Gravitational Field Representation: Why 1-Plane?

[roachth1]

How did you find PF?: Surfing web

Can someone advise on this? In most diagrams showing how mass effects the gravitation field (earth for instance), bending fabric of space, it is demonstrated on one plane. Why is it shown this way and is there any other way of illustrating this?

 

[Dale]

Our member @A.T. has some much better drawings.

 

[Ibix]

It's broadly evocative drawn that way.

Drawn honestly, what they are drawing is called Flamm's Paraboloid. Loosely speaking, they've taken a slice of spacetime which corresponds to what we call "space at an instant in time", and embedded it in a Euclidean space. It's all perfectly fine maths, but it's not actually particularly helpful for understanding the physics (see xkcd).

There are a number of better visualisations. As @Dale notes, @A.T. produced this video:


Proper analytical tools include the Kruskal diagram and Penrose diagram, but they need a lot of mathematical knowledge to interpret.

I should point out that no diagram of spacetime is completely satisfactory for much the same reason that no flat map of the world is ever completely right. But it's even worse than a world atlas in general relativity, since you can't even accurately represent flat spacetime on a piece of paper. Even flat spacetime doesn't obey Euclidean geometry. But such diagrams can often get across important points, as A.T.'s does.

 

[A.T.]

Why is it shown this way and is there any other way of illustrating this?

Because humans cannot visualize flat 4D, yet alone curved 4D. So you reduce the dimensions to two spatial ones, or one spatial and one temporal one, like in the video above.

 

[DaveE]

I was literally just about to ask this question when I saw this post at the top of my feed.

The thing I have never understood about any of these layman's explanations of GR is that they never really address why an object "wants" to move in a straight line after the space is curved. I completely understand (I think) the drawing of curved space, but how does the object moving not also follow the curved space time basis? When you curve spacetime don't you also curve the definition of "straight" with it? It's seems like the explanations start with a flat R² space (or R³ space time), curve it, but then describe the particle as moving straight through some other uncurved R² space.

Can anyone point me to a better explanation that doesn't either require me to know Tensors and buy a copy of MTW, or require me to watch another video of a ball rolling DUE TO GRAVITY (in an extra dimension) on a curved sheet to explain gravity (worst demo ever IMO, it makes you think you understand something without actually explaining anything)?

 

[DaveE]

This is a nice series of explanations. I'm still confused about the curved space drawings, but maybe I should be.

 

[PeterDonis]

they never really address why an object "wants" to move in a straight line after the space is curved

The straight line is not a straight line in space, it's a straight line in spacetime. The curved lines you are thinking about are projections of the straight line in spacetime into space. Those projections will in general look curved even though the line in spacetime is straight.

 

[Ibix]

they never really address why an object "wants" to move in a straight line after the space is curved.

Why should it "want" to move in a straight line in flat space, for that matter? There isn't really an answer to your question as far as I know, although you can state it in different ways. It's just that when we model test particles following geodesics we get accurate predictions.

I wouldn't get too hung up on the "apple goes in a straight line" thing. It follows a geodesic, which is a generalisation of a straight line. It's certainly represented as a straight line in the video, but I'd be wary of taking that too literally.

Can anyone point me to a better explanation

I think A.T.'s video is about as good as it gets. I vaguely recall him saying it's derived from a book, although I don't recall the author or title - perhaps he can provide a reference.

 

[Nugatory]

but how does the object moving not also follow the curved space time basis?

It does, and that’s what causes gravitational tidal effects. Suppose you and I stand ten meters apart at the equator and start walking north at the exact same speed. We are very careful to move each foot on each step exactly the same distance as the other foot on the previous step, so there is absolutely no sideways drift... but if we walk far enough we will become aware of a force pushing us towards one another so that we collide at the North Pole.

 

[A.T.]

I think A.T.'s video is about as good as it gets. I vaguely recall him saying it's derived from a book, although I don't recall the author or title - perhaps he can provide a reference.

The references, that the video is based on, are listed in the video description on youtube.

http://demoweb.physics.ucla.edu/content/10-curved-spacetime
https://books.google.de/books?id=lwZBAQAAIAAJ&hl=de
http://www.relativitet.se/Webtheses/tes.pdf

 

[vanhees71]

I have the feeling that these pictures of a deformed membrane are very misleading attempts to intuitively describe the reinterpretation of the gravitational interaction in GR in terms of a pseudo-Riemannian 4d space-time manifold. It suggests as if there were some flat space the curved spacetime is embedded in. This is mathematically possible, but it doesn't help further to understand GR. Again, a true understanding can only be reached by studying the adequate mathematics, which is the only language suitable to express the real meaning of the theory.

 

[A.T.]

Again, a true understanding can only be reached by studying the adequate mathematics, which is the only language suitable to express the real meaning of the theory.

You can feed the math into a computer that will give you the results. Does the computer thus have "true understanding" of physics?

You need the math to make predictions and to express the theory in the most precise minimal way possible, which is great. But understanding the concepts is possible without a lot of math.

 

[vanhees71]

How can you understand something properly without having the adequate vocabulary to express it?

 

[roachth1]

Thank you all for the dialogue and the illustrations. Appreciate you all helping a novice. I will sift through and try to absorb!

 

[supernova1054]

Why should it "want" to move in a straight line in flat space, for that matter? There isn't really an answer to your question as far as I know, although you can state it in different ways. It's just that when we model test particles following geodesics we get accurate predictions.

I wouldn't get too hung up on the "apple goes in a straight line" thing. It follows a geodesic, which is a generalisation of a straight line. It's certainly represented as a straight line in the video, but I'd be wary of taking that too literally.

I think A.T.'s video is about as good as it gets. I vaguely recall him saying it's derived from a book, although I don't recall the author or title - perhaps he can provide a reference.

Why do objects in classical physics move in a straight line with constant velocity in no forces are acting on the object? The only answer, which begs the why question , is that a straight line is the shortest distance between two points. In curved spaces, the shortest distance between two points is a curve called a geodesic. Foe example, on the Earth, great circle arcs are the shortest distance between two surface points. Flight paths of airliners follow great circles as much as possible on long flights. Great circles are circumferences of a sphere.
If the Earth were ellipsoidal, geodesics wold be arcs of ellipses described by elliptic integrals in general.
Why follow the least distance? That is the way we have found that nature behaves. Why? You would have to ask a “higher power” than us mere humans.

 

[PeroK]

In curved spaces, the shortest distance between two points is a curve called a geodesic.

In SR & GR a geodesic is actually the longest distance between two points, but it's the same principle at work.

 

[supernova1054]

Look it up. A geodesic is the SHORTEST distance between two points.

 

[Dale]

Look it up. A geodesic is the SHORTEST distance between two points.

Actually, it is an extremum. It could be a minimum, a maximum, or a saddle point. For timelike curves in flat spacetime, @PeroK is correct, it is indeed a maximum

 

[PeroK]

Look it up. A geodesic is the SHORTEST distance between two points.

For timelike curves in SR and GR, the geodesic is the curve of greatest proper time (spacetime distance). In these geometries the geodesics are local maxima (rather than local minima).

 

[supernova1054]

Actually, it is an extremum. It could be a minimum, a maximum, or a saddle point. For timelike curves in flat spacetime, @PeroK is correct, it is indeed a maximum

You are correct and I stand corrected.