Cosider a point on a crumpled paper. It has two degree of freedom and its motion path will be along the wraps of its world sheet. Given the mass-energy distribution, Einstein field equations give the curvature of space-time around. So the test body has no other way beside moving on the curve.

The prehelion precession of mercury is about 575 arcsec per century. Newtonian mechanics predicts a precession of about 532 arcsec but Einstein GR gives the observed value. Other planets would be also better described by GR, but the difference between GR and CM would be more negligible.

Actually gravitational force is not replaced by geometry. The crux of GR is the equivalence between gravitational mass and inertial mass. In both cases we have a scalar parameter : a gravitational charge on one hand, an inertial mass on the other hand, and they are strictly equal. The force that we perceive, either as a gravitational attraction, or as an inertial force, are the same. The simplest way to represent this fact is to say that the universe is curved, and the presence of matter is the cause of this curvature. Indeed the attraction of the Sun is felt by all bodies which are close enough : it is not specific to each one.

The equivalence is indeed important. Bodies moving inertially move in straight lines. In GR straight is defined as a geodesic of the metric. Defining what is or is not a straight line is a matter of geometry.

I'm just a casual observer without math, but I think the most critical distinctions are that classical gravitation is a direct material force interaction between two objects of mass, whereas in GR objects of mass are not directly interacting - they affect some change to an undescribed characteristic property of the vacuum that directly correspond to a system of spacetime dimensional coordinates which, in turn, affect the motions of objects of mass.

As I see it, there must be some physical property of spacetime that corresponds to the described abstract geometrical dimensions - but those properties have not been determined. However, the abstracted geometry that represents the changes imparted describe very well the changes imparted to all objects moving within the system of coordinates.

So the most distinct aspect of GR is that, as John Wheeler said so succinctly, "Spacetime tells matter how to move; matter tells spacetime how to curve". The other most critical element, of course, is that GR establishes the direct association between the tension of spatial coordinates and the progression of time...

As I comprehend, then, spacetime provides some universally shared, common physical vacuum media (IMO _not_ to be confused with 'ether' or any form of condensed matter) which determines the motions of objects - as influenced by the kinetic and potential energy of matter.

I can't really go further, except to point out that the fundamental process of accretion not only produces condensed objects of potential mass energy, but correspondingly _extracts_ that dispersed potential energy from the kinetic energy of spacetime - perhaps mechanically 'shaping' that unidentified physical aspect of the vacuum (corresponding to dimensional spacetime coordinates) in the process! As I understand, the initial accretion of particles does not involve gravitation at all - it is based on EM charge attraction. Perhaps it is that initial extraction/contraction interaction between matter and spacetime that _creates_ gravity! I think there is something very crucial illustrated here, but I don't clearly understand what it might be...

I also suggest that, while classical gravity is so familiar to so many because it is so simple to use to solve simple problems, when applied to very complex problems at galactic and cosmological scales, especially involving very large, self-interacting compound objects, it's simply too convenient to produce erroneous results...

Compound objects may be best described by the common conditions of spacetime that they produce, even in weak field conditions, rather than any aggregated virtual object of condensed mass.

Since the beginning of XX° century the concept of force acting at a distance has been replaced by the concept of field, present everywhere and interacting locally with matter according to "charges". From this point of view there is no difference between gravitation and the other 3 fundamental forces (electromagnetic, weak and strong interactions). The traditional (let us say old) formalism of GR, centered on the metric and the Lévy-Civita connection has been replaced by the fiber bundle formalism, used in Yang Mills models (such as the standard model). In this formalism gravitation and the other force fields are on the same footing.. From there one can build a model using lagrangian, principle of least action, and every thing combining gravitation and the other fields. However the key ingredient of the fiber bundles is a group, for gravitation it is the Spin group (3,1) and for rhe other fields it is SU(3)xSU(2)xU(1). So we do not have yet a true unified model : for this we need a great unification theory, which sums up to a way to combine all these groups together. Notice that Quantum Mechanics has some impact, in that it helps to specify the structure of the fields, but is not more involved.

