Four-Velocity and Schwartzchild Metric

In summary, the Schwartzchild metric is used to calculate the 4-velocity of a stationary observer in a spacetime. By solving for the components of the four-velocity, it is shown that u2 = c2, indicating that the four-velocity is always equal to c2 in the rest frame of a particle.
  • #1
DRose87
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Homework Statement


What is the Schwartzchild metric. Calculus the 4-velocity of a stationary observer in this spacetime (u). Show that u2 = c2.

Homework Equations


Schwartzchild Metric
[tex]d{s^2} = {c^2}\left( {1 - \frac{{2\mu }}{r}} \right)d{t^2} - {\left( {1 - \frac{{2\mu }}{r}} \right)^{ - 1}}d{r^2} - {r^2}d{\theta ^2} - {r^2}{\sin ^2}\theta d{\phi ^2}[/tex]

The Attempt at a Solution


I already turned this in... I just want to make sure I understand this properly. I'm pretty sure that [itex]\mathbf{u}\cdot\mathbf{u}[/itex] is always equal to c^2 in the rest frame of a particle... So I think the point of the problem is to figure out the components of the four-velocity based on this fact. So here is my attempt at a solution... how ever I actually arrived at the solution in going from the bottom to the top.

So here's how I attempted to solve it:
From the Schwartzchild metric:
[tex]g_{00} = c^2\left(1 - \frac{2\mu }{r}\right)[/tex]

The four-velocity u of a stationary observer is given by:
[tex]
\begin{gathered}
\left[ {{u^\mu }} \right] = \left( {{{\left( {1 - \frac{{2\mu }}{r}} \right)}^{ - 1/2}},0,0,0} \right) \hfill \\
{c^2}\left( {1 - \frac{{2\mu }}{r}} \right){\left[ {{{\left( {1 - \frac{{2\mu }}{r}} \right)}^{ - 1/2}}} \right]^2} = {c^2} \hfill \\
\hfill \\
\end{gathered} [/tex]

It follows that:
[tex]{\mathbf{u}} \cdot {\mathbf{u}} = {g_{ab}}{u^a}{u^b} = {g_{00}}{\left( {{u^0}} \right)^2} = {c^2}\left( {1 - \frac{{2\mu }}{r}} \right){\left[ {{{\left( {1 - \frac{{2\mu }}{r}} \right)}^{ - 1/2}}} \right]^2} = {c^2}[/tex]
 
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  • #2
Looks fine to me.
 

1. What is four-velocity and how is it calculated?

Four-velocity is a mathematical concept used to describe the movement of an object in four-dimensional spacetime. It is calculated by taking the derivative of an object's position with respect to time, and then normalizing it to account for the curvature of spacetime.

2. What is the importance of four-velocity in understanding the motion of objects in space?

Four-velocity is important because it allows us to describe the motion of objects in a way that accounts for the effects of gravity and the curvature of spacetime. It also helps us understand how objects move relative to one another and how their velocities change as they move through spacetime.

3. What is the Schwartzchild metric and how does it relate to four-velocity?

The Schwartzchild metric is a mathematical equation that describes the curvature of spacetime around a non-rotating, spherically symmetric object, like a black hole. It is used to calculate the four-velocity of objects moving in this gravitational field, taking into account the effects of gravity on the object's motion.

4. Can the Schwartzchild metric and four-velocity be used to study the motion of objects in other gravitational fields?

Yes, the Schwartzchild metric and four-velocity can be applied to any non-rotating, spherically symmetric gravitational field, not just that of a black hole. This includes objects like planets, stars, and other massive bodies.

5. How does the Schwartzchild metric and four-velocity relate to Einstein's theory of general relativity?

The Schwartzchild metric and four-velocity are key concepts in Einstein's theory of general relativity. They help us understand how gravity affects the motion of objects in space and are used to describe the curvature of spacetime. This theory revolutionized our understanding of gravity and has been confirmed by numerous experiments and observations.

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