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Formulation of Einstein-Hilbert Action

  1. Nov 21, 2009 #1
    I am trying to understand where the Einstein-Hilbert action comes from.

    Neglecting a constant multiplier, Wikipedia expresses this action as the integral over all spacetime of

    - (R + L)* sqrt(-g)​

    where R is the Ricci scalar, g is the determinant of the metric tensor, and L describes the matter fields.
    (L is, I believe, though Wiki doesn't exactly say this, the stress-energy tensor multiplied by a Lagrangian multiplier.
    By setting L to 0, you get the vacuum solution for the field equations.)

    By setting variation of this "action" to 0, the field equations of GR result.

    My problem is I don't understand how to connect the Einstein-Hilbert action with the underlying physics which I take to be

    #1 spacetime is a locally Minkowskiian manifold where freely falling particles of matter move along geodesics and

    #2 for every non-rotating frame moving with a freely falling particle the laws of physics are the same, i.e., they can be expressed using coordinate-free scalars, vectors, tensors, and covariant derivatives.

    All this is to the best of my knowledge and belief.
  2. jcsd
  3. Nov 21, 2009 #2


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    From the locally Minkowskian condition you deduce the metric is a dynamical field with fixed signature 2.

    Then you add a postulate that the dynamics of the metric field can be derived from a Lagrangian.

    This will give you a Lagrangian that in principle contains many, many terms (http://arxiv.org/abs/0911.3165, Eq 1, the "completely general generally covariant theory of gravitation"). By luck, we only need the first term!

    At present the theory has no matter, so there is no freely falling matter travelling on geodesics. To add matter which respects the equivalence principle, we have to add a matter Lagrangian, and put some conditions on how the metric enters the matter Lagragian. Try the discussion at http://arxiv.org/abs/0707.2748.
    Last edited: Nov 21, 2009
  4. Nov 21, 2009 #3


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    Ok, so you know that the metric in GR replaces the notion of a gravitational potential in Newtonian gravity. Then you want to formulate the Einstein equations via an action principle. This action should be a scalar under general coordinate transformations and constructed from the metric. First a remark about integration on manifolds; the measure

    d^n x

    gets replaced by the measure

    \sqrt{|g|}d^n x

    which is proven in various texts. Here |g| is the absolute value of the determinant of the metric. This measure is a scalar under coordinate transformations, which you can also prove by taking the determinant of the transformation law of the metric and noting that dx transforms with a Jacobian.

    Then you have to find a Lagrangian which is a scalar constructed from the metric. Now, at every point on a manifold you can choose a metric which is constant, with vanishing first derivatives (but not second derivatives in general! ) So if we would construct a Lagrangian with only the metric and its first derivatives, we can choose a coordinate system in which the metric becomes constant and the first order derivatives vanish. Being a scalar, the action obtains the same form in EVERY coordinate system.

    So, we need second order derivatives of the metric to describe any dynamics at all. The simplest scalar of the metric and derivatives up to second order derivatives is the Ricci scalar. So the proposal is

    S[g_{\mu\nu}] = \int_M \sqrt{|g|} R d^n x

    and luckily, this gives the Einstein vacuum equations obtained from pure physical reasoning. The funny thing is that this action gives the Euler characteristic (a topological invariant) of M for n=2, so in this case it doesn't give dynamics. For n=3, the equations of motion only describes "global gravity"; in three dimensions the Ricci tensor and Riemann tensor have the same number of independent components, so in vacuum a vanishing Ricci tensor means a vanishing Riemann tensor which means flat spacetime. The addition of a cosmological constant, which is allowed by energy-momentum conservation, makes things a little more exciting; it allows for constant curvature. But we still can't have things like gravitational waves.

    It's only in n=4 and beyond that we can get gravitational waves (and when we quantize things: gravitons!).
  5. Nov 29, 2009 #4
    Thank you both for your helpful replies. I admit I am more than a little surprised to find out there isn't a more direct justification for the E-H action. I will carefully read the papers referred to by atyy and report back if I think I find anything of general interest.
  6. Nov 29, 2009 #5


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    What kind of justification did you expect? And why do you think it's not "that much"?
  7. Nov 30, 2009 #6
    I was expecting that someone would have clear definitions of the basic principles, expressions of them as mathematical axioms, and a proof that the evolution of the universe must, because of those axioms, follow a path that minimizes the Einstein-Hilbert Action subject to certain simplifying assumptions being made (disregarding electromagnetic forces, etc.).

    Wikipedia has an article on GR and on the E-H action, but nothing connecting the two. Looking at the papers recommended by ATYY, it appears that I am not going to find the missing link there either, but I haven't really looked at either of these deeply yet.

    In a sense, then, at this point it appears to me that the E-H action IS the axiomatic basis for GR and its justification is, of course, that it leads to predictions that work, (although perhaps there is a problem as you get outside our galaxy with "dark matter", etc.). Alternatively, you could say that the EFE constitute the axioms and that Einstein gave arguments for them that connect up well with the basic principles. I don't know whether this is true, but if you can shed any further light on the subject, please do so.
  8. Nov 30, 2009 #7
    You do need to postulate the EFE and that your spacetime is smooth, curved, four dimensional, and has (-,+,+,+) signature. From that it follows that it must be locally Minkowskian. #1 (travelling along geodesics) can be proved if we postulate the appropriate action of a particle moving through spacetime, or we can go deeper and write down field equations for the underlying field (say, Dirac lagrangian) and, after some algebra, we'd prove that "particles" of the field move along geodesics.

