Einstein said that gravity can be looked at as curvature in space- time and not as a force that is acting between bodies. (Actually what Einstein said was that gravity was curvature in space-time and not a force, but the question what gravity really is, is a philosophic question, not a physical one)
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In the framework of GR, gravity is indeed not a force as it's a consequence of Newton's first law instead of the second one. Each point in space-time comes with its own velocity space attached, and you need the parallel transport (and thus a connection aka gravity field) to be able to even define what you mean when you say a body moves without acceleration. In the more general setting of arbitrary second-order systems (ie if we forget about Newton's laws), the space of acceleration fields carries an affine structure. A connection is one way to choose a zero point and make it into a vector space so you can have the notion of addition of forces (or rather acceleration fields). From this point of view, gravity would indeed be a force like any other, but special insofar as it gets chosen as the one that is called zero. |
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Well, if we're talking about what Einstein said, then the way Einstein defined gravitational field and gravitational force in GTR is that it is given by the connection, with its components by the Christoffel symbols: $$\Gamma^{\alpha}_{\mu\nu} = \frac{1}{2}g^{\alpha\beta}\left[g_{\mu\beta,\alpha}+g_{\nu\alpha,\beta}-g_{\mu\nu,\beta}\right]$$ where commas denote partial derivatives and the metric $g_{\mu\nu}$ plays the role of gravitational potential. But this is quite different from Newtonian gravitational force. In Newtonian mechanics, you have 'real' forces and 'inertial' (aka "fictitious") forces, the difference being that you can make inertial forces disappear by adopting an inertial frame. For example, Newton's laws in a uniformly rotating reference frames introduce centrifugal and Coriolis forces that are proportional to the mass of the object acted upon and can be removed changing to an inertial, and hence non-rotating, frame. In other words, inertial forces are the "fault" of choosing a non-inertial frame of reference. By the above definition, gravity is an inertial force. Similarly to the Newtonian case, it can be made to disappear by changing the reference frame--but there is also a big difference: in the Newtonian framework, inertial frames are global, and so inertial forces disappear everywhere. In GTR, that's no longer the case: there are only local inertial frames in general, and so you can only make it disappear locally. Caution: modern treatments of general relativity do not adopt this definition. Many of them (e.g., Misner, Thorne, and Wheeler) intentionally do not identify either 'gravity' or 'gravitational field' with any particular mathematical object, not the connection, not the curvature, nor anything else. But then (for MTW) it is not technically correct to say that gravity is spacetime curvature either, but rather refers "in a vague, collective sort of way" to all of these geometrical constructs. |
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Websters specifically defines force as the gravitational interaction (definition 4b). We all were taught in high school that gravity was a force. Given the lack of consensus among the authorities, a more edifying, less controversial, and equally true statement might be: In general relativity, gravity is a fictitious force. In classical mechanics, fictitious forces are not considered "real" forces. However, nobody, not even relativists, goes around claiming "the Coriolis force is not a force". The issue of gravity being a force or not has nothing to do with general relativity. If you believe that inertial forces are forces, then gravity is a force. If you believe that inertial forces are not forces, then gravity is not a force. |
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In GR, there are always two points of view--- local and global. In the local point of view, you look in a neighborhood of a point, and make a free-falling frame, and then motion is entirely in straight lines at constant velocity so that you don't see gravity. In this way of looking at it, gravity is not a "force", meaning it doesn't make a generally covariant contribution to the local curvature of the particle space-time paths. In the global point of view, you see an incoming particle from infinity deflected by a field, and you say a force has been acting if the particle is deflected. In this point of view, every deflection is a force by definition. The global point of view is the way in which gravity is treated in quantum field theory or string theory. The local point of view is the insight due to Einstein, and it is no surprise he would emphasize it in his public remarks. The answer is "it depends on your philosophical definition of force, whether you take a local view or a global view." I prefer the global view, since it is more quantum, so I say gravity is a force, but I don't disagree with people who take the other view, since it is also valuable. |
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