Object interactions in relation to space curvature....

[Mr_Phil_Osophy]

I'm a complete rookie in this field so please correct me where I go wrong, I just really want a better understanding of this subject.

So as far as I am aware, mass causes the space surrounding it to curve or bend.
What I want to know is how much does it bend the space? is the bending of space proportional to the curvature of the mass that bends it?

I'm trying to picture in my head what would happen under particular circumstances, and the following example comes to mind:

so imagine you have an object X, and it is constructed under the following conditions:

1. it is perfectly straight - regardless of what scale you look at it, be that from a far, or up close at the atomic scale (as straight as something can be on the atomic scale)
2. it is hundreds of thousands of miles long, and a mile wide. and a mile thick. (like a large support beam)
3. It was constructed out in the depths of space away from any other mass and so away from any curved space.
4. it is made from a very strong material that is in no way flexible and does not change state under the conditions it finds itself in this example (be that heat orgravity etc)

Now imagine that this object was moved towards a large planet. How would the curvature of space effect this perfectly straight and absolutely unchangeable object? if you placed it half way on a rouge planet floating in the depths of space, how would object X be perceived both from the perspective of standing on the object, and from a far?

Would object X appear to remain flat to someone who would walk along it, or would there be some experience of curvature?
Would it appear to be curved from someone looking at it from afar?
I am guessing the curvature would depend on the distance of object x to the planet?

Image A:

I'm going to guess that due to space curvature that it wouldn't look like the above image.

How drastic would the curvature be to object X and how does the size of the objects involved effect this?

Obviously with a ruler and a football it would just look like image A attached above. But how would it look if you have a planet the size of Jupiter for example?

Image B:

Say the planet in image B is the size of Jupiter which is floating in space away from any other massive objects, how would object X be affected as it approached the planet from point A?

I hope this is making sense. Sorry if I'm talking like an absolute noob.

 

[PeterDonis]

So as far as I am aware, mass causes the space surrounding it to curve or bend.

More precisely, it causes spacetime to curve. For an object like the Earth, all of the ordinary effects of gravity that you are used to are (heuristically) due to "time curvature" (which is not really a good term), not space curvature; the curvature of space due to the Earth's mass (or even the mass of Jupiter or the Sun) is tiny.

The best way we have of measuring both of these curvatures (space and time) is actually by looking at the paths followed by light rays (or radar pulses, since many of these experiments are done with radio waves instead of visible light). Observing the bending of light rays from stars that passed close to the Sun during an eclipse was one of the first classic tests of General Relativity; heuristically, half of this bending effect is due to space curvature by the Sun (the other half is due to "time curvature"). We have also done experiments that send radar pulses to spacecraft in various carefully selected orbits in the solar system and measure the return times and paths, in order to look at how the paths bend and how much the pulses are time delayed by the gravity of the Sun and planets.

Some possibly useful references:

http://web.mit.edu/6.055/old/S2009/notes/bending-of-light.pdf

https://en.wikipedia.org/wiki/Shapiro_time_delay

 

[PeroK]

In addition to what @PeterDonis has said, this is probably not the best way to think about curved spacetime. For example, the spacetime around a star or planet is spherically symmetric. In particular, at a given time, the space is spherically symmetric, so there is nothing particularly weird going on.

There's a bit of a myth that space is curved in a specific way around a large object. But, a light ray will follow one path (a slight deflection as it passes the Sun, say) and an object like the Earth will move in a nearly circular orbit. Comets will move in large elliptical orbits and other objects may crash into the Sun.

So, there is not just one "shape" like a rollercoaster or plastic sheet that everything must follow: it's more subtle than that and crucially involves the curvature of the time dimension as well as the spatial dimensions.

 

[A.T.]

1. it is made from a very strong material that is in no way flexible and does not change state under the conditions it finds itself in this example (be that heat orgravity etc)

Such materials contradict Relativity, so Relativity cannot model them.

If you want to have a ruler that is locally straight (a spatial geodesic) then it will be bend globally (the ends will not be parallel) near a mass.