    Then all you need is something to describe how exactly the spacetime is curved by matter. E-H is the simplest possible action that happens to work. It is definitely NOT put in axiomatically, there may be other terms, but laws of physics are such that we're unable to measure other possible terms with any degree of precision, so we just ignore them for the time being.
  9. Dec 1, 2009 #8


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    The way I inderstand it is that you view the Hilbert action axiomatically as the action that gives you the equations of motion for the gravitational field, on basis of the following facts:

    * It gives the same equations of motion as "covariantizing the Poisson equation", like Einstein did
    *It is the simplest action which fullfills all the requirements which your action has to have (second order derivatives of the metric, scalar, and simplicity)

    Ofcourse, you can add all kinds of covariant terms to your Hilbert action, and that's also what string theory predicts: the Hilbert action is just the leading term in an expansion.

    Maybe you should compare it to the Klein Gordon equation: what is the justification for the action which gives you the Klein Gordon equation?

    * It has a frist derivative of the field, which gives you second order equations of motion
    * It's Lorentz covariant
    * The Schrodinger equation can be obtained from the equations of motion

    Ofcourse, you can add higher derivative terms, but then you also have to impose more boundary conditions or introduce more boundary terms in the action (just like you can do for the Hilbert action by adding the Gibbons–Hawking–York boundary term)
  10. Dec 1, 2009 #9
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  11. Dec 3, 2009 #10


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    If I read your original post and the above quote correctly, it seems to me that you are regarding the E-H action as being, in some sense, more fundamental than Einstein's equations. I would assume that you also probably regard the classical L = T -V action as more fundamental that Newton's equation of motion, F = Ma. It might interest you to know that in the 1970s and 1980s there were quite a few papers published on "The Inverse Problem of the Calculus of Variations". Succinctly, the inverse problem answers the question "When I have a system of differential equations, how can I find an action, the variation of which yields those equations?" Given that we can pose this question, and answer it, it seems to me that the existence of an action is no more fundamental than the existence of a well motivated field equation, or equation of motion.

    If anyone is interested in a list of references to the inverse problem, let me know by PM, or post a note here and I will dig them out of my "archives" and post them in this thread.
  12. Dec 3, 2009 #11


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    Yes, that's interesting. Is there a way to enumerate and classify all possible actions? For example, for GR there is the EH action, the Palatini action, the Holst action, and many others. Also how can one tell if there is an action or not, eg. I think the Kadar-Parisi-Zhang equation has no corresponding action?
  13. Dec 4, 2009 #12

    I am interrested by your references on the inverse problem.
    I had asked for such material in a old thread and I am still curious. See there:


    I hope this is not too much work to dig them out!

  14. Dec 4, 2009 #13
    Also keep in mind that E-H action is not really the complete and real action (at least not in the sense that the integral of [itex]-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}[/itex] is the complete action of the electromagnetic field). There's a big elephant in the room known as "the surface term" that makes E-H action relatively useless for practical calculations, and only of interest to perturbative people.
  15. Dec 4, 2009 #14


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    I seriously doubt that there is a way to enumerate and classify all possible actions. In classical mechanics there are an infinite number of Lagrangians yielding the same equations of motion all of which differ by the total time derivative of an arbitrary function of the generalized coordinates and the time. I'm pretty sure this is mentioned in Goldstein's [tex] \textit Classical Mechanics [/tex], but I can't seem to locate the page. This property generalizes to systems of equations such as Maxwell's or Einstein's equations.

    The answer to your second question can be found in the references that I will give in my next post. In brief, differential operators must satisfy certain conditions in order for them to allow the use of a simple algorithm to construct a Lagrangian. However, some (Bessel's equation comes to mind) can be obtained from a variational principle when multiplied by an "integrating factor". It turns out that if you don't mind introducing an "adjoint variable" (see the first reference in my next post) you can obtain any differential equation from a variational principle.
  16. Dec 4, 2009 #15


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    Thanks for this and the next post!
  17. Dec 4, 2009 #16


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    Thank you for the link to your thread.

    A very readable starting point for the inverse problem of the calculus of variations is,

    Atherton, R.W., and Homsy, G.M. "On the existence and formulation of variational principle for non-linear differential equations", Studies in Applied Mathematics, Vol. 54, p 31, (1975).

    A paper that generalizes the above is,

    Ahner, H.F. and Moose, A.E/ "Covariant inverse problem of the calculus of variations" J. Math. Physics, Vol 18, p. 1367 (1977).

    This book has a very readable discussion in one of the later chapters,

    Finlayson, B.A., "The Method of Weighted Residuals and Variational Principles", Academic Press, New York, (1972).

    The following chapter in a book has a brief history and provides some context for different approaches,

    Tonti, E. "Inverse Problem: Its General Solution", in Rassias, G.M., and Rassis, T, "Differential Geometry, Calculus of Variations, and Their Applications", Marcel Dekker, New York (1985)

    There is quite a bit of literature on the subject. Here are two more from the Journal of Math Physics,,

    Hojman, Sergio and Urrrtia, Luis F. "On the inverse problem of the calculus of variations", Vol 22, p1896 (1981).


    Bamopi, F and Morro, A., "The inverse problem of the calculus of variations applied to continuum physics", Vol 23, p2312 (1984).

    There were numerous articles published in the Hadronic Journal between 1978 and 1983 on the subject. One of these is

    Tonti, Enzo "A general solution of the inverse problem of the calculus of variations", Hadronic Journal, Vol 5, p1404 (1982)

    Hope this helps.
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