Complement to the previous answer :

A Yang Mills like model relies on the implementation of the principle of least action : the "mechanic" part of the model lies in a lagrangian, and the eqations of motions, interactions, fields,...are found by taking the extremum of the action which is some integral of the lagrangian. So one gets differential equations. There is no longer concepts of "force" , but there is the concept of impulsion-energy, a tensor which is conserved in the system.

All this seems a bit abstract, but actually the modern formalism of fiber bundle, once you have done the effort to acquire the mathematics, is quite elegant and simple, much better than the old computations of GR.

You can have a look at the basis in my paper on particles and fields.

I agree that GR can be described as ("Yang Mills") gauge theory of the Lorentz group, but there are other "fundamental" differences between this "geometric" symmetry (which mixes space time coordinates) and the SM *internal* symmetries which mix "particles" (i.e., fields). At least the average Earthling does not yet consider space time coordinates to be "of the same kind" as matter and gauge fields. Also, there remain fundamental differences between "matter" and "Yang-Mills" and "Lorentz" gauge fields, e.g. the spin of these fields (and in addition to the spin-statistics theorem there is also the singular role of time in quantization, even in string thy where spacetime coordinates are ("like matter") fields -- maybe you meant this when you said "QM helps to structure..."(?))

Cher Maximilan,

I agree that coordinates iin GR are of a different nature : even in a fiber bundle formalism the base manifold is coordinated by space-time, and there is a unique manifold, whatever the field. So what we are looking at in this formalism is what is "above" the manifold. In a classic Yang Mills field the particle is represented by a vector in a representation of a group : this is a localized vector, a section of a vector bundle (for a matter field). So we have to look for something equivalent for the "inertial part", subject to the graviational field. This is obviously a representation of the motion, and motion shall be understood as comprised of translation and rotation. This representation is a spinor, a special representation of the Spin group. So by taking the tensorial product of the spinor (the motion) and the vector representing the charges carriers, we get a vector bundle which has all the ingredients, localized, and able to interact both with both the gravitational and the other fields. They are themseleves represented as connections acting on the fiber bundle. This is classic stuff, and does not involve in any way QM.

The issues are

1. to find a plausible explanation for the spinor formalism. I have striven on the subject, and my solution is sensible, original without revolution. The key point is that in a relativist picture (SR or GR) the group of displacements if not the right answer. We have fully to accept that the inertial part of a body is represented by a vector upon which acts the field. And this is motivated by the necessity to include the rotational part of the motion. And it gives naturally birth to the spin, which is related to the frame of the observer (who has his own motion).

2. QM is involved in that the representation of the Spin group used for the spinor must also relects the representation of the group SO3. From there we have a restriction. The result is that a spinor belongs to a 10 real dimensional manifold (remembers something ?) but for a given particle, characterized by an intrinsic spinor (which would be seen in a frame attached to the particle), its motion (that is the transformation to go from the intrinsic spinr to what is measured by an observer) is represented by 2x3 dimensional vectors, as in the classic way.

3. The mix of Spin group and SU(3)x... is not satisfying. We should have a unified vector bundle but this requires to find some structure which encompasses both groups, that is a great unification. However the problem is far easier than to invoke extra dimensions ! My feeling is that the solution should be found through Clifford algebras.

4. The meaning of matter fields. They appear naturally as sections of the vector bundle. They can be seen as "general solutions" of the differential equations ruling the evolution of the system, or equivalently as virtual particles, according to our own sensibility. QM brings also here some meaning to the problem.

5. The implementation of the principle of least action in the general case, with fields (defined alll overs the universe) and particles (defined over their own trajectories) is a bit complcated because the usual Lagrange equations do not apply. However with some additional mathematics (distribution over vector bundles) it is easily managed.

The issue of the specificity of time appears only on the group used. I do not consider "observables" associated to the coordinates.

Thanks for this long reply. I won't be able to give an adequate answer in a few words, let me just mention some further elements

- w.r.t your 5°: In a very nice* geometric description of supergravity and matter developed by Girardi and Grimm, chiral matter fields arise as parameters of an invariance of the sugra Lagrangian (Kähler transformations). [*The complete lagrangian which yields all components of the usual sugra + matter + YM, is just the volume element of superspace !]

