Metric of Gravitational Field near Earth's surface I am trying to do a calculation in which I am trying to work out how a scalar field behaves in the earth's gravitational field near the surface.
I know that the Schwarzschild metric would describe the field, however I am only considering a local scalar fiel near the earth's surface.
What metric should one use?
 A: To derive an appropriate metric for use near the surface of the earth, start with the Schwarzschild metric
$$
    d\tau^2=A(r)dt^2
        - \frac{1}{A(r)}dr^2 - (d{\mathbf{x}}^2-dr^2)
$$
with
$$
    A(r) = 1-\frac{\kappa}{r}
\hskip2cm
    \kappa = \frac{2GM}{c^2}
$$
and
$$
    \mathbf{x}=(x,y,z)
    \hskip1cm
    r = \sqrt{x^2+y^2+z^2}
    \hskip1cm
   d{\mathbf{x}}^2 = dx^2+dy^2+dz^2.
$$
(I wrote the metric
 here using $x,y,z$ coordinates instead of the more
traditional spherical coordinates.
The way I wrote it, the last parenthesized term 
corresponds to the "angular part.")
Suppose that we want to approximate this metric in the vicinity of the point $\mathbf{x}_0$. Write $\mathbf{x}=\mathbf{x}_0+\mathbf{X}$ and expand to lowest order in $\mathbf{X}$. Clearly $d\mathbf{x}=d\mathbf{X}$. To handle $r$, use
$$
  r \approx r_0 + \mathbf{u}_0\cdot\mathbf{X}
\hskip2cm
  \mathbf{u}_0=\frac{\mathbf{x}_0}{r_0}
\hskip2cm
  r_0^2=\mathbf{x}_0\cdot\mathbf{x}_0.
$$
Then
$$
  dr\approx\mathbf{u}_0\cdot d\mathbf{X}
\hskip2cm
  A(r)\approx 1-\frac{\kappa}{r_0 + \mathbf{u}_0\cdot\mathbf{X}}.
$$
Finish by using this last expression to expand $A(r)$ and $1/A(r)$ to first order in $\mathbf{X}$. Substituting these approximations back into the Schwarzschild metric gives
$$
  d\tau^2\approx (\alpha+\beta\mathbf{u}_0\cdot\mathbf{X})dt^2
  - (\mu+\nu\mathbf{u}_0\cdot\mathbf{X})(\mathbf{u}_0\cdot d\mathbf{X})^2
  - d\mathbf{X}^2
$$
with $\mathbf{X}$-independent coefficients $\alpha,\beta,\mu,\nu$  determined explicitly by the preceding steps. We can simplify this by choosing, say, $\mathbf{x}_0=(R,0,0)$, so that $r_0=R$ and $\mathbf{u}_0=(1,0,0)$.
You can also look up the Rindler metric. That might be easier, because there are probably lots of references that have already worked out the behavior of a (quantum) scalar field in Rindler spacetime. You can probably find some leads by looking up the Unruh effect, which is a flat space-time analogue of the (eternal black hole version of the) Hawking effect, using the Rindler metric in place of the Schwarzschild metric.
