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I have a question about interpreting (explaining, even) the general theory of relativity.

A common interpretation of GR, as I understand it, is to imagine two-dimensional space represented by flatlanders (ants) living on the surface of a trampoline (or stretched elastic sheet). The third ‘hyper-dimension’ from the point of view of the ants is time.

If a heavy object is placed in the middle of the trampoline, it distorts the surface in space-time, so that ‘straight lines’ of motion become geodesics, which are actually curved. A marble rolled across the trampoline curves around the object, because it’s following the shortest path in space-time. So far so good.

A reader here a while back (Why would spacetime curvature cause gravity?) posted the question: so why do two stationery objects move towards each other, and the answer was that in space-time, they are not stationery - they are moving through time. And it’s their space-time lines which still try to take the shortest/straightest route, as illustrated very nicely in a YouTube video linked in one of the answers.

Now I’m interested in getting back to a more intuitive understanding of why an attractive force seems to be exerted on the second object by the first, without getting tied up in the rather complex tensor representations of geodesics etc. At the same time, I want to know WHY a massive object distorts space-time.

Most texts take the curvature of space-time as an axiom (postulate), without pretending to offer any explanation. Note that (as I understand it) the heavy object on the trampoline is only intended to represent or visualise the curvature - there is no suggestion that it’s a simple matter of heavy objects ‘weighing down’ space time.

But I’m thinking, why not take the analogy literally, and follow through on this idea of weighing down?

In the analogy, it seems obvious that a heavy object distorts the surface downwards - it’s a result of the external, real gravity in the analogy (acting in a hyper dimension from the ants’ perspective). And the marble, of course, is simply rolling downhill.

Now as we know from Einstein’s principle of equivalence, gravity equals acceleration. The third dimension in ant-world is time - so, what if we suppose that time is accelerating and hence generating that hyperdimensional force?

For me, the surface of the trampoline is not simply a snapshot, or slice, of space-time, but far more significant - it is the present universe. (As an aside, because of its distortion, the relative ‘hight’ of points in space time (above ground level) varies. Furthermore different observers (ants on the trampoline) project different tangential planes from their standpoints. Do these two effects explain all the results of special relativity?)

Now assume that time moves inexorably forwards, and the surface of the trampoline is moving upwards, indeed accelerating. As we know, acceleration and force are the same thing: so the surface of the trampoline is pushing up against all objects in the universe.

Suppose also that all massive objects have a type of inertia which resists time - then the trampoline surface is retarded, or flexed backwards! That is what causes the distortion.

Now looking at a stationery marble sitting off to the side of the heavy object in the middle, it too is being pushed forwards (upwards) by time (the surface of the trampoline). Only the surface is now inclined - so there is a sideways component of the force, which pushes the marble towards the heavy object.

Simple, obvious, and indeed an inevitable consequence of the assumptions that a) time is accelerating forwards, and b) heavy objects have an inertia which resists the passage of time.

Is this idea complete nonsense, or is it a valid way of interpreting how space-time is bent (and how that generates the appearance of gravity)?

Is this how some physicists already interpret general relativity - am I wrong to suppose that the analogy is not usually intended to provide a reason for the curvature?

It’s really a question for someone very familiar with the mathematical underpinnings of GR - can the curvature of space-time be successfully modelled by an accelerating timeframe and massive objects having an inertia which resists it?

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closed as off-topic by ACuriousMind, Kyle Kanos, Ryan Unger, Prahar, Neuneck Aug 24 '15 at 13:34

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Does acceleration of time explain gravity (rather than the other way round)?

No. But note that gravity doesn't make optical clocks go slower when they're lower. Gravity is there because optical clocks go slower when they're lower. Because a concentration of energy in the guise of a massive star "conditions" the surrounding space, altering its metrical properties, this effect diminishing with distance. See the Einstein digital papers for more.

If a heavy object is placed in the middle of the trampoline, it distorts the surface in space-time, so that ‘straight lines’ of motion become geodesics, which are actually curved. A marble rolled across the trampoline curves around the object, because it’s following the shortest path in space-time. So far so good.

The rubber-sheet analogy can be misleading but it isn't totally wrong. It depicts curved spacetime or a "curved metric", and a metric is to do with measurement. For example, let's say you place optical clocks throughout an equatorial slice through the Earth and the surrounding space, then plot the clock rates. You depict lower slower clock rates as lower down in a 3D image, and higher faster clock rates higher up. So your plot looks like this:

enter image description here CCASA image by Johnstone, see Wikipedia

That's a picture from the Wikipedia Riemann curvature tensor page. It's the rubber-sheet depiction of curved spacetime. And because it's derived from optical clock rates, it's a plot of the "coordinate" speed of light. You might also say it's a plot of time rates, but note that there isn't any actual time passing through an optical clock. A clock isn't some kind of gas meter. It doesn't literally measure "the flow of time", that's just a figure of speech. Note that the curvature you can see on your plot relates to the tidal force while the slope relates to the force of gravity. But also note that you need that curvature to get the plot off the flat and level in the middle. So if you don't have it, you don't have a gravitational field. This is why spacetime curvature is said to be the "defining feature" of a gravitational field.

A reader here a while back (Why would spacetime curvature cause gravity?) posted the question: so why do two stationery objects move towards each other, and the answer was that in space-time, they are not stationery - they are moving through time.

That's wrong I'm afraid. There is no actual "motion through time". There's no motion through spacetime either, see Ben Crowell's answer here. And spacetime curvature doesn't actually cause gravity. Your optical clock doesn't go slower when its lower because your plot of clock rates is curved. It goes slower because a concentration of energy in the guise of a massive star "conditions" the surrounding space, altering its metrical properties, this effect diminishing with distance.

And it’s their space-time lines which still try to take the shortest/straightest route, as illustrated very nicely in a YouTube video linked in one of the answers.

Light doesn't curve because it tries to take some shortest route through spacetime. You will not find Einstein saying that. See the Einstein digital papers for what he did say.

Now I’m interested in getting back to a more intuitive understanding of why an attractive force seems to be exerted on the second object

IMHO it's simpler than you think. Remember pair production and electron diffraction and think of the wave nature of matter. Think of matter as light going round and round, then simplify it to a square path. What happens to the horizontals? They bend down a little. Draw it.

I want to know WHY a massive object distorts space time.

The clue is in the stress-energy-momentum tensor which describes "the density and flux of energy and momentum in spacetime". Note the energy-pressure diagonal and the shear stress, and google on elastic space time. When you add energy it's like adding more space at that location, so that you create something akin to a "pressure gradient" in the surrounding space. Like this picture but pushing out instead of pulling in. Note that a massless photon causes gravity. A concentration of energy causes gravity, not mass per se.

Most texts take the curvature of space as an axiom

Note this Baez article which says this: "Similarly, in general relativity gravity is not really a 'force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial". A gravitational field can be said to be curved spacetime, but not curved space.

But I’m thinking, why not take the analogy literally?

Because gravity doesn't work like that.

What if we suppose that time is accelerating and hence generating that hyperdimensional force?

Then you start departing from Einstein and general relativity for no good reason. I would urge you to read the Einstein digital papers instead of being some "my theory" guy.

Is this idea complete nonsense?

Sorry, I'm afraid it is. But it's good that you're thinking for yourself.

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