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How does curved spacetime result in an effective gravitational force?

  1. Nov 18, 2008 #1
    So... I am told that mass warps the regional fabric of space and time and that any massive object in that region of curved spacetime will feel this curvature and essentially 'fall into the curvature'. The problem is that I dont exactly understand how curved spacetime implies a gravitational force.

    At this point my friend goes off with the analogy of a bowling ball on a trampoline. The reason that this analogy doesnt do it for me is that is only an analogy and can only be taken so far. I can 'see' how the analogy works because I have taken mechanics and understand that a spherical object an incline with a component of force pointing down the incline will pull it down the incline. But when you get this literal with the analogy, it doesnt really commute to GR, it basically doesnt work beyond the purely abstract explanation. (This is obviously because the trampoline isnt the same thing as spacetime.) Plus the analogy seems, to me, to beg the question; the analogy uses the gravitational phenomenon to explain the gravitational phenomenon...

    So again, if a graviton won't do, what is the mechanism that causes warped spacetime to create an effective force (maybe it can be called a virtual force)?

    And I suppose that leads me to wonder: Does Einsteins relativity require the NON-existence of a particle-interaction explanation of gravity? If we soon find a graviton, would this disprove GR?
  2. jcsd
  3. Nov 18, 2008 #2


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    There is no gravitational force in GR. Free falling objects are force free, and therefore moving straight in a curved space time. See here how locally straight paths in a curved space time (right diagram) translate to the curved paths we perceive by assuming a flat space time (left diagram):

    You are not the first one to notice this flaw:
  4. Nov 18, 2008 #3


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    To make an analogy, consider two airplanes at the equator, one at the prime meridian (0° longitude) and the other at 10° longitude. They each fly northwards along their respective longitude lines. As far as each plane is concerned, it is flying in a "straight line," with a constant heading. Each plane's pilot does not have to steer the plane except perhaps to compensate for weather conditions.

    At the equator, they start out flying parallel to each other, but as they proceed northward, their paths gradually converge until they collide at the North Pole. If the pilots didn't know the Earth is actually a sphere, but assume it's flat instead, they might be tempted to postulate a fundamental "convergence force" that causes initially-parallel straight paths to converge.
  5. Nov 18, 2008 #4
    No one understands "exactly" how any of the four forces works! Believe curved spactime.... or not, you choose. But it's a useful tool for predictions and understanding phenomena so keep it in mind, but don't be blind to other explanations.

    We have mathematical descriptions, we have analogies, and theories but "exactly" how anything works is usually at the fundamental level a mystery. We don't even know precisely what mass,energy,space and time are....how could we understand gravity "exactly" or "perfectly"....not yet.

    The rubber sheet and bowling ball is ONE way of visualizing gravity; Eintein's general relativity goes further, does a better job than Newton's theories...but GR breaks down at high mass densities and massive curvature...so it's not the the final answer either...nor is quantum gravity, yet another perspective, a complete solution yet....but it offers different insights, gravitons for example.

    Experimental "proof" of the graviton would merely reflect the particle nature of another force field (wave) and would not disprove GR. Gravitons are a quantum mechanical construct while GR is a smooth classical wave formulation...again, two different views of similar phenomena. Why do masses even have gravitational effects??...nobody knows that either but it could be due to higgs bosons, higgs fields, yet another attempt to plumb the mysteries of how things are put together. Or maybe, according to string theory, certain vibrational patterns "resonate" with spacetime curving them....and those curvatures are reflected as mass!!

    Two non standard ideas:
    So maybe something like a fundamental version of a Higgs field is what "really" causes gravity!!! Or could it be that acceleration causes gravity??

    We could also conjecture that at the time of the "bang", big or bounce or quantum fluctuation, or whatever it was, when a phase transition moved from an unstable high energy environment to the stable low one we observe today, out popped mass,energy,time,force,etc; in other "beginnings" maybe gravity did not appear...hence no universe formed, it was a "dud", an evolutionary dead end.
  6. Nov 18, 2008 #5
    The easiest way I imagine it is this: Imagine a box filled completely with water, 100% no air. No room for the water to move, even though I realize at the atomic level it is "in motion". If you inflate a balloon inside this box, it's going to increase the water pressure hence putting pressure on the balloon. Bodies in space do the same thing. The pressure is spacetime/subspace/dark matter whatever you want to call it pushing on the object.
  7. Nov 19, 2008 #6
    Awesome explanations, the fog is beginning to clear.

