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What is intuitevely the field part of GR equations?

  1. Jan 15, 2014 #1
    I never seen any intuitive interpretation of the LHS of the equations.
    I realize the question is a tall order.
    But i think something can be extracted.

    -----------------------------------

    *Is this statement correct?

    The true information content of GR is :
    1)The non covariant law of newton reinterpreted as a curvature theory. A GR-like simple noncovariant model of gravity. Valid for slow speeds and weak field
    2) SR in curved space time.

    Lets put it differently

    lets split the metric
    g = η + X
    now, interpret it, that we have the field X inside flat space (η minkosky metric = flat space)
    Its not a crack pot theory, its just a stricked mathematical identity.
    Its an equivalent reformulation, absolutely nothing changes.
    Its easier to visualize stuff in flat space. Rather then curvature in 4D space time....
    In the end of the day, saying that gravity is the curvature of space time, is just a philosophical statement. Here's a perfect bijection, between curved space GR, and "flat space GR".

    Again, its just a reformulation of GR, just for pedagogical reasons. I'm with in the rules of the Forum.
    With this small reinterpretation of the GR.... I'll call it "Flat GR" (FGR)

    *Is this statement correct?
    The true information content of FGR is
    1)newton's law
    2)SR

    ------------------------------------

    *If yes. Can you explain, how this happen in the derivation of the GR equations??

    --------------------------------

    It is meant here by true information content. The analogy with How Electromagnetism is derived from conlombs law and SR. This is done by considering a reference frame, were Comlombs law is correct, then change reference frame. Because coulombs law, is not covariant, it will get messed up. The new messed up comlomb law, is added to the normal comlombs law as a system of equations, and a new source element is define for it. Now the system of equations are covariant.
    You apply the same logic on newton with SR


    *If you have, just one of the 10 equations, the 00 equation. You can derive the other 9 from the first, with just certain Lorentz transformations and differentiation. Right?

    ------------------------------------

    About non linearity: It is meant, that since SR equates energy and mass, and newton law is about the attraction of masses, it follows, that the gravitational field acts on it self also. Making it nonlinear.

    About the stress energy tensor: The Raw electric charge, by it self, has the only propety, that it can't be created or destroyed, you just need to keep the total charge constant. The propertis of mass however, is basically all of mechanics, that the source is the stress energy tensor, simply says, that mass follows the laws of mechanics.

    The elements of the stress energy tensor are energy, momentum, force and tork.
    The continuity equation of the stress energy tensor, is just all of mechanics....
    This is why we need to use that as the source. Its not just random

    --------------------------

    *About FGR:
    Are they some interpretations, about what the elements of X stand for? (potentials?? gravitomagnetism??? torsions ???)
    Interpretations, about what the equations of X represent?

    In general reinterpret some stuff of FGR
    We need to travel along a closed curve, in order to detect curvature. That resembles a loop current in a magnetic field.
    What about the flat Riemann and ricci tensors?
     
  2. jcsd
  3. Jan 15, 2014 #2

    PeterDonis

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    Do you mean the LHS of the Einstein Field Equations? I.e., the Einstein Tensor?

    $$
    G_{ab} = R_{ab} - \frac{1}{2} g_{ab} R
    $$

    I don't know that there is any simple intuitive interpretation of what that tensor means; the intuitive interpretations I've seen are of the Ricci tensor, ##R_{ab}##. For an example of that, see John Baez' article on the meaning of Einstein's Equation, here:

    http://math.ucr.edu/home/baez/einstein/

    The main reason that the Einstein tensor is what appears on the LHS of the Einstein Field Equation is that it obeys the contracted Bianchi identities, i.e., its covariant divergence is identically zero. This means the covariant divergence of the RHS of the field equation must also be zero, which ensures local stress-energy conservation. Misner, Thorne, and Wheeler, in their classic GR textbook, talk about the geometric meaning of the Bianchi identities, as expressing the fact that "the boundary of a boundary is zero", but I haven't found a good discussion of that online.

    1) is not correct, for at least two reasons. First, GR is covariant. Second, GR contains more information than just Newton's laws; Newton's laws are just what comes out in a particular approximation (slow speeds and weak field) from one particular solution of the EFE (the Schwarzschild solution for a vacuum, spherically symmetric spacetime). The EFE has many other solutions with different properties.

    2) is more or less correct: GR can be thought of as a generalization of SR that allows spacetime to be curved. But this is *much* more general than just adding Newton's laws to SR.

