Not Gravity, Geometry?

Ah, now that is a core question. It may take more than a bit of explaining though .... and beware there is no neat/pat answer that will likely satisfy your intuition. It really is quite a paradigm shift. Go and get a cup of your favorite brew before reading. I'll give you Mike's Tour Of GR! :-)

Firstly : one has to include time in the geometry. Just talking about 3-D space alone won't cut it. Try this simple example. Fire a cannonball at some fixed angle into the air. You'll find that where it lands depends ( ignoring lots of other fiddly non-gravitational stuff ) on the velocity with which it was fired. Indeed for given initial and final points even, there are two trajectories - the high lob and the low, flat shot - if you allow both the angle and direction of velocity to change.

The same applies to orbits around a central body, like planets and moons, where the subsequent evolution of the path requires considering the magnitude as well as the direction of the velocity vector. One could imagine the Earth in it's current orbital position around the Sun, with it's usual velocity, and it'll keep circling. But suppose it was at exactly the same position but with say, five times it's usual speed - it certainly won't hang around the Sun for much longer. So there's a time ( rate of change ) required in the explanation. That's as true for GR as it is for classical explanations.

Secondly : when we say geometry it's crucial, though no doubt annoying, to have to consider the basis of measurement. As Einstein found out with Special Relativity, some 'obvious' ideas in classical physics turned out to be wrong, or at least misleading and approximate. So there are many choices of geometric description, and for Einstein the challenge was to come up with a formulation that would be physically true ( same predictions ) regardless of special choices of how/where/when the geometry is defined. Thus the Earth ought continue to orbit the Sun, say, as viewed by a whole range of observers in various positions, with different clocks and measuring sticks.

Fortunately much of this was already done, incidentally, by a chap called Riemann. He studied what geometry would be like if it wasn't according to Euclid/Pythagorus etc. His essential breakthrough was a way of describing geometry locally and within the thing being discussed. So one might look at an apple and say : it is round much like a sphere, more so near the top where the stem is, is dimpled/puckered either end and is pretty smooth overall. These are words which are really 'external' descriptors. Riemann's way of deducing this equivalently was to say : I have a point on the apple and I compare two paths diverging from that point, further down each path I find they meet each other again. He comes up with a 'value' at each point on the apple, such that if one considers the totality of all these values you could arrive at what it would look like if you did see the apple from afar.

Now the reasons for going to a local ( in time as well as space ) rather than a general description is several fold :

- 3D in space plus one in time is 4 dimensions. Hard to visualise per se.

- to split the problem into two parts. Begin by stating the geometry in terms of what distribution of matter/energy produces it, then given that, see what response some object has in that geometry. Hence 'matter tells space how to warp, and space tells matter how to move'.

- spacetime is flat whenever viewed from close enough. This means that for a short enough time and/or for a small enough distance any movement looks Euclidean ( or Gallilean or Newtonian ).

So thirdly : how do you describe movement in detail in this framework? Well those 'values' I mentioned above are really a set of values at each point in spacetime. They 'explain' how you transition from one point to the next if you are freely falling ( only subject to gravity ). This is where the 'warping' business comes in : at each point in spacetime where gravity is acting ( and gravity is everywhere acting on everything ), these 'metric tensors' are a local guide to how directions change and thus which way to go next. I've described in another thread an analogy with small villages on some undulating landscape. At the centre of each hamlet is a signpost where roads intersect that has directions indicating the way to nearby villages. The metric prescribes how these signposts ought vary from place to place. [ The full horror is calculus, infinitesimals and equations with partial derivatives ... ]

If you were describing a globally flat spacetime ( unrealistically meaning no mass or energy was about ) then the metrics will state that the signposts don't vary from village to village. If you grind through the math in this scenario then you'd wind up describing the law of inertia in free space, where things just keep going if they are already. Specifically they wont deviate from a straight line.

Fourthly : I have left light out thus far. In GR the phrase 'straight line' is replaced by 'the path that light follows' or 'null geodesic'. For ordinary life these are easily seen to be the same thing. If you can arrange matters to view three objects by eye and see that they overlap/occult one another simultaneously then we say they are in a line. Bricklayers, surveyors, shooters etc explicitly do this all the time. So one way of mapping the curvature of spacetime is to study what the light rays are up to. The time component comes out as a change in the frequency of the light radiation, and thus is a measure of how time ( ie. clocks ) varies around and about.

Add in that any non-zero rest mass can't match ( or exceed ) light speed then you have an overall rule that matter won't escape the confines of the light paths. Hence the 'cone' analogy to spacetime points, where curvature means the cones are wobbly shaped if compared to the flat case. The way the cones change their shape from here to there is encoded in the metric.

So finally back to Newton's case of a single central mass that influences another one some distance away - your originally query. The good & bad news is that we have an approximate but not exact solution to the GR equations for this. It's still pretty good and has been observationally well confirmed more than a few times though ( eclipses, Mercury .. ). It was figured out by an artillery officer on the eastern European front in WWI, and he died not long after mailing it to Einstein. It's the ( Karl ) Schwartzchild ( maybe without the 't' ? ) solution, and has been expanded upon by others to include rotation and electric charge too ( see Kerr ).

The solution has a special quality with regard to a certain distance from the central body, the Schwartzchild radius. This radius depends on the mass of the body and some fundamental universal constants. If a body happens to lie completely within it's own Schwartzchild radius it will become a black hole. I won't recount all the observational features of black holes bar the prime one - that not even light can travel from within that radius to without ( discounting quantum effects ). For me that radius is about that of a proton I think, for the Earth about an apple size, for the Sun about the width of a mountain. As Mike/Earth/Sun each are not compact or dense enough - insufficient mass within a given volume - then while not black holes, there may be some measurable deviation of light rays passing by ( as viewed from far away ). An eclipse just after WW1 discovered the effect for the Sun, later technology in the 1960's ( plus the GPS more recently ) confirmed that for the Earth. No one has yet come along to demonstrate the effect around me though .....

Personally I try to avoid the word 'curvature' or at least mentally substitute it with the phrase 'observers differ'. That way time can be 'curved' by differently situated clocks progressively disagreeing with each other.

Cheers, Mike.

( edit ) To be more precise, I ought say by 'metric tensors' I mean a metric tensor which is evaluated at many points. A tensor is sort of a multi-functional function. So instead of a single valued function - one number in, one number out - a tensor can have many things both in and out. In a sense we bundle lots of single-valued functions together, for instance how stuff in the z-direction depends on time, or how time depends on stuff in the x-direction. But they work as a group, and reasonable ideas of symmetry ( ie. reasonable universes without surprising behaviours that we haven't yet seen ) contract 4 x 4 = 16 functions to 10 independent ones.

Another way to visualise is : at each point in spacetime ( each moment in space & for each instant ) you have this associated tensor 'gadget' or 'box' that you can crank. We don't of course have an infinite listing of boxes, but Einstein's equation that governs their character. What remains is initial/starting/boundary conditions. We may well know how things vary from ( spacetime ) point to point but that still leaves some freedom in choice of the 'baseline'. Alot of discussions I read about GR astronomical problems divulge alot of assumptions upon these conditions. In a way one might solve say two neutron stars circling each other, however they aren't really alone so you have to 'connect' their behaviour to the rest of the universe at the 'boundaries'. Boundary also applies to the time co-ordinate, thus from whence and until whence is quite relevant.