Why didn't Einstein just invent the concept of gravitational field?

[Swapnil]

Some say that the reason Einstein developed the GR is because he wanted to avoid the spooky "action at a distance," since two massive bodies exerting forces on each other at a large distance with no "mediator" seems rather odd. Therefore Einstein invented the concept of spacetime and gravity could be defined as the curvature of spacetime.

Now my question is that why didn't he just invented the concept of gravitaional field? That would resolve the spooky "action at a distance" right?

 

[dmoravec]

a big part of his reasoning behind the curves in space-time come from his equivalence principle. Gravity and Acceleration feel the same. An accelerating object has a curved space-time line. Therefore it is only natural to assume that gravity must exist as curves in spacetime.

 

[MeJennifer]

Some say that the reason Einstein developed the GR is because he wanted to avoid the spooky "action at a distance," since two massive bodies exerting forces on each other at a large distance with no "mediator" seems rather odd. Therefore Einstein invented the concept of spacetime and gravity could be defined as the curvature of spacetime.

Well I am not at all familiar with those "some" you are referring to. Perhaps you could provide some sources, otherwise this seems nothing more than rumor spreading.

Anyway, even in GR there is no explanation for the "pulling" action of gravity, since the pulling is not part of the curvature, only the volume reducing and tidal effects (Ricci and Weyl tensors) are.

 

[pervect]

Some say that the reason Einstein developed the GR is because he wanted to avoid the spooky "action at a distance," since two massive bodies exerting forces on each other at a large distance with no "mediator" seems rather odd. Therefore Einstein invented the concept of spacetime and gravity could be defined as the curvature of spacetime.

Now my question is that why didn't he just invented the concept of gravitaional field? That would resolve the spooky "action at a distance" right?

How do we measure an electric field? We take a test charge, and measure the force on it, relative to an uncharged particle.

How do we generalize this to a "gravitational field"? We take a test mass and we want to measure the force on it due to gravity. But by the principle of equivalence, we can't distinguish forces due to gravity from forces due to acceleration. And there is no "gravitationally neutral" particle that we can use as a reference. So we are stuck, right from the get-go, in trying to define the gravitational field in the manner we are used to. The problem is that gravity acts on everything, so there isn't any reference about what force is due to "gravity" and what force is due to "inertia". And if the equivalence principle is to work properly, this is as it should be.

So traditional ideas of gravity as a field, just like the classical electromagnetic field, don't pan out, just because gravity is universal and affects everything. Einstein's solution to this dilemna was to say that the gravitational force was due to geometry.

I believe that Einstein, in his later years, came to regard the metric tensor as a sort of "gravitational field". It's not a field in the usual sense, but it is a tensor, and from it everything you want to know about gravity can be calculated if you know the metric. so it makes some abstract sense to call it a "field". Personally I think this is confusing, so I prefer to call the metric tensor the metric tensor, and let people think of "gravitational fields" as the traditional Newtonian sort of field.

One thing that one can't do is attribute energy to any specific location of the "field", regardless of whehter one choses to call the metric tensor a field or just the metric tensor. This happens because of Noether's theorem. It turns out that GR is a bit general for its own good - because it is diffeomorphism invariant, it in general lacks the sort of single parameter time translation symmetry that usually results in a conserved energy. Because the time translation symmetry is an infinite group rather than a single parameter group, energy conservation does not automatically arise in general relativity.

 

[notknowing]

One thing that one can't do is attribute energy to any specific location of the "field", regardless of whehter one choses to call the metric tensor a field or just the metric tensor. This happens because of Noether's theorem. It turns out that GR is a bit general for its own good - because it is diffeomorphism invariant, it in general lacks the sort of single parameter time translation symmetry that usually results in a conserved energy. Because the time translation symmetry is an infinite group rather than a single parameter group, energy conservation does not automatically arise in general relativity.

The topic of energy density in the "gravitational field" is of special interest to me. Indeed, it seems not possible to define an energy density similar to the energy density of the electromagnetic field. But would it maybe not be possible to talk (or define) about a gradient in energy density in the gravitational field ?

 

[LURCH]

I thought the "spooky action at a distance" with which Einstein was uncomfortabe was quantum entanglement, not gravity. Have I got my history mixed up?

 

[pervect]

The topic of energy density in the "gravitational field" is of special interest to me. Indeed, it seems not possible to define an energy density similar to the energy density of the electromagnetic field. But would it maybe not be possible to talk (or define) about a gradient in energy density in the gravitational field ?

If you could determine the gradient of the energy density of the field, you could determine the energy density up to a scalar, so no, you can't find the gradient of the density either.

