How does curved spacetime result in an effective gravitational force?

[morenogabr]

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 don't 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 doesn't 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 doesn't really commute to GR, it basically doesn't work beyond the purely abstract explanation. (This is obviously because the trampoline isn't 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?

 

[A.T.]

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 don't exactly understand how curved spacetime implies a gravitational force.

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):
http://www.adamtoons.de/physics/gravitation.swf

At this point my friend goes off with the analogy of a bowling ball on a trampoline.(...) Plus the analogy seems, to me, to beg the question; the analogy uses the gravitational phenomenon to explain the gravitational phenomenon...

You are not the first one to notice this flaw:
https://www.physicsforums.com/showpost.php?p=1961959&postcount=28

 

[jtbell]

I don't exactly understand how curved spacetime implies a gravitational force.

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.

 

[Naty1]

The problem is that I don't exactly understand how curved spacetime implies a gravitational force...Does Einsteins relativity require the NON-existence of a particle-interaction explanation of gravity? If we soon find a graviton, would this disprove GR?

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.

 

[iRish_waKe]

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.

 

[morenogabr]

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?

 

[bockerse]

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.

Gerrit

 

[DrGreg]

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.

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.

 

[bockerse]

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.

Gerrit

 

[A.T.]

"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).

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

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.

 

[gonegahgah]

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?

 

[A.T.]

It is fairly easy to do the trampoline model with one gravity source and respondents to that force (ie light or a smaller ball).

But it is not a model of gravity in GR, as the OP noticed and was https://www.physicsforums.com/showpost.php?p=1961959&postcount=28". 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.

 

[Mentz114]

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,

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

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

$$-i\frac{\partial}{\partial x}\psi(x,t) = \left(k+\frac{\partial\omega}{\partial x}t\right)\psi(x,t)$$

We've ignored the rulers here, but it does show that it's possible for 'space-time curvature' to cause motion.

 

[meteor9]

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 .

 

[Mentz114]

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.

 

[meteor9]

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 spacecraft s traveling according to the light speed , we can just use the molecular transfer to explore the most distant galaxies.

 

[feynmann]

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.

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

 

[atyy]

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.

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

 

[meteor9]

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 , referring 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.

 

[Jonathan Scott]

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 , referring 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.

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.

 

[DrGreg]

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 , referring 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.

In fact physicists already know Einstein's theory isn't perfect. General Relativity and Quantum Theory are not fully compatible with each other, so at least one, and very likely both, must be defective in some sense. Nevertheless, the two theories are both extremely good, very well tested and all the evidence indicates they both work very well indeed except in the most extreme circumstances such as the very centre of a black hole or the moment of the Big Bang. A huge amount of effort is being made by theoretical physicists to come up with a better theory to encompass both General Relativity and Quantum Theory, but no-one has succeeded yet.

So Einstein's theory is held in such high regard because it works so well and no-one has yet found anything that works better.

 

[meteor9]

I agree with you Jonathan Scott and DrGreg , and i see perfectly what you mean , but my point was not 'fighting' Einstein theories or all the others , or presenting a better theory or formula, simply i wanted to suggest that the students who come here to get help for their homeworks , don't become short-sighted by their 'fascination' for some formula in a way that they neglect their own mind and thinking because of all the mass and volume of the academic manuals of physics. The most important thing a student should always consider, is thinking independantly after reading all the manuals. I'm sure that a student of 17 years old is able to see and to solve a very complicated equation much better than an accredited and acknowledged professor of physics . In brief my point is that physicists encourage students to think for themselves , researching , experimenting everything by their own , after considering all the theories and formula in the existence.

 

[Dale]

Certainly, and should one of those students come up with something better then their ideas will be adopted by the scientific community.

 

[feynmann]

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):
http://www.adamtoons.de/physics/gravitation.swf

If there is no gravitational force in GR. Then where the force come from that we feel tired when stand on the ground in gravitational field?

 

[Dale]

Then where the force come from that we feel tired when stand on the ground in gravitational field?

From the ground, it is called the http://en.wikipedia.org/wiki/Normal_force" .

