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Thread: gravity and acceleration

  1. #1
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    gravity and acceleration

    I'm trying to learn how you simulate gravity by spinning, and I think I have the basics down: gravity and acceleration are equivalent, and circular motion is constant acceleration.

    But I'm very confused. Some sources say it's centrifugal force that generates the gravity (which makes intuitive sense to me) while others say it is centriptetal force, which doesn't.

    Also, did it really take Einstein to figure this out? This seems like something Newton would have understood.
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    Each are both.
    Centripetal force is what accelerates an object radially to keep it in circular motion. So if you're inside a rotating habitat, centripetal force is what the floor applies to your feet, which feels like gravity.
    But from your point of view, in a reference frame rotating with the habitat, there's no apparent source for this force you experience between floor and feet. So you invoke centrifugal (pseudo)force to explain why you are apparently being pressed against the floor.
    Centripetal force is what is observed by someone in an inertial reference frame, watching you whirl around. Centrifugal (pseudo)force (and its pal, Coriolis) is what you need to make the laws of motion work in a (non-inertial) rotating reference frame.

    What it took an Einstein to figure out was that the similarity between acceleration and gravity is telling us something deep about the nature of the Universe, rather than being merely a superficial resemblance.

    Grant Hutchison

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    Centripetal force is the force pulling something into circular motion (we will ignore geodesics right now....), so it’s the force pushing up from the floor in your spinning space station. Centrifugal force is the reaction to that force.

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    In my view this question is an excellent entry point into understanding what the differences between Einsteinian and Newtonian gravity are-- and what they aren't. If you have a space station in deep space where there are no important local gravitational effects, then you might think Newton and Einstein have nothing very different to say-- they both know perfectly well how to transform from a non-rotating to a rotating coordinate system, and they know how to calculate how fast to make it rotate to be comfortable to humans. One would be tempted to say the answer is then that using rotating coordinates forces you to introduce a centrifugal term that looks a lot like gravity, and both Newton and Einstein would agree on that if they first agree on what the inertial coordinates actually are. You might then say that it is the use of rotating coordinates that "generates" the artificial gravity (and the reason you use those coordinates is due to the acceleration of the observer due to the centripetal force) , and the artificial gravity "is" the centrifugal force.

    However, even in that scenario there are fundamentally important differences as to how those men would address the issues, which has a lot to say about their different views of the nature of dynamics. Newton would say that if there are no important gravitational action-at-a-distance forces to be considered, then gravity is playing no role, we simply take Galileo's straight paths as our inertial paths, and transform to any non-inertial coordinates we like. Einstein would say that even though there are no local gravity terms, gravity is still playing a crucial role in deciding the inertial paths through spacetime, and with it, the overall geometry of that spacetime (which would be spatially flat and Minkowskian in time, though that latter would not matter). Newton might be surprised by that claim-- how could gravity be playing any role at all? Einstein might respond that gravity is doing nothing locally, but its global effects are always crucial in determining the inertial paths-- one cannot simply assume Galileo had it right and take that as an axiom for one's theory. Hence gravity is fundamentally the starting point of any dynamical theory, not something you tack on at the end if you think you need it.

    So what is the role of the equivalence principle here? I think many people (including myself!) have in the past mixed up the message of the equivalence principle. It is common to hear (I suspect I also said it!) that the equivalence principle is the discovery that gravity is actually a kind of fictitious force, because the equivalence principle says you can't (locally) tell the difference between gravity and being in an accelerating reference frame without knowing it. Normally if you are accelerating and don't know it, you imagine "fictitious forces" to make sense of what you see. Thus it sure sounds like gravity is to be viewed as one of those. But on further thought, I conclude that this is exactly what the equivalence is not saying! Rather the opposite-- the equivalence principle (Einstein's "happiest thought") was the stunning recognition of Newton's mistake-- it was Newton who was treating gravity as a fictitious force, by treating it as a force at all. Newton was using accelerating coordinates without knowing it, so he inadvertantly introduced fictitious forces that he was forced to associate with gravity. The equivalence principle allows us to banish the connection between gravity and fictitious forces, in concert with banishing fictitious forces altogether.

    Let me explain. Einstein did two very different things in general relativity that are easily conflated into a single thing. One of those things he could have done even in a world without gravity, which is to find a form for the dynamical equations of physics that are coordinate independent. Newton didn't do that-- he found a form that is invariant under transformations between inertial observers only, so you must apply his laws for an inertial observer, and then transform to noninertial coordinates (like a spinning space station). That was not viewed as a problem because it was easy enough to do, but nevertheless it is a kind of chink in its armor that Einstein never liked-- he wanted the equations to apply in any coordinates. Einstein couldn't do that in special relativity, so all special relativity does is allow for high speeds, it still must apply its axioms in an inertial frame and then it knows how to transform to noninertial frames (acceleration is treated as a series of short steps at constant speed and you continuously transform into those instantaneously inertial frames). Einstein wanted to be able to do that in a more automatic way that would be embedded into the laws themselves, so you never even needed to ask what the coordinates were when you wrote the laws (that's what's "general" about it). He would have done that even if there had been no such thing as gravity, and he would have been able to apply the result to spinning space stations-- banishing fictitious forces because they appear when you transform coordinates, but Einstein's laws are coordinate independent. In Einstein's way, you simply don't care what the laws look like in any given coordinates, because what they are is coordinate independent!

