Relativity and the curvature of the earth.

[insane_alien]

So, i was reading up some article about black holes the other day and a thought struck me.

 

according to relativity, light being bent by gravity is following a straight line on the geodesic(if i'm wrong here then the rest is gibberish) . so, if we havea case where the gravity is strong enough to curve light into a circle, this surface would in effect, be flat. So, black holes are flat from a space-time perspective.

 

so if we then extrapolate this out to the earth, this means the earth is slightly more flat than it would be if it didn't have mass. This effect would of course be tiny for the earth and over-powered by oblateness, terrain and the population density of chiuauas.

 

nothing particularly profound here, just want to know if my random tangent thought was right.

[timo]

I don't get what "if we have a case where the gravity is strong enough to curve light into a circle, this surface would in effect, be flat" means. Considering the light following "a straight line on the geodesic": to make things short, the solutions for paths that light (or any other object on which no force -except gravity- is acting) are called geodesics. Those paths are sometimes referred to as something like the extension of a straight path to a curved space. I'd not focus on this statement, tough.

Regarding your statement "black holes are flat from a space-time perspective". If you consider a black hole being a physical object: I wouldn't know what defines whether an object is flat or not. If you consider a black hole being the spacetime of a mass that is concentrated on a single spot: that is not flat in the sense that the curvature is non-zero.

[insane_alien]

right, so i got some terminology mixed up, i though geodesic was actually the shape of spacetime or soemthing.

 

theres a reason i didn't study relativity.

[md65536]

right, so i got some terminology mixed up, i though geodesic was actually the shape of spacetime or soemthing.

 

theres a reason i didn't study relativity.

I'm (still) not a physicist, so take this with salt:

 

A geodesic is the shortest path between 2 given points in curved space. In Euclidean geometry, a geodesic is a straight line.

With curved spacetime, the shortest distance is not always a straight line. In fact, geodesics will appear to have different curvature, depending on the observer.

I have a feeling that if you were to travel along the path of light as it curved through a strong gravitational field, it would appear to you that you were always following a straight line (though you would see space warping around you as you change between weak and strong gravitational fields), while an observer in a weaker field would see you travel a curved path.

 

Using the rubber sheet analogy, imagine that from your perspective you always see the rubber sheet as flat, even though someone else might describe it being deformed by a large mass pulling it into a "bump". The shortest path would not be straight over the bump, but would be to go around it somewhat, with a curvature that depends on how deep the bump is. With a typical bump, you would never see the shortest path across the bump being a full circle. For a really steep bump, the limit would probably be a semi circle.

 

So, mass does not pull light or the "lines of space" into it the way that gravity pulls matter. It curves space, the way pushing into a rubber sheet might.

 

 

Suppose you want to describe a geodesic from point A to point B, that is nearly a full circle, around a planet or a black hole or something. What you are describing is that the shortest path between A and B goes all the way around the circle. So it must be no longer than the straight line distance between A and B. To do this, you would need to stretch the space that exists between A and B into a circle.

 

Or another example: Suppose you are describing some gravitational phenomenon that allows you to shine a flashlight in your hand towards some planet-sized blackhole-like object, and have the light curve along a geodesic around the object that brings it back to your eye. Then the geodesic from flashlight to eye describes the shortest path between the two. This would only go "around the object" if you could take the spacetime between your flashlight and eye, and pull on it and wrap it around the object. If that were possible to observe, any observer that still saw the object as planet-sized would probably see your arm being at least planet-sized (most likely many many times larger).

 

 

I didn't quite express what I intended to there, at least not very clearly.

 

Another analogy might be if you imagine placing a ruler along a geodesic.

If you travel along the ruler it will appear to be straight.

If the ruler is 1m long and straight according to one observer, yet another observer sees the same geodesic as a planet-sized near-circle, then what they will see is the ruler and surrounding space stretched into a planet-sized circumference.

 

I doubt this describes anything realistic, however I believe that warping of similar scale (or larger or even infinite?) occurs with black holes.

To us, a black hole may seem like a small sphere. To light that cannot escape it, the same distances seem infinite.

 

---

 

Must edit... since I totally went on a tangent from the original post.

 

Yes, something horizontal on earth that we see as pretty flat appears flatter to us than it would be observed by a distant observer in weaker gravity (they would see it being slightly/unnoticeably curved).

However, since the earth is sphere it's pretty much always going to be round. A distant observer will observe length contraction due to the curvature of space, so they would see the earth slightly/unnoticeably smaller than we would observe it. This is pretty much the same as saying it's less flat: a given surface area on a larger sphere is flatter than the same surface area on a smaller sphere.

Edited November 12, 2010 by md65536

[ajb]

so, if we havea case where the gravity is strong enough to curve light into a circle, this surface would in effect, be flat. So, black holes are flat from a space-time perspective.

 

 

I doubt you can in general find a space-time that has null geodesics that are circular without loosing some "sensible" properties like being globally hyperbolic, free of CTCs or without some non-trivial topology (something like "periodic in time"). Though I open to being corrected. (I am thinking 4-d here)

 

For example, the space-time around a black hole (no matter how massive) does not have circular (or even bound) null geodesics (not stable ones for sure). You cannot make light follow a circular orbit.

Edited November 12, 2010 by ajb

[insane_alien]

I doubt you can in general find a space-time that has null geodesics that are circular without loosing some "sensible" properties like being globally hyperbolic, free of CTCs or without some non-trivial topology (something like "periodic in time"). Though I open to being corrected. (I am thinking 4-d here)

 

For example, the space-time around a black hole (no matter how massive) does not have circular (or even bound) null geodesics (not stable ones for sure). You cannot make light follow a circular orbit.

 

http://en.wikipedia.org/wiki/Photon_sphere < this seems to contradict that. and is where my idea came from

[ajb]

For the Schwarzschild metric circular orbits of photons are not stable, but I think they may still be a useful concept. Any small perturbation from the circular orbit will send the photon crashing into the massive object or send it off to infinity.

 

The Kerr metric, which describes a rotating black hole has CTC's and I think circular orbits of photons here are also unstable.

Edited November 12, 2010 by ajb

[insane_alien]

i suppose that makes sense as the photon is always travelling at uniform speed and space-time is probably quantised.

 

ach well, not all my ideas can be brilliant.

[lemur]

Does gravitation determine the distance to the horizon from a given altitude? In other words, if the Earth was 20 times as massive as it is now, would the horizon from an altitude of, say, 20km appear at a different distance because of the gravity difference? Put another way, could light reflecting off a point in Africa make it to an observer somewhere above America that would otherwise be beyond the horizon, just because the Earth's gravity bent the path of the light more? So, in general could you say that higher gravity decreases the altitude of a given vantage point, in that the trajectory of the light to that vantage point would get bent more toward the source of gravity?