Exploring Curved Space and LP Spaces in L2: The Curvature of Black Holes

[granpa]

In L² space c² = a² + b²
http://en.wikipedia.org/wiki/Pythagorean_theorem
http://en.wikipedia.org/wiki/Lp_space#Motivation
http://en.wikipedia.org/wiki/Triangle_inequality

If we draw a large triangle on a curved surface like the Earth then the Pythagorean theorem won't work. But if we shrink the triangle down the curvature becomes less and less until it approaches being completely flat and the Pythagorean theorem works again.

If we draw a large triangle in L^P space where P<>2 then the Pythagorean theorem also won't work. The difference is that it still won't work even if one shrinks the triangle down to nothing.

My question is what happens in the curved space of a black hole? Is it like the curved surface of the Earth or like the intrinsic curvature (no, that probably isn't right. maybe 'intrinsic distortion from L² space) of LP space? Does a small enough piece of space near a black hole behave like regular flat L² space (yes i know, technically it would be minkowski space but i am only interested in the space component and nonrelativistic speeds)

 

[nickthrop101]

no one knows the answer to that question, mathmatically then yes it should work
it depends if you are past the event horizon ( point of no return) yet
becouse at the center of a black hole, the singularity, then there is said to be infinate mass and therefore infinate curviture, so physically there you could not work like that :)

 

[granpa]

Thanks for the response.

Thats what I assumed but when I was reading the wiki article about LP space I suddenly realized that I didnt really know.

 

[DrGreg]

My question is what happens in the curved space of a black hole? Is it like the curved surface of the Earth or like the intrinsic curvature (no, that probably isn't right. maybe 'intrinsic distortion from L² space) of LP space? Does a small enough piece of space near a black hole behave like regular flat L² space (yes i know, technically it would be minkowski space but i am only interested in the space component and nonrelativistic speeds)

Anywhere in spacetime, whether near a black hole, inside or outside the event horizon, or anywhere else, behaves locally "like the curved surface of the Earth" and not "like L^p space". The equivalence principle means that if you zoom in close enough to any event, the surrounding spacetime looks almost like the flat spacetime of Minkowski space, which is analogous to (but not identical to) L² space.

An L^p space, when p ≠ 2, has a notion only of "distance" and no notion of "angle" unlike spacetime. (Technically it is "normed vector space", but not an "inner product space".)