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I Tidal forces on a spring-mass system

  1. Feb 18, 2017 #1
    Say I have two masses connected by an unstretched massless spring at some height above a planet with a strong gravitational field.

    Once I let go, both masses would follow geodesics in spacetime towards the center of the planet. Because the masses travel radially, the spring would compress and store spring potential energy.

    Where does the spring energy come from? It doesn't come from a force since gravity isn't a force in GR.
    Classically, we can say that the cosine of the gravitational force vectors compresses the spring. Classically, it alines with F=-dU/dt. But in GR, it seems that the compression just happens out of nowhere, simply because space gets narrower.

    I thought about the potential energy due to height being converted into kinetic and spring energy as an explanation. But this sounds too classical for me.

    I also thought of the energy coming from spacetime itself. But I'm not so sure. If this were the case, then the surrounding spacetime would lose energy. This might make sense, as the mass falls, the spacetime it leaves behind becomes slightly less warped due to the two masses simply not being there. But then again, it enters a new region, and the masses simply warps that region as well. This reminds me somewhat of the rocket propulsion problem but instead of trading chemical PE for motion, it trades potential energy due to position in height for spring compression energy and kinetic energy. This reasoning is still seems a bit classical, so I'm not certain.
  2. jcsd
  3. Feb 18, 2017 #2
    Would that be true - is towards the centre of the planet?
  4. Feb 18, 2017 #3
    Hmm well, the spring pushes back, so I guess the force on the masses from the spring would make the masses deviate from geodesic paths

    but initially, at the instant of letting go, it should be on geodesics. Also, if the spacetime warping is large enough, shouldn't the masses travel close to geodesic paths, so close to radial?
  5. Feb 18, 2017 #4
    Now, is that the answer. I don't know wrt to GR.
    Interesting question that you have proposed.
  6. Feb 18, 2017 #5


    Staff: Mentor

    This is a good question.

    Remember why gravity is considered to not be a force: it can be transformed away locally by a coordinate transform. One key word in that is "locally" which means over a region small enough that tidal effects are negligible. By construction, your scenario is large enough that tidal effects cannot be neglected, and the tidal forces cannot be transformed away.

    Additionally, as long as the spacetime is asymptotically flat you can define a conserved energy.
  7. Feb 18, 2017 #6


    Staff: Mentor

    Actually, the spring would be stretched, not compressed; radial tidal gravity causes free-falling objects to separate, not converge.

    Also, the masses would not remain in free fall, because the spring would pull on them.

    From gravitational potential energy. At least, that's the best simple heuristic. It's no different than asking where the kinetic energy comes from when you drop a rock off a cliff.
  8. Feb 19, 2017 #7
    Ah, so gravity here is not fictitious. The equivalence principle doesn't hold.

    Since gravity cannot be transformed away, then what is gravity? It isn't a fundamental force in GR, its just movement in curved spacetime. It's hard to imagine gravity being made of other fundamental forces here. Is the tidal force from the constraints of the curved space?
  9. Feb 19, 2017 #8
    I meant seperated horizontally. So the left and right ones in the diagram, not the top and bottom ones.
    Screen Shot 2017-02-18 at 10.04.24 PM.png

    Yeah, I understand that it is from potential energy. But in the classical case, the energy is stored by a force from within the system.

    If I consider the two masses and the spring in between as an isolated system while excluding everything else, we see the spring compress, yet something external should be doing work on the system.
  10. Feb 19, 2017 #9


    Staff: Mentor

    Ah, ok.

    No, you won't, because excluding everything else means excluding the Earth and its tidal gravity. In the presence of tidal gravity, the masses and the spring are not an "isolated system" any more, and reasoning as if they were will give wrong answers.
  11. Feb 19, 2017 #10
    I meant if we designate the system as mass and spring only, the earth would be external to the system and should influence it. Isolated system was a bad choice of words.

    If I drop a mass, I can approach it in two ways. Conservation where the mass and earth are part of the same system or the mass as its own system and the earth being external, doing work on the mass.
    Last edited: Feb 19, 2017
  12. Feb 19, 2017 #11


    Staff: Mentor

    That is correct.
    Spacetime curvature is essentially the same thing as tidal gravity.
  13. Feb 19, 2017 #12


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    In order to answer "where the energy comes from", we'd have to localize the energy of the gravitational field. To answer "where", energy would need to have a location in the first place.