See picture 2 here:
http://demoweb.physics.ucla.edu/content/10-curved-spacetime

 

[Ibix]

it is made from a very strong material that is in no way flexible and does not change state under the conditions it finds itself in this example (be that heat orgravity etc)

I very much doubt it is possible to describe such a thing coherently. "Not flexible" alone implies an infinite speed of sound, which isn't possible. So I don't think this is a useful idea for exploring gravity.

Edit: I see A.T. beat me to it.

 

[pervect]

I'm a complete rookie in this field so please correct me where I go wrong, I just really want a better understanding of this subject.

So as far as I am aware, mass causes the space surrounding it to curve or bend.
What I want to know is how much does it bend the space? is the bending of space proportional to the curvature of the mass that bends it?

As others have remarked, you need to substitute space-time for space here.

I'm trying to picture in my head what would happen under particular circumstances, and the following example comes to mind:

so imagine you have an object X, and it is constructed under the following conditions:

1. it is perfectly straight - regardless of what scale you look at it, be that from a far, or up close at the atomic scale (as straight as something can be on the atomic scale)
2. it is hundreds of thousands of miles long, and a mile wide. and a mile thick. (like a large support beam)
3. It was constructed out in the depths of space away from any other mass and so away from any curved space.
4. it is made from a very strong material that is in no way flexible and does not change state under the conditions it finds itself in this example (be that heat orgravity etc)

While GR is about curved space-time, as I've said, we can use an example of curved space to show that what you are trying to imagine isn't possible. To do this, we need to have some shared idea of what curvature is. Rather than go into mathematical details, we'll just present some basic facts about what is curved and what is not that we will use, and hope that you can agree with them without too much arguing.

1) Planes are flat, because the obey Euclidean geometry
2) The surface of a sphere isn't flat, because it doesn't

It'd be helpful to add the following, but perhaps it's only helpful and not needed.

3) The surface of a football isn't flat.

These three points aren't very difficult to imagine, but if your idea of curvature doesn't mach these three points, then your idea of curvature isn't the same as mine. The only way I know of to address this idea of different notions of curvature is to get very precise about what we mean by curvautre, which means highly mathematical, and a great deal of time and study. It's not realistically something that can be understood by reading a post, it would be the subject of a graduate level course in differential geometry, plus possibly any missing prerequisites that are needed to understand the graduate level course.

We'll need some more definitions as well.

4) We will call a "straight line" on our curved surface a geodesic. All the points on the line must lie on the surface, no point can leave the surface. A geodesic curve is the equivalent of a "straight line" on a curved surface.

5) We need to know that geodesics exist on any curved surface, and that in the simple case of a sphere, geodesics are great circles, and that on a plane, they are straight lines in the sense of Euclid.

As an aside, we are talking about curvature here by imagining embeddings. This is done for convenience and ease of exposition. There are better ways of talking about curvature without imagining an embedding, but they are more abstract.

Now we can start imagining "very strong objects" that lie on a surface, and ask - can we move the objects freely around on the surface, without distorting their shape. In the case of a plane, and in the case of a sphere, the answer is yes.

There's a fairly simple demonstration that we can't do this in the general case, though we've seen that there are special cases where we can. Let us imagine a surface of a hemisphere joined to a plane. So we have a geometry where part of it is curved (the part on the hemisphere), and part of it is flat (the part on the plane). Now, we draw a triangle on the curved part of the geometry, and ask if it can be moved without distortion to the triangle on the flat part of the geometry. To make the statement a bit more mathematically precise, we can ask "are the two triangles simliar".

The answer is no. The sum of the angles of a triangle on a sphere is greater than 180 degrees. See for instance the wiki on spherical geometry , https://en.wikipedia.org/w/index.php?title=Spherical_geometry&oldid=820419798

The sum of the angles of a triangle on a plane is equal exactly to 180 degrees.

