Gauss-Bonnet term extrinsic curvature

[sourena]

Gauss-Bonnet term extrinsic curvature calculations?

In General Relativity if one wants to calculate the field equation with surface term, must use this equation:
S=$\frac{1}{16\pi G}$$\int\sqrt{-g} R d^{4} x$+$\frac{1}{8\pi G}$$\int\sqrt{-h} K d^{3} x$
The second term is so-called Gibbons-Hawking boundary term and K is extrinsic curvature.
If one is about to use another Lagrangian, for instance Gauss-Bonnet term, must calculate the new extrinsic curvature, K, associated with this new Lagrangian.
I want to know is there a standard method for calculating K?
I would be grateful if anybody can help me in learning this procedure. Please introduce references if you know some.

 

[Bill_K]

According to Wikipedia, what Gibbons-Hawking-York call "extrinsic curvature K" is the trace of the second fundmental form. Nonstandard terminology! But if this is correct, I can answer your question.

To describe a surface embedded in a larger manifold you need two sets of coordinates, hence two types of indices. Let xⁱ be the coordinates in the outer manifold and u^α coordinates in the submanifold, the surface. A basic quantity that transforms between them is

xⁱ_α = ∂xⁱ/∂u^α

This is a hybrid quantity, sometimes called a bitensor, with both kinds of indices present. The covariant derivative of such an object requires both kinds of Christoffel symbols,

xⁱ_α;β = ∂²xⁱ/∂u^α∂u^β + _(outer)Γⁱ_jkx^jx^k - _(inner)Γ^δ_αβxⁱ_δ

It's easy to convince oneself that xⁱ_α;β represents a set of vectors orthonormal to the surface. Hence there must exist a surface tensor b_αβ such that

xⁱ_α;β = b_αβnⁱ

where nⁱ is the normal vector. b_αβ is called the second fundamental form.

The curvature you are after is the trace of b_αβ.

 

[Matterwave]

The second fundamental form was defined to me as the vector-valued one-form (on the tangent bundle to the hypersurface) $\nabla n$ where n is the normal vector field to the hypersurface in question, and the nabla operator is the covariant derivative operator on your manifold (restricted to directions in the hypersurface obviously). In this way, the second fundamental form (which is also sometimes called the extrinsic curvature!) measures the amount by which the normal field varies from point to point on the hypersurface.

Alternatively, the second fundamental form can be found, after some calculation, to be:
$$K=\frac{1}{2} L_n g'$$

Where L is the Lie derivative, and g' is the induced 3-metric on the hypersurface.

This equation is the one found in Wald.

 

[Bill_K]

Sorry sourena, I think I misunderstood your question! You're not asking, "How do I calculate the extrinsic curvature", you're asking, "What if the Lagrangian is something else besides R, then what will K be?".

I don't have an immediate answer, but here are some thoughts. For a general Lagrangian L which is a function of fields φ and their derivatives φ_μ,

δL = (∂L/∂φ)δφ + (∂L/φ_μ)δφ_μ

When we vary the action W = ∫L d⁴x,

δW = ∫δL d⁴x = ∫((∂L/∂φ)δφ + (∂L/φ_μ)δφ_μ)d⁴x

we need to integrate the second term by parts:

δW = ∫((∂L/∂φ - ∂_μ(∂L/φ_μ) )δφ d⁴x + ∫∂_μ((∂L/φ_μ)δφ)d⁴x

and write the total divergence as a surface term:

∫((∂L/φ_μ)δφ) d∑_μ

What you want then is to find a K that produces this, i.e. such that

δK/δφ = (∂L/φ_μ)δφ n^μ

where n^μ is the normal to the surface.

Is this getting warm, do you think?

(For relativity, L = R and φ = g_μν)

 

[sourena]

Dear Bill_K and Matterwave

First of all, thank you so much for your time and attention. I value it a great deal.

Yes Bill_k, this is what I want to do. To be more precise I'm going to explain what exactly I want:

Consider the action

S_M=(1/2k²)$\int d^{D}x\sqrt{-g}${R-2$\Lambda$+$\alpha$ L_GB}

where L_GB=R²-4 R_abR^ab+R_abcdR^abcd

Varying the action with respect to metric gives

$\delta$S_M=(1/2k²)$\int d^{D}x\sqrt{-g}$$\delta$g^ab(G_ab+$\Lambda$g_ab+2$\alpha$H_ab)-(1/k²)$\int d^{D-1}x\sqrt{-h}$n_a(g^a[cg^d]b+2$\alpha$P^abcd)$\nabla$_d$\delta$g_bc,

where h_ab=g_ab-n_an_b is induced metric,

P_abcd=R_abcd+2R_b[cg_d]a-2R_a[cg_d]b+Rg_a[cg_d]b,

and H_ab=RR_ab-2R_acR^c_b-2R^cdR_abcd+R_a^cdeR_bcde-(1/4)(R²-4R_cdR^cd+R^cdesR_cdes) the Lovelock tensor.

From the second term in $\delta$S_M one is able to calculate the following action

S_$\Sigma$=$\int d^{D-1}x\sqrt{-h}$(K+2$\alpha${J-2$\hat{G}$^abK_ab})

Please look at the following paper, page 2

http://arxiv.org/abs/hep-th/0208205

I want to learn the procedure by which I can calculate the extrinsic curvature, J and so forth from the boundary term.

I hope I could explain what exactly I want.