- Also: the notion of "vector" is slightly ambiguous when dealing with "Yang Mills" thy "acting on" Lorentz indices, since usually the vector is a 1-form potential, but since the Lorentz connection carries other spinorial indices its components have higher spin.

- And there is also the "problem" of the anomalous mass dimension of the associated "Yang Mills fields" and the fact that L_YM ~ F² but L_Einstein ~ R (not R²)...

Cher Maximilian

I am not sufficiently versed in string theories to bring much in your answers. Just some remarks

1. Adding more dimensions is, from my point of view, not a real improvment. The fiber bundle formalism can do most of the job at a lesser cost. And warped dimensions bring only more troubles than they are worth.

2. The "anomalous" mass dimension, as I undertand it, comes when one uses the scalar curvature. But there is no good reason to stick to scalar curvature and Lévi-Civita connection in a general fiber bundle framework (on the contrary,it increases the discrepancy between gravitation and the other fields).

3. Spinor is a tensor combining (as far as we have no GUT) gravitation and the other fields.It does not involve the potential, which is linked only to the "force fields". And this spinor, because it sums up the inertial part, actually does not incorporate any derivative. This is not a vector of a jet bundle.

I thought the number of potential gauge groups is finite? Why haven't all such "fiber bundle" theories all ready been considered?

Dear Paul,

Fiber bundles in GR are not new (see for instance Advanced classical field thories by Giachetta and Sardanachily). The kind of principal bundle that can be build on a given manifold is effectively some sort of topological invariant, in the Chern theory. This can be useful to restrict the topology of the universe once we have the right answer for the group, but so far it is not very useful, from my point of view.

Is it impossible to stick to classical notions, without going into

quantization ?

When gravity is described by a local field theory of the spin group (3,1)

is it a classical field theory or a quantum field theory ? Concept of

spin is quantum mechanical, so I guess it is a quantum field theory ?

When it is a quantum field theory it will describe short distance

phenomenon. We cannot then use it to calculate orbits of planets.

So let us discuss Einstein's gravity at the classical level and

try to understand how to describe gravitational attraction between

two bodies without using the concept of force.

Consider earth's motion around sun. It is due to a balance of

gravitational attraction and centrifugal force away from the sun.

If we forget the concept of force, how can we make this balance

using Einstein's classical gravity.

Biswajoy,

While, as I mentioned previously, IMO there is a largely undescribed element of the physical vacuum that is inherent in GR that corresponds to the abstract dimensional coordinates of spacetime that constrain the motions of objects of mass.

While my ability to explain is very limited, perhaps it would help to refer to http://en.wikipedia.org/wiki/Stress_energy_tensor - which begins:

"The stress–energy tensor (sometimes stress–energy–momentum tensor or energy–momentum tensor) is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. The stress–energy tensor is the source of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity."

In my simplistic conceptual view, it is the local energy density of the vacuum, described by the dimensional coordinates of spacetime, that directs and constrains the motions of material objects. While some may object, IMO objects are not moved by abstract dimensional coordinates...

To Bisjawoy,

1 .One can build a perfectly viable Yang Mills theory, incorporating gravitation, spinor, and the other fields, in the fiber bundle formalism, without QM. This is classic stuff.

2. Spin is a relativist feature, not a quantic one.

3. There is no quantic world (which is by the way a metaphysical concept). Actually the basic "axioms" of QM (Hilbert space, operators,...) do not mention a scale : the Planck constant appears only in the Schrödinger equation.

4. I have proven that most, if not all,these axioms come from thekind of mathematical models that we use. So they are valid at any scale, as far the model meet some general conditions. In this case the formalism can be used for any kind of objects, including galaxies.

Not all the problems are solved in this framework, and gravitation is not yet fully integrated with the other fields, but the issues are quite clear, there is no mysteries, additional dimensions or metaphysical concepts. The probabilities (such as "random behavior" of particles) have a sensical interpretation, which does not involve any supposed bizarre feture of the real world.

See my papers on Systems and Hilbert space (for QM) and on Particles and fields

@Dutailly,

In elelentary particle physics spin is introduced by the

Dirac equation. Schrodinger equation, which is non-reativistic

cannot describe the concept of spin. So I guess I agree that

spin is coming from relativity. Or in other words as you

have put it "spin is a relativistic feature".