Therefore, we cannot move a large triangle from the curved section of the geometry to the flat section of the geometry, without distorting the triangle. The sum of the angles of the triangle must change, this means the triangle on the straight part of the geometry is different than the triangle on the curved part. The idea that there can be such a thing as a "very strong object" that can move around without distortion is not compatible with a geometry where the curvature changes. A sphere is symmetrical enough to allow us to do this even though it's curved, But in the case of our hemisphere+joined plane, or in the case of a football shaped geometry, we cannot even imagine a "very strong object" because we realize the object has to distort when we move it from one section of the geometry to another.

Large rigid objects just aren't compatible with curved geometries, and they aren't compatible with GR, either.

Your thought question is asking "what happens if we imagine the impossible". The short answer is that if we don't realize what we are imagining is impossible, we become very confused.

 

[PeterDonis]

In the case of a plane, and in the case of a sphere, the answer is yes.

More generally, if you are assuming that the "undistorted shape" of the object already conforms to the local curvature in the area where the object starts, then for any manifold of constant curvature, you can move the object to anywhere and it will retain its undistorted shape. If we are talking about ordinary 2-surfaces, there are only three such manifolds: the plane, the sphere, and the hyperboloid of constant negative curvature. In 4-d spacetimes there are more possibilities (though still not too many). But any realistic spacetime (like any realistic 2-surface) will not have perfectly constant curvature everywhere, so, as you note, in any realistic case, you will not be able to move objects without distorting them.

 

[sweet springs]

• it is perfectly straight - regardless of what scale you look at it, be that from a far, or up close at the atomic scale (as straight as something can be on the atomic scale)

• it is hundreds of thousands of miles long, and a mile wide. and a mile thick. (like a large support beam)

• It was constructed out in the depths of space away from any other mass and so away from any curved space.

• it is made from a very strong material that is in no way flexible and does not change state under the conditions it finds itself in this example (be that heat orgravity etc)

You can make a perfect circle, say diameter AB, in similar way. You measure length of diameter AB in the depths of space. Then you put a planet in the center and measure AB. You will find that length of diameter AB increase as GR says.

 

[Mr_Phil_Osophy]

More precisely, it causes spacetime to curve. For an object like the Earth, all of the ordinary effects of gravity that you are used to are (heuristically) due to "time curvature" (which is not really a good term), not space curvature; the curvature of space due to the Earth's mass (or even the mass of Jupiter or the Sun) is tiny.

Ok, so the bending of space isn't proportionate the the shape of the mass that bends it. Is the degree to which the space can be considered curved increase with regards to distance to the mass? And how does time curvature affect an object? Is this to imply that time varies in some way depending on the location of the object?

Observing the bending of light rays from stars that passed close to the Sun during an eclipse was one of the first classic tests of General Relativity

Yeah I remember reading about this, its the experiment that Einstein suggested?

Some possibly useful references:

Thank you I'll check them out when I get a moment.

One last question, how similar is the curvature of spacetime to the displacement of water by an object? Or is it not useful to look at it that way?
I'm trying to get my head around whether the void of space should be considered something like Descartes suggested with his denial of the vacuum, rather than being seen as nothing at all. Does it make sense to say that a mass, displaces spacetime to some degree?

 

[Mr_Phil_Osophy]

For example, the spacetime around a star or planet is spherically symmetric. In particular, at a given time, the space is spherically symmetric, so there is nothing particularly weird going on.

Can you elaborate on what you mean by this, I'm not sure I follow? what do you mean by the spacetime around a mass is spherically symmetric?

There's a bit of a myth that space is curved in a specific way around a large object.

Yeah I think my problem was that I was seeing it as a sort of displacement. In my head I don't see the vacuum of space as nothing, but as something which can be displaced, although not in any atomic sense (considering its a vacuum). I read a bit of Descartes' critique of the vacuum, and then a few papers on how we shouldn't see the vacuum as empty. This seems to be influencing my perception of the problem, and maybe making it murkier? I'm not sure.

 

[Mr_Phil_Osophy]

Such materials contradict Relativity, so Relativity cannot model them.

How so? Yeah I wasn't suggesting they existed, I was just trying to visualise how a hypothetical long object would interact with a large mass as it approached it length ways. I understand at least that spacetime is not a uniform thing and differs in its relation to a mass. So I was trying to figure out how these differences affect hypothetical objects just out of interest to see if it might improve my understanding of the subject.

