1. Unpublished draft prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1
Conserved Integral Quantities in Spacetime
S. J. Fletcher1†
1
Bureau of Meteorology Training Centre, Bureau of Meteorology, Melbourne, Victoria,
Australia
(Received xx; revised xx; accepted xx)
We construct semi-explicit expressions for the vector fields, Z, under which a specified
p-form field, Gp
, is invariant in spacetime. This guarantees conservation of the integral
of Gp
over any p-dimensional submanifold moving with the flow of Z. Applications are
presented.
1. Introduction
1.1. Definitions
Definition
Spacetime is a 4-dimensional manifold, M4
, with a pseudo-Riemannian metric tensor
field. Using an orthonormal basis the metric has diagonal components (−1, 1, 1, 1).
Definition
Let Z be a vector field and Z3
a 3-form field in spacetime. These fields are said to be
associated if:
Z3 ≡ Z1 ≡ g (Z, ) (1.1)
is the Hodge star operation and g the covariant metric tensor.
Knowledge of Z3 is equivalent to knowledge of Z since the mappping between them is
1-1.
Definition
Let Gp denote a p-form field in spacetime. Gp is invariant with respect to Z if:
£Z
Gp
= 0 (1.2)
£Z
denotes the Lie derivative along Z.
1.2. Motivation
Consider the integral quantity:
I =
V
P
Gp
(1.3)
where VP is a p-dimensional submanifold of M4
moving with the flow of a spacetime
vector field Z.
Condition (1.2) is an important one in physics since, by the generalised Reynolds’
† Email address for correspondence: s.fletcher@bom.gov.au

2. 2 S. J. Fletcher
transport theorem, it implies conservation of the integral:
dI
dτ
= 0 (1.4)
d/dτ denotes the derivative along the trajectories of Z.
1.3. Construction
For a specified p-form field, Gp , the problem under consideration is to find all 3-form
fields, Z3
, whose associated vector fields satisfy (1.2). We assume that Gp
= 0 and that
all fields are sufficiently differentiable for the problem being considered.
We recall the following expression for the Lie derivative of a differential form:
£Z
Gp
= iZ
d Gp
+ d iZ
Gp
(1.5)
iZ
denotes the interior product of Z with Gp and d the exterior derivative of a differential
form.
It is always possible to find a (p − 1)-form field, Fp−1
, such that:
iZ
Gp = (−1)
p−1
Fp−1 (1.6)
(1.2), (1.5) and (1.6) imply:
iZ
d Gp
= (−1)
p
d Fp−1
(1.7)
Applying identity (A 1) equations (1.6) and (1.7) become:
Z3
∧ d Gp
= d Fp−1
(1.8)
Z3
∧ Gp
= Fp−1
(1.9)
A necessary condition for solution of (1.8) and (1.9) is:
Z3
∧ d Gp
= d Gp
∧ Z3
(1.10)
2. Semi-explicit expressions for Z3
Let Gp
be a specified non-zero p-form field and Nq
an arbitrary q-form field. By applying
the identities given in appendix A we rearrange (1.8) and (1.9) to give:
(i) Tensor equation(s) for Fp−1
which must be satisfied independently of Z3
. When
a particular coordinate system is chosen these will become a (system of) first order
partial differential equation(s) for the components of Fp−1 and thus will generally possess
multiple solutions.
(ii) An expression for Z3
in terms of Gp
, Fp−1
and Nq
.
2.1. p=4
d F3
= 0 (2.1)
( G4 ) Z3 = F3 (2.2)

3. Conserved Integral Quantities 3
2.2. p=3
2.2.1. d G3 = 0
d F2
= 0 (2.3)
( G3
∧ G3
) Z3
= G3
∧ F2
+ (N0
) G3
(2.4)
2.2.2. d G3
= 0
( d G3
) F2
= d F2
∧ G3
(2.5)
( d G3
) Z3
= d F2
(2.6)
2.3. p=2
2.3.1. d G2
= 0 ; G2
∧ G2
= 0
d F1
= 0 (2.7)
(G2
∧ G2
) Z3
= 2 G2
∧ F1
(2.8)
2.3.2. d G2
= 0 ; G2
∧ G2
= 0
G2
∧ F1
= 0 (2.9)
d F1
= 0 (2.10)
( G2
∧ G2
) Z3
= G2
∧ F1
+ N1
∧ G2
(2.11)
2.3.3. d G2
= 0 ; G2
∧ G2
= 0
(G2 ∧ G2 ) d F1 = 2 (G2 ∧ F1 ) ∧ d G2 (2.12)
(G2
∧ G2
) Z3
= 2 G2
∧ F1
(2.13)
2.3.4. d G2
= 0 ; G2
∧ G2
= 0
G2 ∧ F1 = 0 (2.14)
( G2 ∧ G2 ) d F1 = ( G2 ∧ F1 + N1 ∧ G2 ) ∧ d G2 (2.15)
( G2 ∧ G2 ) Z3 = G2 ∧ F1 + N1 ∧ G2 (2.16)

4. 4 S. J. Fletcher
2.4. p=1
2.4.1. d G1
= 0
d F0
= 0 (2.17)
( G1
∧ G1
) Z3
= (F0
) G1
+ N2
∧ G1
(2.18)
2.4.2. d G1 = 0 ; d G1 ∧ d G1 = 0
( d G1 ∧ d G1 ) F0 = 2 G1 ∧ d G1 ∧ d F0 (2.19)
( d G1 ∧ d G1 ) Z3 = 2 d G1 ∧ d F0 (2.20)
2.4.3. d G1
= 0 ; d G1
∧ d G1
= 0
d G1 ∧ d F0 = 0 (2.21)
( d G1 ∧ d G1 ) F0 = G1 ∧ ( d G1 ∧ d F0 + N1 ∧ d G1 ) (2.22)
( d G1 ∧ d G1 ) Z3 = d G1 ∧ d F0 + N1 ∧ d G1 (2.23)
2.5. p = 0
( d G0 ∧ d G0 ) Z3 = N2 ∧ d G0 (2.24)

5. Conserved Integral Quantities 5
Appendix A. Identities
The following identities may be established in an orthonormal basis in spacetime.
Let Z be a vector with associated 3-form Z3
. Let Aq
denote a q-form, W1
, X1
and Y1
1-forms, R2
and S2
2-forms.
−iZ
Aq
= Z3
∧ Aq
(A 1)
X1
∧ Y1
= Y1
∧ X1
(A 2)
[W1
∧ (X1
∧ Y1
)] = ( W1
∧ Y1
) X1
− ( W1
∧ X1
) Y1
(A 3)
( X1
∧ Y1
) R2
= (X1
∧ R2
) ∧ Y1
+ [ (Y1
∧ R2
) ∧ X1
] (A 4)
(X1
∧ R2
) ∧ R2
= (X1
∧ R2
) ∧ R2
(A 5)
2 (R2
∧ S2
) X1
= (X1
∧ R2
) ∧ S2
− (X1
∧ R2
) ∧ S2
+
(X1
∧ S2
) ∧ R2
− (X1
∧ S2
) ∧ R2
(A 6)
(R2
∧ R2
) X1
= 2 (X1
∧ R2
) ∧ R2
(A 7)
(R2
∧ R2
) X1
= (X1
∧ R2
) ∧ R2
− (X1
∧ R2
) ∧ R2
(A 8)