In your description of relativity, how do you replace the concept

of gravitational force between two massive bodies and how

is your description different from Einstein's gravity, which

introduces deformation of space time. The metric there, is a

function of x,y,z,t.

@James and Charles

Stress energy tensor is the source of gravitational field. The

result is unique deformation of space time time specified

by the metric.

So can we say that Einstein equation is a rule by which if you

specify stress energy tensor then you can obtain the metric ?

The equation is G^{ab}= k T^{ab}

If T^{ab} is a function of (x,y,z,t) then so will be G^{ab}. Then

from G^{ab} how do I get the metric tensor.

To Biswajoy

1. Spin : this is actually similar to the rotation for a solid. The motion of a solid is expressed in classic geometry by two vectors (spatial speed, and axis of rotation). In relativist geometry we have a spinor, which gives eventually 6 real parameters plus a choice for the orientation of the "axis of rotation", with respect to the observer, so we have the usual spin. The way to go there is a bit abstract, but the result is quite intuitive and very similar to the common sense. Moreover the result holds for any"solid" body (meaning which is considered as having n internal structure).so it is valid for stars in galaxies.

2 In the fiber bundle formalsm, which is not new, the force fields are represented by connections, so for gravitation this is not necessarily the Lévi Civita connection, and the metric, which is still present, is actually a sub product. The computations are much easier.

Charles,

Nicely done!

@ Charles.

Thank you very much for the links.

1)

Is G^{ab}=k T^{ab} a differential equation ?

At the first sight it seems to be equating two

matrices. If I know elements of T^{ab} then

multiplication of k to all the elements of

T^{ab} gives G^{ab}.

2)

Tell me now how can I solve Einstein's

equations to calculate the trajectory of earth

around the Sun.

http://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity might also be helpful...

Let us try to understand motion of earth around sun

as a motion through a geodesic. We know that the motion

is elliptic with sun at one of the foci. Then this ellipse should

be a geodesic of the gravitational effect of the sun. Mercury

also moves in another elliptical orbit, then that orbit is

again a geodesic. Then how can we differentiate this

two geodesics. How do we know that these two geodesics

do not intersect each other. Any simple explanation is

welcome.

An example of experimental measurements of deformations

of space-time:

Deformations of space-time can be measured also by laser

interference methods. In LISA experiment three satellites will

aim laser beams at each other. The three satellites form an

equilateral triangle. By laser interference deformations in space

time can be measured. This experiment is motivated in the context

of gravity waves. These gravitational waves are predictions of

Einstein's gravity. According to Wikipedia,

" In physics, gravitational waves are ripples in the curvature of

spacetime that propagate as a wave, travelling outward from

the source. Predicted in 1916 by Albert Einstein to exist on the

basis of his theory of general relativity"

Therefore we see that, according to Einstein's gravity not

only force is replaced by deformations of space-time around

a massive object; an additional consequence is that ripples

of space time curvature can propagate as a wave outwards

from a possible source. The possible source should have

a "specific type of mass distribution" in space-time, such that

the source have a non-vanishing time varying quadrupole

moment. Possible sources are binary star systems composed

of white dwarfs, neutron stars, black holes. Note that an

arbitrary mass distribution cannot emit gravitational

waves, even though it can deform space-time.

Continuing with my previous post, let us examine qualitatively,

the types of mass distributions which are capable of emitting

gravitational waves.

Let the mass-energy density be Ω(r). The

monopole moment is

int Ω(r) d^3r,

This is a conserved quantity being total mass-energy..

The static dipole moment is

int Ω(r) r d^3r

this is center of mass-energy. In the center of mass

frame it cannot change, so there is no radiation in the

CM frame. Because existence of radiation is frame

independent, gravitational radiation is not emitted in

other possible frames also.

Next higher moment of a mass distribution is

quadrupole moment which can change with time.

It is not a conserved quantity. Hence it can radiate

gravitational radiation.

As an example earth itself is a mass distribution.

This distribution has a quadrupole moment which

can affect trajectories of artificial satellites.

reference:

http://www.astro.umd.edu/~miller/teaching/astr498/lecture24.pdf