The problem is not just an additive scalar. See for instance the following article about Noether's theorem:

http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html

In contemporary terminology the general theory of relativity is a gauge theory. The symmetry group of the theory, is a gauge group. It is the group of all continuous coordinate transformations with continuous derivatives, often called the group of general coordinate transformations. It is a Lie group that has a continuously infinite number of independent infinitesimal generators. In Noether's terminology such a group is an infinite continuous group. The symmetry group of special relativity, the Poincare' group4 is a Lie subgroup of the group of general coordinate transformations. It has a finite number (7) of independent infinitesimal generators. Noether refers to such a group as a finite continuous group. This distinction between a Lie group with a finite (or countably infinite) number of independent infinitesimal generators and an infinite continuous group is what distinguishes Noether's theorem I and theorem II in I.V.. Theorem I applies when one has a finite continuous group of symmetries, and theorem II when there is an infinite continuous group of symmetries. Field theories with a finite continuous symmetry group have what Hilbert called `proper energy theorems'. Physically in such theories one has a localized, conserved energy density; and one can prove that in any arbitrary volume the net outflow of energy across the boundary is equal to the time rate of decrease of energy within the volume. As will be shown below, this follows from the fact that the energy-momentum tensor of the theory is divergence free. In general relativity, on the other hand, it has no meaning to speak of a definite localization of energy. One may define a quantity which is divergence free analogous to the energy-momentum density tensor of special relativity, but it is gauge dependent: i.e., it is not covariant under general coordinate transformations. Consequently the fact that it is divergence free does not yield a meaningful law of local energy conservation. Thus one has, as Hilbert saw it, in such theories `improper energy theorems.'

 

[notknowing]

Thank you Pervect for this detailed answer. I'll need some time to study it in more detail.
What puzzles me though is that Einsteins field equations are nonlinear which seems to imply that there exists something like "gravitational energy". And if this can not be localized, one would naively expect that the strength or direction of the field would also not be defined correctly. How can one reconcile this ?

 

[turbo]

I thought the "spooky action at a distance" with which Einstein was uncomfortabe was quantum entanglement, not gravity. Have I got my history mixed up?

In his 1924 essay "On the Ether", Einstein rejected Machian concepts in which gravitation and inertial effects are the result of "action at a distance". He insisted that gravitation and inertia arise from matter's interaction with the space in which it is embedded - an entirely local effect with no consideration of force acting over a distance. This "etheric" concept did not sit well with his contemporaries, though there a people pursuing it today. For instance, Thanu Padmanabhan, who is quite interested in the intersection of gravitation and quantum theory models vacuum as if it is an elastic solid with which matter interacts. Links to some of his writings on this subject are on his homepage.

http://www.iucaa.ernet.in/~paddy/

 

[pervect]

Thank you Pervect for this detailed answer. I'll need some time to study it in more detail.
What puzzles me though is that Einsteins field equations are nonlinear which seems to imply that there exists something like "gravitational energy". And if this can not be localized, one would naively expect that the strength or direction of the field would also not be defined correctly. How can one reconcile this ?

Defining gravity as a field does have similar problems. What works in GR is geometry, not fields.

If you have an observer at a particular point, there exists an observer at that point who sees no "field". This is an observer following a geodesic, i.e. in free fall, at that point.

This implies that "fields" can't transform as a tensor. If all components of a tensor are zero at a pint for one observer, they must be zero at that point for all observers.

When we translate the math of GR to talk about "fields", this translation requires us to have some particular symmetry about the problem. Usually this is a static metric, one that isn't changing with time.

If we could single out some particular background frame on physical reasons, we could use this to define the symmetry needed to define a field unambiguously in all cases. In GR this isn't possible - all frames are just as good, and there isn't any way to talk about the "field" at a point. This goes back to my earlier remarks, about how we lack a "gravitationally uncharged" reference parrticle. If we had such a particle, it would create the sort of symmetry one needs to define a field.

So this is a short version of why, in GR, we have geometry rather than fields.

I'm not sure if I should add this, but I will anyway. There is another approach to the issue. This is to assume that space-time is flat, and that gravity is a field. (Just ignore the fact that I just got through saying one couldn't do that :-)).

The result one gets from this approach is that these "fields" cannot actually be observed. So we see that there is a point, we can "create" a field with the right assumptions, but we can't define one without making some extra assumptions. The problem is that we have to spell out all our assumptions to work with this notion of "field".

In other words, this "field" description is coordinate dependent.

This is a bit like assuming that there is some sort of special frame of reference in special relativity that defines "absolute motion", and then finding that there is no way to observe it.

For a tutorial but rather technical paper on this, see http://xxx.lanl.gov/abs/astro-ph/0006423.

 

[Swapnil]

a big part of his reasoning behind the curves in space-time come from his equivalence principle. Gravity and Acceleration feel the same. An accelerating object has a curved space-time line. Therefore it is only natural to assume that gravity must exist as curves in spacetime.

Since I found out about this equivalence principle, I have been trying to understand it. I looked up some books and several online articles/texts on it, but I was still confused...until I found this text:
http://www.einstein-online.info/en/spotlights/equivalence_principle/index.html

I think this is a great online text for any beginners of GR who want to know what Einstein's equivalence principle is all about. A very through and clear text.

edit1: Actually, I have one more article that might be very useful. I think that it would be best if the beginner reads the above article first and then read s the following article, which shows how the equivalence principle leads to Gravitaional time dilation.
www.upscale.utoronto.ca/GeneralInterest/Harrison/GenRel/TimeDilation.pdf

edit2: OK, after reading the above two articles, I think that the following article would be the best way to go for a beginner. It shows that why does it make sense for gravity to bend light according to the equivalence principle. Its really amazing!
http://www.etsu.edu/physics/plntrm/relat/general.htm