    So is it safe to assume the most of the rest of physics fits into this GR perspective in a semi-neat way? I guess Einstein/Campbell showed that EM is compatible... what kind of examples of large mass/extreme curvature result in a GR breakdown?
  8. Nov 19, 2008 #7
    I just thought I'd add something as I understand it from a book Einstein wrote entitled "Relativity": Einstein states that light passing a massive body through space(time) exerts and feels a gravitational influence, and is bent in accordance with that, BUT ALSO bent in additional accordance with spacetime curvature. Meaning that the presence of "mass" is BOTH the source of the 50% 'bending' of light that is gravitationally governed and the 50% 'bending' represented by the modified direction (that light 'interprets' as straight) that is spacetime warping in light's path.

  9. Nov 20, 2008 #8


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    Sorry, there is no "ALSO" about it. "Feels a gravitational influence" and "bent in accordance with spacetime curvature" are two different ways of describing the same effect.

    It's true that Einstein's theory predicts double the bending that Newton's theory predicts. But Einstein's theory replaces Newton's theory, it doesn't supplement it. So the bending is 100% Einstein's and 0% Newton's.
  10. Nov 20, 2008 #9
    Dr Greg,

    You said what I thought I'd said. You are correct in that the 2 components can't be disentangled any more than can space and time. I should've just quoted Einstein referring to deflection of light by a gravitational field:

    "It may be added that, according to the theory, half of this deflection is produced by the Newtonian field of attraction of the sun, and the other half by the geometrical modification ('curvature') of space caused by the sun." ("Relativity" by A. Einstein, p. 124-5).

    I just meant to remark on how strange it is that 50% of the deflection is due to each component.

  11. Nov 20, 2008 #10


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    Note that Einstein speaks about the other half of bending being caused by curvature of space (not spacetime as you wrote in your previous post). This is essential because curvature of spacetime causes the entire 100% of the bending not just 50%.

    I doesn't really make sense to split the effect this way. It is often mentioned for historical reasons, as Einstein initially did not consider space to be curved too, and replaced Newtons law, with curvature of the time dimension only. But in a manifold like spacetime, you cannot have just one curved dimension.
  12. Nov 20, 2008 #11
    It is fairly easy to do the trampoline model with one gravity source and respondents to that force (ie light or a smaller ball).

    How do you do the trampoline model with two gravity sources and respondents?
    Do you find the centre of gravity between them, treat them as a bigger combined ball for the purposes of space-time curvature and make that the lowest dip point?

    The centre of gravity would be closer to the larger object.
    So on a trampoline this would make the slope between the bigger ball and the centre of gravity steeper then the slope between the smaller ball and the centre of gravity.
    But the bigger ball moves slower towards the centre of gravity so this should be the opposite way round.

    How would you model a two body example using the trampoline analogy?
    Or is their a better physical analogy to explain the idea?
  13. Nov 20, 2008 #12


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    But it is not a model of gravity in GR, as the OP noticed and was already explained. The two body problem is tricky in GR and I don't know if there is a exact solution for it, like the Schwarzschild solution for one body.
  14. Nov 21, 2008 #13


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    Going back to the original question. 'Space-time is curved' is another way of saying that clocks and rulers change from one place to another. Consider a simple solution of Schroedingers equation for a particle not in a field,

    [tex]\psi(x,t)=Aexp\left[\frac{i}{h}\left(Et+px\right)\right] = Aexp\left[i\left(\omega t+kx\right)\right][/tex]

    where A is a normalization constant. It is clear the the operator -id/dx gives the momentum, p. If space-time is curved then [itex]\omega[/itex] depends on x, and the momentum operator gives an extra, time-dependent term, meaning that the particle is gaining ( or losing ) momentum.

