symmetric, linear transformation of the space of second-order differential forms AgM). Now for each x E A4 we can regard g, and TX as endomorphisms of A;(M), the space of first-order differential forms. Their exterior product gJT, is then a symmetric linear transformation of AZ(M). The corresponding tensor of curvature type is denoted by gAT. 0378-4371/82/ooo(Mooo
BO2.75 @ 1982 North-Holland
144
KISHORE
We can now associate defined by
B. MARATHE
with triple (M, g, T) a tensor
W=R+gAT,
W of curvature
type
(3)
where R is the Riemann curvature tensor on M. The tensor W has been used by the author’.*) to obtain various characterizations of gravitational field equations. The algebraic and differential properties of W have been studied by ,Modugno3) which lead in particular, to new proofs of some classical results of Lichnerowicz. The gravitational sectional curvature function is defined on the space of non-degenerate cotangent 2-planes to the space-time manifold M and is denoted by f. If P is a non-degenerate 2-plane with orthonormal basis (U, V) then f is defined by f(P) = f(U, V) = s(U)s(V)G(W(U, where s(U) induced by orthonormal The mean
V), U, VI,
(4)
= g( U, U) is the sign of U and G is the inner product on AgM) g. It can be shown that f does not depend on the choice of basis of the plane P. curvature m(U) along a unit covector (l-form) is defined by 1
i=4
m(U) = j F, fW’, Vi), where U = U’, U*, U3, U4 is an orthonormal basis of A:(M). The average curvature or bending m(K) of a non-degenerate k-plane (k 5 2) is defined by 1-9 i f(U’, Vi), m(K) = k(4- k) i=l j=k+r
(6)
where the first k covectors form an orthonormal basis for K. This definition is a generalization of the definition of mean curvature given by Tachibana4) for the case of Riemannian manifolds. The calculation of m(K) is simplified by the observation that the plane K is completely determined by a unit decomposable k-form U’ A.. . A Uk. Now if (M, g, T) is a gravitational field, the field equations can be expressed by the vanishing of the trace-free part of Ric (W), the Ricci tensor of W. Using this result in (5) and (6) we get: Theorem 1. Let U be a unit covector and K the 3-plane which is the orthogonal complement of U. Then m(U) = m(K) = -Tr(Ric( W))/12, where Tr denotes the trace operator.
(7)
THE MEAN CURVATURE
OF GRAVITATIONAL
FIELDS
145
The converse of this theorem is also true and provides a new characterization of spaces admitting gravitational fields. We state it as Theorem 2. If eq. (7) holds for any unit covector or non-degenerate triple (M, g, T) defines a gravitational field. For 2-planes the sectional curvature function characterization of gravitational fields.
3-plane, the
itself provides the following
Theorem 3. (M, g, T) defines a gravitational field if and only if f(P) = f(P’) holds for all non-degenerate 2-planes P and their orthogonal complements P’.
References