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

[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?

 

[PeroK]

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?

 

[PeroK]

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.

 

[Ibix]

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!

 

[Ibix]

It's literally your choice. Imaginary surfaces abide by rules you personally have set, and which you haven't shared with us.

Perhaps instead of asking strange abstract questions you should talk about what physics you are trying to understand.

 

[victorvmotti]

It looks like I have the answer myself. Given this definition of $w$ dimension it is not actually an extra dimension. Instead it is some scaled time dimension and drops from the five dimensional metric that I was looking for.

 

[Muu9]

Perhaps it would clear the confusion if you instead had asked about a 3d hyperplane parallel to the xyz hyperplane that moves in the w axis