- #1

physicus

- 55

- 3

## Homework Statement

My question is about a step in the lecture notes [http://arxiv.org/abs/hep-th/0307101] on page 6, and it is probably quite trivial:

I want to see why a lightlike particle in [itex]AdS_5\times S^5[/itex] sees the metric as plane wave background. The metric is

[itex]ds^2=R^2(-dt^2 \cosh^2\rho+d\rho^2+\sinh^2\rho \,d\Omega_3^2[/itex][itex]+d\psi^2\cos^2\theta+d\theta^2+\sin^2\theta\,\Omega_3'^2)[/itex]

In order to study the metric close to a lightlike geodesic we make the follwoing change of coordinates:

[itex]{x}^+=\frac{1}{2\mu}(t+\psi), {x}^-=\frac{\mu R^2}{2}(t-\psi), \rho=\frac{r}{R}, \theta=\frac{y}{R}[/itex]

I am supposed to get in the [itex]R\to\infty[/itex] limit

[itex]ds^2=R^2(-\mu^2(dx^+)^2+\mu^2(dx^+)^2)+(-2dx^+dx^--\mu^2r^2(dx^+)^2+dr^2+r^2d\Omega_3^2[/itex] [itex]-2dx^+dx^--\mu^2y^2(dx^+)^2+dy^2+y^2d\Omega'{}_3^2)+\mathcal{O}(R^{-2})[/itex]

This is not the final result, but from there on I know how to continue.

## Homework Equations

[itex]\cosh x=1+\frac{1}{2}x^2+\mathcal{O}(x^4), \cos x=1-\frac{1}{2}x^2+\mathcal{O}(x^4)[/itex]

[itex]\Rightarrow \cosh^2 x = 1+x^2+\mathcal{O}(x^4), \cosh^2 x = 1-x^2+\mathcal{O}(x^4)[/itex]

## The Attempt at a Solution

I can expand in [itex]\rho, \theta[/itex], since they will be small in the [itex]R \to \infty[/itex] limit:

[itex]ds^2=R^2(-dt^2 \cosh^2\rho+d\rho^2+\sinh^2\rho \,d\Omega_3^2+d\psi^2\cos^2\theta+d\theta^2+\sin^2\theta\,\Omega'{}_3^2)[/itex]

[itex]=R^2(-dt^2(1+\rho^2)+d\rho^2+\rho^2d\Omega_3^2+d\psi^2(1-\theta^2)+d\theta^2+\theta^2d\Omega'{}_3^2)+\mathcal{O}(R^{-2})[/itex]

[itex]=R^2(-dt^2+d\psi^2)+(-dt^2r^2+dr^2+r^2d\Omega_3^2-d\psi^2y^2+dy^2+y^2d\Omega'{}_3^2=+\mathcal{O}(R^{-2})[/itex]

Now I use:

[itex] dx^+dx^-=\frac{1}{2\mu}(dt+d\psi)\frac{\mu R^2}{2}(dt-d\psi)=\frac{R^2}{4}(dt^2-d\psi^2)[/itex]

So the first term above is [itex]R^2(-dt^2+d\psi^2)=-4dx^+dx^-[/itex].

However, I do not know where all the [itex](dx^+)^2[/itex] in the solution are coming from, since

[itex](dx^+)^2=\frac{1}{4\mu^2}(dt^2+2dt\,d\psi+d\psi^2)[/itex]

Where do these mixed [itex]dt\,d\psi[/itex] terms come from?