    Within its domain of validity, it's a mathematical identity, yes. But that domain is narrower than you might think; it imposes some very strict conditions on the spacetime:

    * The "correction" X to the flat metric can't be of the same order of magnitude as the flat metric. Many solutions to the EFE contain regions where this is not the case; for example, close to the event horizon of a black hole.

    * The spacetime must be asymptotically flat, i.e., at large enough distances from the origin the "correction" X must go to zero. Many solutions to the EFE are not asymptotically flat: for example, the FRW solutions used in cosmology.

    * The spacetime must have topology R^4, as Minkowski spacetime does. Many solutions of the EFE have other topologies (for example, the Schwarzschild solution has topology R^2 x S^2).

    So once again, what you are describing is not "GR" in general; it's a particular approximation that's only valid for a particular, restricted class of solutions.

    Within its domain of validity, yes. See above.

    Yes, but there's no guarantee that a physically correct theory must be easy for humans to visualize.

    No, it isn't, because curvature of spacetime can be measured; it's just tidal gravity. You could say that calling tidal gravity "curvature of spacetime" is a philosophical statement, I suppose, but that's a question of terminology, not physics.

    No, it isn't a perfect bijection; "curved space GR" includes many solutions that "flat space GR" does not. See above.

    This is not correct even within the limits of FGR as you've defined them, again for at least two reasons. First, Newton's law of gravity is inconsistent with SR; you can't make a consistent theory out of those two elements. So to build an FGR theory at all, you have to take some other route. Second, FGR includes effects that Newton's law of gravity does not; two examples are gravitational waves and gravitomagnetism (or at least the weak-field version of it).

    Can you give a reference for this "derivation" of EM from Coulomb's law and SR? Some people may use this as a heuristic understanding, but this does not look like the actual derivations of Maxwell's Equations that I'm familiar with.

    Not really. I have seen some statements that sort of sound like this (for example, Kip Thorne in Black Holes and Time Warps, IIRC, says something sort of like this), but you have to be careful in interpreting what they mean.

    In the general case, the EFE is 10 independent equations. However, there are ways of "repackaging" the 10 independent equations so they look different. One way, for example, is to take one of the equations, say the 00 equation, and impose the condition that it must hold in *every* local inertial frame at a given spacetime event. Since the local Lorentz transformations at a given event form a 6-parameter group, this gives you 6 independent equations, which can be used instead of 6 of the EFE components. Plus, you can always impose the condition that the covariant divergence of the EFE must be zero, which gives 4 independent equations; so now we have 10 independent equations, but they're a different (though mathematically equivalent) set of equations from the 10 we started with. We haven't reduced the number of equations; we've just reformulated them in a different way.

    In the full theory of GR, yes, gravity is nonlinear, because the EFE is nonlinear. But if you're using a linear approximation to GR (such as the "linearized" framework that is often used to study gravitational waves, or the weak-field, slow-motion limit), you are ignoring the nonlinear effects, so *in that approximation* gravity is not nonlinear.
     
  4. Jan 15, 2014 #3

    Mentz114

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    I must object to this because the curvature is precisely, geometrically defined.

    Also, the linearised theory you call 'FGR' is not a replacement for the full equations.

    There are other theories of gravity that give the same predictions as GR .

    1. Tele-parallel gravity, where gravity arises (like the Lorentz force) from enforcing a local symmetry ( invariance under space-time translations)

    See for instance arXiv:gr-qc/0603122v1 30 Mar 2006 http://arxiv.org/abs/gr-qc/0603122

    2. Field theory gravity (FTG) where the background is Minkowski and the fields are rank-2 symmetric tensors

    See for instance arXiv:gr-qc/9912003 v1 1 Dec 1999 http://arxiv.org/abs/gr-qc/9912003
     
  5. Jan 15, 2014 #4

    PeterDonis

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    I'm not sure this is actually true, at least for the second theory (FTG). FTG, as noted in the abstract of the paper you linked to, matches GR predictions for weak fields but does not necessarily do so for strong fields. Also, FTG is subject to the same issues I listed in my previous post, concerning asymptotic flatness, global topology, and the size of the "correction" of the field to the background Minkowski metric. (There is also the issue that in FTG the background Minkowski metric is unobservable; all actual distances and times are given using the curved metric produced by the field.)

    I'm not familiar enough with teleparallel gravity to know to what extent it is subject to issues like these, when compared with GR.
     
  6. Jan 15, 2014 #5
    X = g - η
    that seams well defined all the time. o_O
    You can even squeeze in a worm hole....
    Its like representing the curved surface of the earth on a 2D map. No?
    There are even infinite ways in doing so.
    I even read once, about the abstract bijection (not differentiable), between the surface of a square and a line. It had a name that a forget, it used the diagonalization of cantor.....
    You can even make a theory of the universe, with the earth been flat.....
    How you actually do this, the correct way???

