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Given that a worldline, worldsheet, worldvolume, are representation in a 4D-spacetime of a point particle, a string or a brane, respectively, I was wondering if those objects necessarily have to be static in spacetime, or if they are allowed to move.

If they can move, what dimension is being used to measure their changes in position?

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They are allowed to move. In fact, if you think about it for a minute you can see that Lorentz invariance requires that they be allowed to move because given a static brane I can always boost to a reference frame where the brane is moving. In order for this to be an equally good description of the physics we must allow branes to move.

As an aside, the same logic is often used when constructing solitons. We start by determining the solution for a static soliton because it is easier, but we know that it must be able to move in order for the Lorentz invariance to be respected.

In practice we tend to ignore the dynamics of D-branes because they are so massive, but in principle they can move and be accelerated.

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If branes experience acceleration, or move at a given velocity, which dimension is being used to measure the rate of changes in their position. I mean time would change a what rate compared with what. –  rraallvv Dec 25 '12 at 21:38
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