Gravity and Strings

518 Extended objects
The equations of motion of the spacetime fields are
Gµν + 2T ϕ µν
(−1)p+1
+
2 · (p + 1)! e−2aϕT A(p+1)
µν
N T(p)
√ d
|g|
p+1 ξ |γ | e −2bϕ γ ij ∂i X µ ∂ j X ν gρµgσν δ (d) (x − X (ξ)) = 0,
− 8πG(d)
∇2ϕ + (−1)p+1a 4 · (p + 2)! e−2aϕ F 2
(p+2)
− 4πG(d)
N T(p)b
√ d
|g|
p+1 ξ |γ | e −2bϕ γ ij ∂i X µ ∂ j X ν gµν δ (d) (x − X (ξ)) = 0,
−2aϕ
∇µ e F(p+2) µν1···νp+1
− 16πG(d) N µp
√ d
|g|
p+1 ξɛ i1···i p+1∂i1 X ν1 ···∂i p+1 X νp+1 (d)
δ (x − X (ξ)) = 0,
(18.72)
and those of the worldvolume fields are
∇ 2 (γ )X µ + γ ij ∂i X ρ ∂ j X σ Ɣρσ µ − 2b∂(ρϕgσ) µ
γij − e −2bϕ gµν∂i X µ ∂ j X ν = 0,
+ (−1)p+1 µp/T(p)
(p + 1)! √ |γ | e2bϕ F(p+2) µ µ1···µp+1 ∂i1 X µ1 ···∂i p+1 X µp+1 ɛ i1···i p+1 = 0.
(18.73)
The first of the worldvolume equations can be used immediately in all the other equations
to eliminate the worldvolume metric. Furthermore, using the static gauge for the first
(p + 1) p-brane embedding coordinates that we denote by Y i ,
Y i (ξ) = ξ i , (18.74)
and the following Ansatz for the transverse embedding coordinates;
X m (ξ) = 0, (18.75)
which corresponds to a p-brane at rest at x m = 0, it is possible to perform the worldvolume
integrals in the equations of motion of the spacetime fields and only ( ˜p + 3)-dimensional
Dirac δ functions remain as sources.
Our Ansatz for the spacetime fields is given by the extreme (ω = 0) p-brane solutions
Eqs. (18.65), H being now a function of the transverse coordinates x m to be determined.
In the absence of sources (or outside of them) H can be any harmonic function of those
˜p + 3 transverse coordinates, satisfying
∂m∂m H(x( ˜p+3)) = 0. (18.76)
In general, H, and therefore the solution, has singularities that can be understood as originated
by sources that are not included explicitly in the action. When we include source
terms in the action, the singularities of H have to match them. In this case, the sources are