    This is not linearized gravity. I don't try to cut out high order stuff.....
    Linearized gravity is a ploy to cut out small stuff...
    I keep here ALL of X. The space time is flat, and ALL of X is a "thing" inside flat space.
    Its not linearized gravity, its flattened up gravity, what sticks out is converted in a "thing" inside space time.
    When you'll plug that in the equation, you don't drop the products of X, or what ever other approximation.
    Nothing really changed.....

    If newton was covariant, that would defeat the whole tric with taking lorentz transformations of it. They would just give back it self. How you think EM is derived from conlombs law???
    http://en.wikipedia.org/wiki/Relativistic_electromagnetism

    I think John Baez simply drops the ricci scallar part.
    Explain, why the equations should describe anything in the real world????
    Why should G actually work??? I don't see from the derivation why it should concur with reality.
    Does the derivation insures that G should be unique??? Why start with the ricci curvature, it could have been something else, so that the divergence is zero....

    "tidal gravity"
    You can explain that the newton way. No need of curvature here....

    "10 independent equations"
    The system of equations you rederive this way, are a corollary of the old one. It enters my definition of "same" equations. You could use your new system of equations, in place of the old one. You do realize, that one equation alone is not covariant right? This is what i'm trying to say about conlombs law and EM, but you got too literal again .... Yea, "you impose the condition" where is valid, and as you add equations, the validity of the system of equations expands, untill it covers everything.....
     
  7. Jan 15, 2014 #6

    jcsd

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    The issue of topology seems like the biggest problem and it's problem you can't get around. Whilst there are more flat spacetimes than Minkowski space, I believe unless the spacetime has a Euler characteristic of zero you're going to introduce arbitrary singularities by mapping it on to a flat space because the map won't be a homeomorphism.

    Also whilst I don't believe Pete's other conditions are a strict prohibition against re-writing the metric in the form of g = η + X, when they are not fulfilled you have the problem of which "η" to use and the choice it seems will be entirely arbitrary.
     
  8. Jan 15, 2014 #7

    PeterDonis

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    No, it isn't. Formally, yes, you can always subtract out ##\eta## from ##g##, but the result you get might not meet the requirements for being a valid metric. (For example, as a matrix, it might not have an inverse. This happens at the horizon of a black hole, for example.)

    Of course you can switch coordinate charts, meaning switching the definition of ##g##, in order to cover regions that can't be covered by the original chart. However, that also changes the definition of X, so you still don't have a single valid definition of X that covers the entire manifold. See below.

    Also, the actual observable is always ##g##; neither X nor ##\eta##, as you've defined them, are actual observables. Nor do you need either X or ##\eta## to make any physical predictions; you only need ##g##. So even if X and ##\eta## are well-defined mathematically, their physical interpretation is problematic, particularly in cases like the FRW spacetimes which are not asymptotically flat, so there is no region where X becomes small.

    Show your work, please.

    Yes, and with the same issues with there not necessarily being a single such map that covers the entire manifold. For example, it's impossible to represent the entire surface of a sphere, like the Earth, with a single 2-D map; any such map will have to leave out at least one point. (The most common such map, the Mercator projection, does not cover either of the poles.)

    Sure, you can also make a bijection between a line and the interior of a cube. What does that have to do with anything we're discussing here? You can do a lot of things mathematically that don't necessarily have meaningful physical interpretations.

    How do you know? You obviously haven't done it or seen it done, since you go right on to ask:

    How should I know? You're the one who said it could be done, not me. I don't think it's possible.

    In other words, you are not assuming that X is small. But then you need to deal with what happens when X isn't small.

    This may be your personal opinion, but it's not a very well-informed one. Linearized gravity has plenty of physical applications. A major one today is studying weak gravitational waves, which is a busy field since we are trying to detect GWs from astronomical events.

    In which case it isn't always well-defined. See above.

    No, this is not correct; even when X is well-defined, the spacetime--the actual physical spacetime--is not flat. The "background" ##\eta## is flat, but ##\eta## is unobservable. You need to get a better understanding of how the scheme you are talking about actually works.

    You can't take Lorentz transformations of Newton's Laws; they're not Lorentz invariant. You have to *change* the kinematics.

    This is a teaching strategy, not a rigorous derivation. Try again.

    Can you give any specific references that make you think this?

    Because that's what the experiments say.

    That's a problem with your intuition, not the derivation.

    Yes.