If you want to have a ruler that is locally straight (a spatial geodesic) then it will be bend globally (the ends will not be parallel) near a mass.

Ok, that's interesting, and to some degree approaches the problem I have been trying to put forward in this thread. So a perfectly straight (and really long) ruler, on earth, would curve with the earth? rather than slowly veer off into the sky as you move further along the ruler?

See picture 2 here:
http://demoweb.physics.ucla.edu/content/10-curved-spacetime

This is interesting. So if time varies depending on the distance from a mass, does that not distort the concept with have for the age of the universe? How can you say something is X years old, if time is relative to the distance from objects?

 

[Mr_Phil_Osophy]

"Not flexible" alone implies an infinite speed of sound, which isn't possible. So I don't think this is a useful idea for exploring gravity.

Could it not possibly imply that sound wouldn't travel down such an object? Or that the object wouldn't vibrate? or am I misunderstanding what you're saying here?

 

[PeroK]

Can you elaborate on what you mean by this, I'm not sure I follow? what do you mean by the spacetime around a mass is spherically symmetric?

At a given time, the space around a spherical mass is given by:

$ds^2 = (1 - \frac{2M}{r})^{-1}dr^2 + r^2(d\theta^2 + \sin^2 \theta d\phi^2)$

Which can be compared with Euclidean space, which is:

$ds^2 = dr^2 + r^2(d\theta^2 + \sin^2 \theta d\phi^2)$

The only difference is the relationship between the coordinate $r$ and the spherical surface at that coordinate. What this means is that:

In Euclidean space, the area of the spherical surface at a distance $r$ from the centre is $4\pi r^2$, which can be calculated from integral calculus, with which you may be familiar.

In the curved space, as you move away in space from the centre of the mass, the area of the sphere varies differently in relation to the radial distance you are moving. The calculations get a bit messy, but as you move outwards the area of the sphere increases by more than you would expect in Euclidean space.

So, the $r$ coordinate does not represent a radial distance. The radial distance between $r = 10^6 km$ and $r = 10^6 + 1 km$, say, is not $1km$. If you physically measured the radial distance between two spheres of the appropriate areas, you would find the radial distance between them to be less than $1km$.

This is one example of "curved" space. You either take the mathematical definition of distance between any two points that I gave above. Or, you could embed your 3D space as some shape in four dimensions.

Either way, you probably need a bit of differential geometry to get a real feel for it. And, even then, you are heavily reliant on the mathematics to see what is going on.

 

[A.T.]

Yeah I wasn't suggesting they existed

It's not about whether they exist. It's about whether Relativity can make predictions about them.

So a perfectly straight (and really long) ruler, on earth, would curve with the earth? rather than slowly veer off into the sky as you move further along the ruler?

Depends on what you mean by "perfectly straight" and "curve". You need to understand geodesics on curved manifolds. A simplified version was in the link I posted.

 

[Ibix]

Could it not possibly imply that sound wouldn't travel down such an object? Or that the object wouldn't vibrate? or am I misunderstanding what you're saying here?

If you tap one end of the bar the whole thing must move instantly to avoid flexing. That means mechanical shocks propagate at infinite speed. That includes sound.

 

[PeterDonis]

so the bending of space isn't proportionate the the shape of the mass that bends it

No. The shape of the mass itself (like the Earth) is not determined solely by the geometry of space. It's determined by the non-gravitational interactions in the matter that makes up the mass (like the Earth), in hydrostatic equilibrium in the gravitational field.

its the experiment that Einstein suggested?

One of them, yes.

 

[PeterDonis]

At a given time, the space around a spherical mass is given by

It's worth pointing out that this is for a particular choice of coordinates, Schwarzschild coordinates. It's true that these are the "natural" coordinates for an observer who is at rest relative to the massive body, but they're still not the only possible choice, and other choices of coordinates will make "space" look different. For example, in Painleve coordinates, which are the "natural" coordinates for an observer who is free-falling radially inward towards the massive body, "space" is flat (Euclidean).