How does space-time bend?
Does gravity cause space-time to bend or IS gravity to bend space-time? [closed]
I think the correct answer should be that what we call gravity is a fictional force that we experience when we live in an accelerated frame of reference (as opposed to a sluggish one). In contrast to other forces, gravity disappears when the coordinates are changed. When a person is in a falling elevator, they experience a free fall, that is, they feel floating and they would conclude that they are not exposed to gravity. However, we at the surface of the earth would say that gravity is clearly causing the elevator to tumble towards the ground faster and faster.
Of course, the solution to this strange condition is that gravity is not a force at all. We live in a four-dimensional universe with a pseudo-Riemannian geometry, in which freely falling objects move along the geodesics or lines of the shortest space-time distance. Since the geometry (like the surface of a sphere) intrinsic may be curved, these geodesics are not what we consider straight lines. The person in the elevator moves along a geodesic, while we are accelerated on the surface of the earth and we are not moving along a geodesic. The space-time paths (or world lines) of the elevator and the floor below are not straight lines and will therefore cross at some point. This intersection is the point in spacetime where the elevator touches the ground.
One way to imagine this is to look at two ants walking along the lines of longitude on a globe. Longitudes are large circles and geodesics of the sphere. The two ants start at the equator at different degrees of longitude and travel due north at the same speed. Their paths are initially parallel to each other, but as they move along the curved surface the distance between them decreases until they eventually collide at the North Pole. It seems like there is a force that is pulling them together, but in fact the force is fictional. The reason they got closer is that on the sphere the geodesics converge and cross, unlike in flat space, where the geodesics are straight lines that never cross. If the globe is very large, the ants will never know they are moving on a curved surface and would conclude from this that there must be some force that is attracting them. This is the basic picture of how "gravity" works from a general relativity perspective.
Now for your question, the difference is subtle. While what we call "gravity" is subject to semantics, something deeper is afoot. The general theory of relativity is commonly referred to as the "theory of gravity". In this case, we can think of the answer as the latter: by definition, gravity is the bend in space-time. On the other hand, if we look at gravity as a force, apparent gravity becomes essential by the fact caused that spacetime is curved. But we can essentially look at this logic in circles if we think too much about it. It all depends on what we define as "gravity".
But deeper than that is the question what gravity causes . In classical mechanics we are told that gravity is caused by mass, in the sense that massive bodies have a gravitational field that attracts them. But we know that's not the right picture. To generalize your question: will space-time curvature by Dimensions caused ? In a sense yes, in a sense no. Einstein's equation is
where κ is a constant, the tensor Gμν is a function of the metric that encodes the curvature of spacetime, and T.μν is the stress energy tensor that encodes the matter / energy content of the universe.
Since general relativity theory is fundamentally four-dimensional, and there is no preferred direction to call "time," we need to solve Einstein's equation essentially "all at once." It is clear that the matter content of the universe determines the curvature of the universe, while the curvature of the universe tells matter how to move. So you have a kind of chicken and egg problem: matter tells space how to bend, and space tells matter how to move.
There is a Hamilton formalism (ie an initial value) for GR that works for globally hyperbolic spacetime (ie it does not hold for all possible spacetime). It's called the ADM formalism (named after Arnowitt, Deser, and Misner). It allows one to set initial conditions for a spacetime (initial curvature and matter / energy state) and to calculate the evolution of this spacetime and its matter content over "time" in a way that is generally covariant (does not violate the relativity of the observer )). However, this still does not sever the inherent connection between space-time curvature and matter / energy content.
As an interesting related question, one might wonder whether a massive particle moving through space can gravitationally interact with itself. That is, the mass of the particle distorts space-time and therefore changes its trajectory. At the end of Jackson's "Classical Electrodynamics" there is a similar question about the acceleration of charged particles that interact with their own radiation. I think his conclusion is that such processes are not really considered because they would create such small corrections. In the context of GR, I would suspect that such questions fall into the realm of quantum gravity.
In relation to your last question, you may mean "without Space-time curvature In this case the answer is no, the apple would not fall, all objects would move on straight space-time paths that never cross and would therefore always remain at the same distance from each other.
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