There is a very fundamental flaw in the common explanation given of the space-time curvature due to massive objects. It is said that a massive object curves space time just like a bowling ball on a rubber sheet, and another object near the massive object simply rolls down-hill on the rubber sheet, that is the reason why we observe that the other object is experiencing gravitational attraction to the massive object which curved the space-time.

The flaw is this:

Even if the massive object did bend the spacetime downwards, and there was a second object in the vicinity, this second object just would not roll down-hill automatically, unless there was a SECOND EXTERNAL source gravity, other than these two objects pulling everything downwards. In other words, even if the space time is curved, there is no particular reason for the second object to move closer to or be attracted by the first, because there is no other external force that would force it to move down-hill.

As a visualization, consider the bowling ball on a rubber sheet and another object on the same rubber sheet in outer space where there is no other source of gravity. Now, neither will the bowling ball curve the rubber sheet downwards (because it is not being pulled downwards), nor will the other object move down-hill on the rubber sheet even if the rubber sheet was depressed downwards, because there is no downward gravitational force from earth or another external source which would cause either of these two effects. So, there would be no reason for the other object to be attracted towards the first object. This is inconsistent with observation because objects do really experience gravitational attraction towards each other. But it certainly can't be explained by the bowlig ball and rubber sheet example. What is the explanation for all this?

Flaw 2: If there are 2 massive objects on the rubber sheet, they will both bend spacetime around themselves, and they will both remain trapped in their own depressions, and would never move towards or be attracted to each other. This is again contrary to observation. Since both masses are in reality attracted to each other, so they can't possibly bend spacetime and can't be trapped in their own depressions. Why?

Flaw 3: All pictures of bent spacetime on the web show spacetime being bent downwards. Why? In open space, with no other object nearby, all directions are equivalent to all other directions, and "downward" is undefined. So if spacetime is really bent, in which direction is it really bent?

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    $\begingroup$ The rubber sheet analogy really is a terrible one for exactly the reasons you mention. It's unfortunate that I can't think of any more suitable analogy. $\endgroup$ Mar 6 '14 at 21:58
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    $\begingroup$ I believe this has already been covered here: physics.stackexchange.com/q/7781 $\endgroup$ Mar 6 '14 at 22:33
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    $\begingroup$ OK, a part of the answer is given here physics.stackexchange.com/q/7781 but the MAJOR thing still unanswered (or I could not understand) is as follows. Could anyone please explain just this: Even if spacetime is curved by a mass such as the earth, why would this curve cause an apple to move towards the earth? Why would it not just sit in its original position on the curved spactime? The apple following a curved geodesic rather than a straigt line may be one thing, but why would the apple get attracted to the earth at all? Why does space time curvature translate to an attraction force? $\endgroup$ Mar 7 '14 at 5:07
  • $\begingroup$ @user1648764: Now that's essentially a duplicate of this question instead; see especially Marek and Luboš's analogy of geodesics on a sphere. A more technical presentation for spacetime curvature would connect geodedesic deviation to the "gravitoelectric" part of Riemann curvature. Newton's law can be derived fairly simply from the Einstein field equation, but this required quite a bit of prerequisites to understand. See here if you're brave. $\endgroup$
    – Stan Liou
    Mar 7 '14 at 6:37
  • $\begingroup$ @user1648764 I've had the exact same question but I would preface it with this: If we posit that the apple is stationary with respect to the central mass, what force would act upon it such that it would begin to follow the curvature of space created by said mass? If the apple were moving, it is easy to see that it would follow the path defined by spacetime's curvature, but what force acts upon it if it were not moving? $\endgroup$ May 30 '18 at 14:08

The deformed rubber sheet is hyperbolic geometry. All its triangles' three interior angles sum to fewer than 180 degrees (down to an epsilon degrees). A gravitational well is elliptic geometry. All its triangles' three interior angles sum to more than 180 degrees (certainly up to 540 degrees).

"Circular orbits on a warped spandex fabric"


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