    So this is what I'm saying in answer to the question-- the big difference between Newton's treatment of a rotating space station and Einstein's is analogous to, yet unrelated to, their different treatments of gravity. It is only Newton who would have any reason to call what is happening in the space station "artificial gravity" or a "fictitious force", because it appears as as additional term when Newton's laws are written in non-inertial coordinates yet interpreted from the lens of inertial coordinates, which is also what his version of gravity does. The analogy to gravity is only there in Newton's approach! In Einstein's general approach, you write the laws in a coordinate-free way, so there are no fictitious forces at all, nothing to compare to some other type of coordinates because all coordinates are treated exactly the same and you always expect to get funky looking terms because that's just what coordinates do to you-- they are not to be mistaken for the physics, which is written quite intentionally in coordinate-free language. You simply never expect inertial motion to cross your coordinates in straight lines because there's never anything special about straight coordinates!

    So how does gravity and the equivalence principle get into the act? The equivalence principle is saying that it is easy to mistake gravity for a fictitious force, and this statement is most profound when you have already banished fictitious forces! Had Einstein been saying that gravity was in essence a fictitious force, then it would have disappeared along with all the other fictitious forces when Einstein found a way to write the laws in coordinate-free form. But it didn't! Instead, gravity is the way the geometry of spacetime is determined, it sets the inertial paths and must therefore be embedded in the coordinate-free laws in no uncertain terms. It is the starting point of laws of dynamics, the first law. The equivalence principle pointed Einstein to the mistake Newton had made (we should all make such mistakes), because it said that Newton was calling the wrong coordinates inertial coordinates, and Newton's approach required that if you did that, you would need to fix your mistake with fictitious forces. The equivalence principle does not say gravity is a fictitious force, it says using the wrong inertial paths forces you to treat gravity as a fictitious force. And since gravity isn't a fictitious force (but it does tell you the inertial paths), you can't really get it right that way but you can often come so close that you never realize you didn't get it right.
    Last edited by Ken G; 2018-May-08 at 04:25 PM.

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    Quote Originally Posted by swampyankee View Post
    Centripetal force is the force pulling something into circular motion (we will ignore geodesics right now....), so it’s the force pushing up from the floor in your spinning space station. Centrifugal force is the reaction to that force.
    Perhaps worth pointing out (in Newtonian terms) that this sort of "reactive centrifugal force" is a real force, which falls under Newton's Third Law, and which would be detected by inertial observers.
    But our rotating observers also need the centrifugal pseudoforce, to explain why a dropped object falls to the floor of their habitat. Centrifugal and Coriolis have to be written into their laws of motion, although these forces are not evident to an inertial observer.

    Grant Hutchison

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    Right, the "reactive centrifugal force" is a bit like saying that the scale you stand on measures the force of gravity on you. But even in Einstein's treatment of gravity, the force the scale measures is a real force, and is strictly not a force of gravity, but it does equal it. Similarly, the "reactive centrifugal force" is a real force that could be measured by a scale, and equals the fictitious centrifugal force, but isn't the same thing because the real force would go away if you poke a hole in the floor of the space station but the fictitious force would still be there to make you fall through it!

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    Quote Originally Posted by Ken G View Post
    Similarly, the "reactive centrifugal force" is a real force that could be measured by a scale, and equals the fictitious centrifugal force, but isn't the same thing because the real force would go away if you poke a hole in the floor of the space station but the fictitious force would still be there to make you fall through it!

    I'm going to write that one down.

    Grant Hutchison

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    Quote Originally Posted by parallaxicality View Post
    Also, did it really take Einstein to figure this out? This seems like something Newton would have understood.
    Let me summarize what I said above to address this specifically. If you, like Newton, take the approach that you first must find inertial coordinates and then apply your laws, and then transform to noninertial coordinates, you will see every term that appears in that transformation as a "fictitious force." But if you, like Einstein, simply apply coordinate-free laws, and only then choose your coordinates to suit your convenience, you don't have to interpret anything you see in those coordinates-- the interpretations are in the coordinate-free language, the calculation in the coordinates has all kinds of coordinate-dependent terms that are not even worthy of naming or regarding as fictitious. So you have banished the concept of fictitious forces in the latter approach. Enter gravity. Newton thought it was obvious the correct inertial coordinates, so gravity had to be an additional effect. Einstein said Newton's treatment of gravity is actually a fictitious force that Newton needed as a correction to his laws, that was there only because Newton had gotten the inertial coordinates wrong, so was mistaking a term he needed to transform to the correct inertial coordinates for a real force. So in short, Einstein's view is not that gravity is a fictitious force, it is that Newton's picture of gravity is actually a picture of a fictitious force! But Newton's picture of gravity is not the correct one and does not appear in Einstein's laws, which are written to be coordinate independent so involve no reference to any fictitious forces. Einstein's laws do reference gravity, in the form of that which determines the inertial paths-- he simply does not regard the inertial paths as obvious. You need to specify the inertial paths, but you do not need to specify that you are using inertial coordinates-- you don't need to make the inertial paths look like straight lines in your coordinates, Einstein's approach is coordinate independent so you can make inertial paths look like anything you want in your coordinate language, but you do have to know what those paths are, and that's what gravity tells you. The equivalence principle was Einstein realizing that Newton was treating gravity wrong because he was (without realizing it) mistaking it for a fictitious force, when in fact it is not a fictitious force and cannot correctly be treated that way (globally), but it acts (locally) so much like a fictitious force that Newton could mistakenly treat it like one.
    Last edited by Ken G; 2018-May-08 at 06:35 PM.

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    Quote Originally Posted by grant hutchison View Post

    I'm going to write that one down.
    Nothing you didn't already know!

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    Quote Originally Posted by Ken G View Post
    Nothing you didn't already know!
    But said in a way that picks out the essence of the difference. Rem acu tetigisti, as Jeeves would have said.

    Grant Hutchison

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    Ah yes, rem acu tango, "it takes two to tango." You know, Newton's third law.

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