    But it is not in general possible to localize gravitational energy. Therefore we can't answer the "where" question. For a textbook reference, try google search for the exact phrase "Why the energy of the gravitational field cannot be localized". You should get a hit on a textbook, Misner, Thorne & Wheeler's "Gravitation" from google books. This is the title of section $20.4 in said textbook.

    So the best answer we can give you is the one that's in the textbooks, the one that says that, contrary to your assumptions, it's not generally possible to localize gravitational energy.

    This is different from electromagnetism in flat space-time, where we can localize energy in the electromagnetic (EM) field. See all the threads about the energy, and rest mass, of a charge capacitor, for an example where we can localize energy in a charged capacitor, and that energy is stored in the field between the plates.

    (Some posters may not be convinced of this, but that's the textbook answer. I don't have a reference handy, alas, but it's widely known.)

    Unfortunately while this approach to energy being stored in fields works fine for E&M, it doesn't work for gravity, as we can't define a "field" of gravity that stores energy in the same manner as we do for electromagnetism.
  14. Feb 19, 2017 #13


    Staff: Mentor

    Yes, although as long as the spacetime is asymptotically flat we can say that there is a conserved amount of energy, even if we can't say where it is located.
  15. Feb 19, 2017 #14


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    Also, if metric is stationary, you can specify a well defined potential function; if the masses in this example are considered small enough to ignore their own fields (test masses), then the potential energy analysis is well posed for this simple case. This is also textbook.
  16. Feb 20, 2017 #15


    Staff: Mentor

    But if you use the second approach, then the answer to your question "where does the energy come from?" is still "gravitational potential energy". That's where the Earth gets the energy that it applies as work to the mass-spring system.
  17. Feb 20, 2017 #16
    Well, it makes sense in that for a system consisting of mass, earth, we can define a quantity called potential energy and observe that this quantity is converted to other types of energy.

    But for the second approach, how can it come from gravitational potential energy if the system is single object? Potential energy is a property of multi body systems.

    Classically, we can consider a mass as a one object system and use the work energy theorem. W=change in KE. It turns out that W=F*distance gives raise to mgh which merely turns out to be the potential energy for the two body system of earth and mass. But it doesn't logically have to be that way. We can consider the earth and mass as a system, and define a potential energy called mgh and observe that this is numerically the same as the kinetic energy. I feel like the only reason both approaches give the same answer is that the existence of a force allows the work to be computed as mgh and that it is this coincidence that allows the same results.

    But without a defined force, can both approaches work in GR? Is work meaningful?
  18. Feb 20, 2017 #17
    For the capacitors, I can say that for a system consisting of both plates and the field between them, the potential energy of the system comes from the field, being that the formula has E and d. Even though the PE is a property of the system as a whole, it is in some sense localized since it has d in the formula. It's like how mgh doesn't literally give the potential energy of a object at a height, but rather the energy due to the configuration of the whole system, which turns out to depend on h.

    So in GR, if we define a system consisting of masses earth and spacetime itself, then the formula for potential energy wouldn't have terms that describe location?

    Are gravitational fields fictitious in GR? So the only thing is the curvature of spacetime, which is not a vector field with well defined direction. Then it would make sense that energy is not the same as for the capacitors.
  19. Feb 21, 2017 #18


    Staff: Mentor

    Potential energy is a property of systems with internal degrees of freedom and a time independent Lagrangian.
  20. Feb 21, 2017 #19
    I recall in classical mechanics, if a Lagrangian doesn't have a parameter, then certain aspects are conserved with respect to that parameter. So the Lagrangian in CM is L=K-V where K=.5mv^2 and V=mgh, none of which depends explictly on time. Then the lagrangian is time independent and energy is conserved.(i.e since L is constant, V+const=K for all time. L=const is just the change in energy. )

    What do you mean by degrees of freedom? Do you mean the number of generalized coordinates has to be finite?
    Last edited: Feb 21, 2017
  21. Feb 21, 2017 #20


    Staff: Mentor

    Hmm, I may have stated it wrong.

    What I meant is that the internal configuration of the system must be able to change. For example, in a hydrogen atom the orbital is an internal configuration.
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