Hypersurface Definition Confusion in General Relativity

In summary, the conversation discusses the definition of hypersurfaces and how they are related to functions with constant values. It is clarified that the requirement for the partial differential to be non-zero everywhere is not necessary, as it only needs to be non-zero at the hypersurface being described. An example of this is given with a sphere in standard Euclidean space.
  • #1
tm33333
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In my notes on general relativity, hypersurfaces are defined as in the image. What confuses me is that if f=constant, surely the partial differential is going to be zero? I'm not sure if I'm missing something, but surely the function can't be equal to a constant and its partial differential be non-zero?

thanks.
 

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  • #2
It's requiring that ##\partial_af## be non-zero everywhere, then saying the subset of points with the same value of ##f## define a hypersurface. Analogously, you can define a function ##f(x,y)## on a two dimensional Euclidean plane and the lines of constant ##f## are the contour lines (1d analogues to 3d hypersurfaces). The gradient on a contour isn't zero, it is perpendicular to the contour line.

(Note that geographical contour lines can close, but a closed contour line encloses at least one point where the gradient is zero, so the definition of a hypersurface excludes this possibility).
 
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  • #3
Well, the requirement that ##\partial_a f=0## everywhere is a bit strict. It is sufficient that it is non-zero at the hypersurface being described by the particular constant. (Although you will need the full requirement if you intend to make a foliation of the manifold.)

As an example, consider the sphere in standard Euclidean space with ##f = x^2 + y^2 + z^2##. For ##R>0##, ##f = R^2## defines a sphere of radius ##R##, which is a level surface of ##f## in ##\mathbb R^3##.
 
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  • #4
Thank you both. That definitely clarifies things!
 
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  • #5
Moderator's note: Thread title edited to be more descriptive of the specific question.
 
  • #6
tm33333 said:
Thank you both. That definitely clarifies things!
I guess you can say that they broke it down for you. :wink:
 

Related to Hypersurface Definition Confusion in General Relativity

1. What is a hypersurface in general relativity?

A hypersurface in general relativity is a mathematical concept that refers to a three-dimensional surface in a four-dimensional spacetime. It is used to represent a boundary or dividing line between different regions of spacetime.

2. How is a hypersurface defined in general relativity?

In general relativity, a hypersurface is defined as a set of points in four-dimensional spacetime that satisfy a specific equation or condition. This equation is known as the "hypersurface equation" and is used to determine the location and properties of the hypersurface.

3. What is the significance of hypersurfaces in general relativity?

Hypersurfaces play a crucial role in general relativity as they are used to define boundaries between different regions of spacetime, such as between a black hole and its surrounding space. They also help in understanding the dynamics of spacetime and the effects of gravity.

4. How does hypersurface definition confusion arise in general relativity?

Hypersurface definition confusion can arise in general relativity when there are multiple possible ways to define a hypersurface. This can lead to different interpretations and predictions, causing confusion and debate among scientists.

5. How do scientists address hypersurface definition confusion in general relativity?

Scientists address hypersurface definition confusion in general relativity by carefully examining the mathematical equations and assumptions used to define a hypersurface. They also conduct experiments and simulations to test different definitions and determine which one best fits the observed data.

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