In almost general case, the space-time metrics looks like:(adsbygoogle = window.adsbygoogle || []).push({});

\begin{equation}

ds^2 = g_{00}(dx^0)^2 + 2g_{0i}dx^0dx^i + g_{ik}dx^idx^k,

\end{equation}

where ##i,k = 1 \ldots 3## - are spatial indeces.

The spatial distance between points (as determined, for example, by the stationary observer):

\begin{equation}

dl^2 = \left( -g_{ik} + \frac{g_{0i}g_{0j}}{g_{00}}\right)dx^idx^j = \gamma_{ik}dx^idx^j,

\end{equation}

where

\begin{equation}

\gamma_{ik} = -g_{ik} + \frac{g_{0i}g_{0j}}{g_{00}} ,

\end{equation}

And we can rewrite merics in a form:

\begin{equation}

ds^2 = g_{00}(dx^0 - g_idx^i)^2 - dl^2,

\end{equation}

in the last expression ##g_i## is:

\begin{equation}

g_i = -\frac{g_{0i}}{g_{00}}.

\end{equation}

The proper time in such metrics (when ##dx^i = 0##):

\begin{equation}

d\tau = \sqrt{g_{00}}dx^0.

\end{equation}

How do the absolute value of velocity of a certain particle are determined by stationary observer?

I thaught that is to be ##v = \frac{proper distance}{proper time}##

\begin{equation}

v = \frac{dl}{\sqrt{g_{00}}dx^0},

\end{equation}

but it it wrong answer.

The correct answer should to be

\begin{equation}

v = \frac{dl}{\sqrt{g_{00}}(dx^0 - g_idx^i)},

\end{equation}

and I don't undertand why.

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# I Velocity measurement by a stationary observer in GR

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