# Metric of a Moving 3D Hypersurface along the 4th Dimension

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victorvmotti
Consider a hypothetical five dimensional flat spacetime ##\mathbb{R}^5## with coordinates ##x, y, z, w, t##.

Now imagine that the hypersurface ##\Sigma =\mathbb{R}^3## of ##x, y, z## moves with constant rate ##r## along the coordinate ##w##, i.e. ##dw/dt=r##. Assuming that ##t \in (-\infty, + \infty)## what is the metric that describes such an evolution or dynamics of the three dimensional hypersurface in the five dimensional spacetime?

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This doesn't make any sense to me. It's like asking what happens if the x-y plane moves long the z-axis over time.

Where did you get this question?

victorvmotti
Yes, that's the idea in the x-y case. How can we write a metric for it?

I didn't get it from anywhere. Imagined and created it.

So you say that this makes absolutely no sense even in mathematics let alone physics?

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I didn't get it from anywhere. Imagined and created it.
i thought so.
So you say that this makes absolutely no sense even in mathematics let alone physics?
It would make sense if you specified it as a coordinate transformation.

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The point is that mathematical surfaces don't move because they don't exist outside mathematical models. You can define a mathematical surface and have a definition that varies with time (or whatever), but the dynamics of that are more or less whatever you want.

If you actually want to know about the dynamics of a physical sheet then you need to specify physical laws and appropriate quantities like mass density etc. Or if you want to know about some foliation of the spacetime itself (c.f. ADM formalism) you need to specify how you are doing the foliation.

victorvmotti
My question is given that mathematical surface or model defined or imagined how can we write a metric that describes such a dynamic? Have no clue at all!