Vrftsjtryk

In the Schwarzschild metric, the geodesic Lagrangian is

$$L=frac{1}{2} left[ left( 1-frac{2m}{r} right) dot{t}^2-frac{dot{r}^2}{1-2m/r}-r^2(dot{theta}^2+sin^2thetadot{psi}^2) right]$$

Therefore the $t$ equation for the geodesic motion of a free particle is

$$frac{d}{dtau}left(frac{partial L}{partial dot{t}}right)=0$$

because the Lagrangian is independent of t. Consequently

$$E=(1-2m/r)dot{t}$$

is constant along the particle worldline.... Suppose that the particle has four-velocity $V$ and unit mass. Then relative to an observer at rest at some point in the particle's history, the particle has speed $v$ given by

$$gamma(v) = frac{1}{sqrt{1-v^2}}=U^aV_a=g_{00}U^0V^0=dot{t}sqrt{1-2m/r}$$

1) How do we get $t$ equation so obviously?

2) How do we get the equation for $E$? There was no justification in my book for this step

3) What is the interpretation of the quantity $U^aV_a$? Is it like a dot product?

4) How do we know that $U^aV_a$ equals $g_{00}U^0V^0$? Where have all the other components of the metric gone?

The world lines that extremize the proper time between A and B are those which satisfy the Lagrange equation:
begin{equation}
-frac{d}{d sigma} left(frac{partial L}{partial({dx^{alpha} / d sigma})} right)+frac{partial L}{partial x^{alpha}}=0
end{equation}
Where the Lagrangian is:
begin{equation}
L= left(-g_{alpha beta} frac{d x^{alpha}}{d sigma} frac{d x^{beta}}{d sigma} right)^{frac{1}{2}}
end{equation}
The Schwarzschild metric in geometrical units is
begin{equation}
left(
begin{array}{cccc}
-(1-frac{2 m}{r}) & 0 & 0 & 0 \
0 & (1-frac{2 m}{r})^{-1} & 0 & 0 \
0 & 0 & r^2 & 0 \
0 & 0 & 0 & r^2 sin ^2(theta ) \
end{array}
right)
end{equation}
The parameter t is therefore called world time. If m = 0 the Schwarzschild metric reduces to the Minkowski metric; if $m neq 0$ it is singular for $r = 0$. For r large with respect to m the Schwarzschild metric coincides with the Newtonian approximation, since $2m/r$ is the Newtonian potential outside a spherical body of mass m centered at $r = 0$. In addition, when
$1−(2m/r)$ vanishes for $r = 2m$, the metric appears to be singular there. This singularity can be removed by choosing a different coordinate system.
The geodesic equations based on the Schwarzschild metric above are
begin{equation}
left(
begin{array}{ccc}
frac{d u_0}{d tau} = frac{2 m u_0 u_1}{2 m r-r^2} \
frac{d u_1}{d tau} = frac{m (2 m-r) u_0^2}{r^3}-frac{m u_1^2}{2 m r-r^2}-(2 m-r) u_2^2-(2 m-r) sin ^2(theta) u_3^2 \
frac{d u_2}{d tau} = cos (theta) sin (theta ) u_3^2-frac{2 u_1 u_2}{r} \
frac{d u_3}{d tau} = -frac{2 (u_1+r cot (theta ) u_2) u_3}{r} \
end{array}
right)
end{equation}
where $u_{alpha}$ are the component of the four proper velocity vector while $tau$ is the proper time. When looking at the metric you can see that is independent of $t$, thus there is a Killing vector associated with this symmetry under displacement of the time coordinate t with components $psi=(1,0,0,0)$. The first equation can be written as:
begin{equation}
frac{d}{d tau} left((1-frac{2 m}{r}) u_0(tau) right) = 0
end{equation}
which is integrated to give
begin{equation}
left((1-frac{2 m}{r}) u_0(tau) right) =- g_{00} frac{dt}{d tau}= E
end{equation}
with E constant.
Now you have to remember that the dot product is obtained by computing the first fundamental form.The first fundamental form is the restriction of the usual dot product in $mathbb{R}^3$ to the tangent plane $T_p S$. Given two vector a and b
the first fundamental form $Ip (a, b)$ at the point $p$ can be expressed as the bilinear form $a^T g b$, where g is the matrix. In the case of the xy-plane or a cylinder $g$ is a diagonal matrix with diag(g) = (1,1). In this case the dot product of $a$ and $b$ is
begin{equation}
(a_1,a_2) left(
begin{array}{cc}
1 & 0\
0 & 1
end{array}
right) (b_1,b_2)^T= a_1 b_1+ a_2 b_2
end{equation}
However is the matric is not flat we have to account for the terms $g_{alpha, beta}$. So yes, $U^aV_a$ is a dot product, with the index a running from 1 to the 4, since you are in a 4 dimensional space. This dot product can be written as
begin{equation}
g_{00} u_0 v_0-g_{00}^{-1} u_1 v_1+r^2 u_2 v_2+ r^2 sin^2(theta)u_3 v_3
end{equation}
However, The 4-velocity of a stationary observer is $(u_0,0,0,0)$, so for a stationary observer we have only the term $g_{00} u_0 v_0$.
Consider now the lapse of proper time τ between two events at a fixed spatial point (differentiation with respect the position is zero) in Schwarzschild space-time:
begin{equation}
ds^2=-d tau^2=-g_{00} dt^2
end{equation}
hence
begin{equation}
d tau=sqrt{g_{00}} dt
end{equation}
The element of proper time dτ is measured by a clock at the particular point, while the element of world time dt is fixed for the whole manifold. Since $g_{00}<1$, so $d tau < dt$ i.e. clocks go slower in a gravitational field. In fact time itself goes slower in a gravitational field. In flat spacetime $m$ turn down to zero so that $dt=d tau$, and a clock records the coordinate time $t$.