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

In summary: That would have been a well understood problem.In summary, the conversation discusses a hypothetical five dimensional flat spacetime with a moving hypersurface. The question asks for the metric that describes the dynamics of the three dimensional hypersurface in the five dimensional spacetime. However, the concept of a moving hypersurface in a mathematical model may not have a physical interpretation.
  • #1
victorvmotti
155
5
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?
 
Physics news on Phys.org
  • #2
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?
 
  • #3
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?
 
  • #4
victorvmotti said:
I didn't get it from anywhere. Imagined and created it.
i thought so.
victorvmotti said:
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.
 
  • #5
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.
 
  • #6
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!
 
  • #7
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.
 
  • #8
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.
 
  • #9
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
 

1. What is a moving 3D hypersurface?

A moving 3D hypersurface is a mathematical concept that describes a surface that is constantly changing or moving in three-dimensional space. It is often used to model dynamic systems or processes.

2. What is the 4th dimension in this context?

In this context, the 4th dimension refers to time. The moving 3D hypersurface is being observed and measured over time, which adds an additional dimension to the analysis.

3. How is the metric of a moving 3D hypersurface along the 4th dimension calculated?

The metric of a moving 3D hypersurface along the 4th dimension is calculated using mathematical equations and formulas that take into account the changing surface and its position over time. It is a complex calculation that requires advanced mathematical techniques.

4. What is the significance of studying the metric of a moving 3D hypersurface along the 4th dimension?

Studying the metric of a moving 3D hypersurface along the 4th dimension can provide valuable insights into dynamic systems and processes. It can help scientists better understand how these systems change and evolve over time, and how different variables may affect their behavior.

5. Can the metric of a moving 3D hypersurface along the 4th dimension be applied to real-world scenarios?

Yes, the concept of the metric of a moving 3D hypersurface along the 4th dimension has many real-world applications. It can be used in fields such as physics, engineering, and computer science to model and analyze various systems and processes.

Similar threads

I Infinitesimal Movement Along 3-d Geodesics
    • Differential Geometry
Replies
1
Views
1K
I If a 4th dimension existed would universe expand anisotropically?
    • Cosmology
Replies
2
Views
925
A Proper Volume on Constant Hypersurface in Alcubierre Metric
    • Special and General Relativity
Replies
7
Views
2K
I Einstein Definition of Simultaneity for Langevin Observers
    • Special and General Relativity
Replies
12
Views
1K
I Geodesics using 2 Variables: Time & Radius from Mass
    • Special and General Relativity
Replies
1
Views
876
B Moving along one axis in a four-dimensional plane
    • General Math
Replies
13
Views
1K
A Differential movement along a curved surface
    • Differential Geometry
Replies
3
Views
2K
I Tangent space basis vectors under a coordinate change
    • Differential Geometry
Replies
12
Views
3K
A Two Questions about Novikov Coordinates
    • Special and General Relativity
Replies
8
Views
1K
I Trying To Conceptualize Two Dimensions Of Time
    • Beyond the Standard Models
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top