 

[feynmann]

From the ground, it is called the http://en.wikipedia.org/wiki/Normal_force" .

But how to explain it in term of curvature of spacetime?
Since A.T. claims there is no gravitational force in GR,

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):
http://www.adamtoons.de/physics/gravitation.swf

[/URL].

 

[Dale]

But how to explain it in term of curvature of spacetime?
Since A.T. claims there is no gravitational force in GR,

Yes, A.T. is correct, that is why I said that the normal force tires out your legs, not the force of gravity.

The GR explanation is that an inertial frame is one that is free-falling, so due to the curvature of spacetime the ground is a non-inertial frame that is accelerating upwards at 9.8 m/s².

 

[akvikram_rk]

i am a student and i have a doubt. when a object is kept stationary or at rest at a point in the curvature of space time, the object has to remain stationary, instead it starts moving towards center of the curvature. now what force makes the object to move from rest/ stationary. its weird that an object will start following the geodesics without any force applied on it.

 

[Jonathan Scott]

i am a student and i have a doubt. when a object is kept stationary or at rest at a point in the curvature of space time, the object has to remain stationary, instead it starts moving towards center of the curvature. now what force makes the object to move from rest/ stationary. its weird that an object will start following the geodesics without any force applied on it.

If only space were curved, stationary objects would not be accelerated. However, it is space-time which is curved. Here's an analogy: Think of how lines in a plane are curved when they pass a massive object in space. Now instead of a space plane, consider a space-time plane where the lines are going in the time direction, but are curved in the same way. Those paths represent accelerated motion of slow-moving particles.

 

[A.T.]

From the ground, it is called the http://en.wikipedia.org/wiki/Normal_force" .

But how to explain it in term of curvature of spacetime?
Since A.T. claims there is no gravitational force in GR,

The ground normal force pushing you upwards is caused by the impenetrability of the ground by your feet (electromagnetic repulsion). It is preventing you from following a geodesic in space time, which would be the default behavior, in the absence of any forces.

when a object is kept stationary or at rest at a point in the curvature of space time, ...

Nothing is stationary or at rest in spacetime. Everything moves trough space or time or both.

 

[Pete62]

[/QUOTE] From A.T. Nothing is stationary or at rest in spacetime. Everything moves trough space or time or both.[/QUOTE]

Getting your head around the time part of space-time was the biggest hurdle for me. Remember, the only way to not be moving in time (not including black hole event horizons) is to be moving at the speed of light.

 

[meteor9]

exactly Pete62 , not only at the speed of light but also at the 'speed' of 'stationary dimension' . By this i mean that if we consider the 'parallel worlds' which have been mentioned so many times in 'quantum physics' , we even would not need any 'speed' , you can 'move' or 'enter' easily to lot of other 'worlds, and why not to 'other' planets or galaxies , without spending energy to produce 'speed' . In my opinion the theory of einstein about the 'curved ' space-time , has stopped all sort of 'different considereations' of space-time concept and has made the physics to remain at a stagnation point. Since then we don't search elswhere to find some other sort of explanations of the 'physics' , 'quantum physics or rather the 'metaphysics' and why not the 'metarealism'

 

[altonhare]

I think the original question has been answered but I thought I'd add my own analogy in case it helps the poster more.

The geometric aspect of acceleration/force in GR is explained more or less by our inability to "see" the "temporal dimension". By this I mean the object may appear to be taking a linear path to us only because we cannot see its path "in time".

As always, let's consider Flatland. There's two stick figure Flatlanders here. There is a big depression in Flatland and one of the sticks is down in the depression, moving out. The other stick is in a flat region watching the other stick coming toward him. These Flatlanders cannot perceive nor conceive of the "up" direction, however the stick at the bottom of the depression is doing most of his motion "upward". The observing stick, then, sees the traveler moving very slowly at first, then faster and faster as the traveler moves out of the depression. The up component of the traveler's velocity is decreasing as its rightward velocity component is increasing. The whole time the magnitude of the traveler's velocity was actually constant, but it appeared to change to the observer because s/he could not see the decreasing velocity in the "up" direction, but only the increasing velocity in the "rightward" direction.

Does that make sense?