    [tex] -i\frac{\partial}{\partial x}\psi(x,t) = \left(k+\frac{\partial\omega}{\partial x}t\right)\psi(x,t) [/tex]

    We've ignored the rulers here, but it does show that it's possible for 'space-time curvature' to cause motion.
    Last edited: Nov 21, 2008
  15. Nov 21, 2008 #14
    Exactly , not only 'space-time curvature' causes motion , but also it gives this incredible opportunity of the confusion of time and space. Based on the equations above , the particle loses its properties to become mere energy .
  16. Nov 21, 2008 #15


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    OK, it's a massless solution, but I think the result would be the same. Energy that would be in the field normally is now in the internal energy.
  17. Nov 21, 2008 #16
    This particularity of internal energy resulted from the confusion of time and space which has gravitational force in the curved schedule, is necessarily 'intelligent', e.g. has the potential of reversing the formula ,consequently , in different parameters ,energy recovers mass and becomes a mass . We can use this potentiality of this particularity in molecular transfer in space travels, so instead of using spacecrafts travelling according to the light speed , we can just use the molecular transfer to explore the most distant galaxies.
  18. Nov 27, 2008 #17
    That's called "geodesic deviation", which describe "tidal force" of a gravitational field, Not the effective gravitational force. For example, there is gravity in uniform gravitational field, but no tidal force
  19. Nov 27, 2008 #18


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    Yet another way to say this is that 50% is due to local "equivalence principle" bending, and 50% is due to global bending: http://www.einstein-online.info/en/spotlights/equivalence_deflection/index.html
  20. Nov 28, 2008 #19
    Here I just have a question to ask ,addressed to all the physicists : why should we all refer always to Einstein and gargling his formula to gain some credibility? As scientists are we unable to innovating and presenting new formula and theories adjusted with the needs of the new era we live in ? It's almost one century that all physicists speculation and formula adopted Einstein as a point of departure , as if it was the 'word of god' or the 'word of gospels' which are immutable and 'sacred'. Without intending depreciate Einstein's 'theories' and formula , we can consider that he was just an autistic cultured person who was able to speculate on things and formulating his speculations by using the mathematic language while the classic scientists sticked on academic knowledge. Now sticking to Einstein will forbid us to see new horizons . Of course we always use and rely on all the past physicists just to build our own theories and formula and for this why not considering Democritus, Aristoteles, Newton and all the others. The concern of our new era is 'time travel' , 'immortality', 'galaxies exploration' and 'traveling to all the other galaxies' and for all of this , refering always to einstein would be a sort of shortcoming , because we just need to 'doubt' all and making new formula and theories which would answer to the needs of our era.
  21. Nov 28, 2008 #20

    Jonathan Scott

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    You seem to be suggesting that physicists are merely taking Einstein's word for it. This is not the case. Einstein discovered simpler ways of looking at things which are now understood by many physicists, and so far they are the best explanation we have for the experimental evidence.

    Special Relativity is mainly about extending the ordinary mathematics of rotations into the Lorentz group, which also includes boost transformations that are like imaginary rotations between space and time axes. Apart using complex numbers instead of pure real ones, the mathematics is very similar to that for ordinary rotations. Einstein's major contribution was to explain the interpretation of what is happening in terms of space and time and their unification as space-time. At present, there are no known experimental violations of Special Relativity, although the theory of quantum entanglement doesn't have any local deterministic explanation which is compatible with it (which may simply mean it doesn't have any local deterministic explanation anyway).

    General Relativity is not so trivial mathematically, but the principle is very neat, in that Einstein's equations simply say that the "curvature" of space-time, expressed in a certain way, is directly determined by the distribution of energy, momentum and pressure, and this idea works extremely well to explain how gravity works, solving previously observed anomalies such as the perihelion precession of Mercury. On larger scales, GR doesn't seem to directly account for the experimentally observed properties of galaxies and of the universe, and there are a few other theoretical oddities as well, such as incompatibility with Mach's principle. Because GR is known to be so astonishingly accurate in solar system experiments, and because no-one can come up with a better theory which makes mathematical and physical sense, it is usually assumed that GR still holds at the scale of galaxies and the universe, and that the explanation is that galaxies contain invisible "dark matter" and that in between the universe contains "dark energy". However, there are alternative theories being promoted, such as the MOND model which suggests an empirical modification to gravity theory to explain galaxy rotation curves. I personally suspect that GR isn't yet the complete story, but MOND isn't really yet even a consistent theory but just a way of making some numbers fit.

    Wishful thinking such as about being able to travel faster than c does serve to drive some forms of scientific investigation, but physical laws are mostly rather boringly limited from a science fiction point of view.
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