    Because the Ricci curvature is what appears in the Lorentz invariant action for gravity. See here:

    http://en.wikipedia.org/wiki/Einstein–Hilbert_action

    Textbooks like MTW go into more detail about what picks out that action, physically, as the right one.

    You can explain the tidal gravity around a static, spherically symmetric source like the Earth using Newton's law, yes. But there are other types of tidal gravity that can't be explained by Newton's law. For example, the observed accelerating expansion of the universe is a manifestation of a form of tidal gravity.

    All this is true, but I'm not sure what it has to do with what you were claiming. See below.

    Yes, but that means you can't just Lorentz transform "one equation alone", nor can you just differentiate it. You have to apply those operations to *all* the equations in order to keep the system self-consistent. So if all you have is one equation out of 10, you can't "derive" the other 9 just by Lorentz transforming and differentiating, which is what you were claiming. That's why I answered "not really" to that claim.

    I did make a bad choice of words with the phrase "impose the condition", however; see below.

    And it's not valid there, either, as far as I can tell. See above, and further comments below.

    "Impose the condition" was probably a bad choice of words, since it appears to have led you into a misunderstanding. I used that phrase in two places: regarding "taking one component", and regarding the covariant divergence.

    To take the second point first, because it's easier, the covariant divergence of the EFE is always zero; that's a mathematical identity. By "impose the condition" here I just mean use that identity to rearrange the system of equations. It's often helpful to do that to make the equations easier to solve. But it's not "imposing" any condition that doesn't already apply; you're not adding anything to the system, you're just using what's already true to help rearrange it.

    The other case, "taking one component" of the EFE, works the same way: we are not "imposing" any condition that's not already true, we're just using it to rearrange the equations. But how it works is a bit more complicated and I didn't describe it very well. Here's what I hope is a better description: we take the EFE,

    $$
    R_{ab} - \frac{1}{2} g_{ab} R = 8 \pi T_{ab}
    $$

    and look at a particular event. Then we pick a local inertial frame at that event, in which the metric, at that event (but *only* at that event--this is a local chart only, not a global chart), takes the Minkowski form, ##g_{ab} = \eta_{ab}##. In this local inertial frame, there will also be some particular unit timelike vector, ##u^a##, that picks out the "time axis" of the frame--i.e., the 4-velocity of an observer who is at rest, in that frame, at that event.

    Given all this, what I described before as "picking the 00 component" of the EFE is really contracting the EFE with the timelike 4-vector we picked as the "time axis" of our local inertial frame:

    $$
    R_{ab} u^a u^b - \frac{1}{2} g_{ab} R u^a u^b = 8 \pi T_{ab} u^a u^b
    $$

    But in the local inertial frame we picked, we have ##u^a = (1, 0, 0, 0)## (i.e., only the 0 component is present), and ##g_{00} = -1## (because the metric is Minkowski at our chosen event), so the contracted equation is just

    $$
    R_{00} + \frac{1}{2} R = 8 \pi T_{00}
    $$

    (Baez' article shifts the trace term to the RHS to get an equation for ##R_{00}## in terms of the components of ##T_{ab}##, but we don't need to go into that here.)

    When I said we "impose the condition" that the 00 component must hold in every inertial frame at our chosen event, what I really should have said is that the contracted equation I gave above must be valid in every local inertial frame at our chosen event. But all that means is that we can pick any timelike 4-vector ##u^a## we like as the "time axis" of our local inertial frame. Since any such timelike 4-vector at an event can be transformed into any other by a Lorentz transformation, this means that there must be a 6-parameter group of possible timelike 4-vectors at our chosen event, any of which we can contract with the EFE at that event. But that condition is already true by virtue of the EFE being covariant; it's not something we had to add in or "impose" (which is why "impose" was a bad choice of words on my part, as I noted above).

    Looking at what happens if we take our contracted equation in one local inertial frame, and then Lorentz transform it into another local inertial frame, will give us equations that tell us what contracting *other* 4-vectors with the EFE must result in. Another way of saying this is that these Lorentz transforms are telling us what other "components" of the EFE must look like. But as I noted before, we're not "deriving" the EFE just from one component; we're just looking at the system of equations from a different perspective, one in which the "degrees of freedom" in the equations come from the degrees of freedom in a local Lorentz transformation. Again, we're not adding anything to the system that wasn't already there.
     
  9. Jan 15, 2014 #8

    Dale

    Staff: Mentor

    We need a reference for this. I don't know of anyone who has done this.

    If any of the participants knows of such a reference please PM me and I can re-open the thread and we can discuss that reference specifically.
     
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