I am sorry; but I feel this discussion ought to stop now! The confusion created by various contributors is increasing just like the entropy of a physical system, meaning that it becomes increasingly impossible to retrieve reliable information!

(To give an example of what I mean: Only the integral of curvature over a surface is a topological invariant! If you only worry about the topology of a surface there are infinitely many inequivalent topologies labelled by the Euler number (# handles, punctures)! The problem is that there isn't any nice possibility of equipping surfaces with Lorentz metrics.)

The experience we are making with Research Gate is that scientific information disseminated via the Internet will become increasingly unreliable, and it will become very dangerous to rely on information downloaded from the Internet - SAD!

first of all, thank you Nadeem. i wondered about the problem as i read an article in some magazine, probably Scientific American, that discussed the seriousness of the problem, again not in detail. It was considered as a 'not understood completely' problem. why is that so?

There has been an immense amount of work in this area, the subject is so big it has it's own name: String Theory. The only catch is that the usual GR kinetic term coming from the scalar curvature is purely topological in backgrounds with a constant dilaton. Thus, it doesn't bear much resemblance to higher dimensional GR.

William, if i am not wrong, string theory also takes into account the quantum mechanics. how, then one can approach the problem in purely classical way?

In string theory, before quantizing, you must first solve the classical problem. With regards to the 1+1 dimensional gravitational sector, this simply becomes a problem of choosing a guage and defining a measure on the moduli space. Arguably, that is all 1+1 GR is, i.e., it's an assignment of an integration measure on the moduli space corresponding to a "spacetime" of every allowable topology. However, for GR in 1+1 dimensions the interesting dynamics do not come from the 1+1 dimensional gravitational sector, rather, it comes from the "matter fields", which correspond to ordinary spacetime coordinates (and their grassman valued superpartners) in one's usual way of speaking.

There has over the years been a lot of activity in this field, motivated by string theory, quantum gravity and 3+1 GR. The hope is that lower dimensional theories can shed light on difficult problems in "more realistic dimensions".

As already stated, the Einstein-Hilbert action in 1+1 is topological. It has no equations of motion.

One way round this is to consider 2+ \epsilon dimensions and then study the limit of \epsilon ->0. It is known that this limit gives Liouville gravity, which is a specific kind of dilaton theory.

Dilaton theories are also motivated by string theory, and so have received some attention.

There is no formulation of general relativity in one dimension, because intrinsic curvature does not exist for a curve. In two dimensions, the curvature of a space or spacetime is described by only one degree of freedom (the Ricci scalar) and the Einstein equation will force this to zero so the only 2d solution is flat space.

As Andrew Bruce has pointed out above, the approach goes for 2+ epsilon dimensions, and the limit of epsilon tending to zero is considered, which is what is done in dilation and string theory (about which i have no knowledge at all).

However, what i intend to ask is that is there a possibility to approach the problem in a different direction? If so, is it possible without including the Quantum Theory?

"However, what i intend to ask is that is there a possibility to approach the problem in a different direction? If so, is it possible without including the Quantum Theory?"

What do you have in mind?

Other approaches to gravity usually focus on a quantum theory. For example causal dynamical triangulation, loop quantum gravity, causal sets and so on. I assume that these can be formulates in 1+1 d (?)

For a lot of quantum gravity theories it is not at all clear that you get general relativity in the classical limits, assuming we are in 3+1 d. I have no idea what can be said about 1+1 d.

There is no gravity in one dimension. There cannot be any curvature...

"There is no gravity in one dimension. There cannot be any curvature..."

You have non-trivial Riemannian curvature tensors and Ricci scalars (essentially the Gaussian curvature) in 1+1 d, which is what we are talking about. It is the Einstein curvature that is identically zero.

Gaussian curvature doesn't lead to any gravity

In spite of being topological, the Einstein-Hilbert action leads to non-trivial global metrics due to the presence of world-sheet moduli. But since it has no degree of freedom, there is no dynamics, and therefore there is no (gravitational) interaction.

Maybe i will be a little redundant, i apologise in advance for that, but certainly i dont agree with some comments. Specifically, i think that the statement "Ricci tensor is identically cero in 2D (i.e. for any 2-manifold)" is wrong, because this would imply the vanishing of scalar curvature as well. In particular, the overall curvature measured by its integral, equals cero. By Gauss-Bonnet, this fact says that all 2-manifolds have cero Euler characteristic, wich is absurd. Then, gravity in 2D is essentially a topological theory, that doesn't mean that it's dynamically trivial, just as it was already pointed out in discussion above.

Perhaps one could get a 1-space dimension result just by averaging over the two angles in spherical-polar coodinates, leaving the radial dimension.

but, do that imply a single dimension? you are just keeping the limit of the polar and the azimuthal angle as zero; in fact, the value of azimuth can be arbitrary. So, basically you are taking 3 dimensions only

For many purposes in physics tghe same approach works fine. Of course it is an approximation. In scattering theory, the radial dimension is sufficient for lower precision experiments, typical of first attempted measurement. As precision goes up, one adds first one angle, then the other, then maybe polarization. These layers just naturally peel off. The result for one dimension is not expected to change other than by improving precision when angular info is added.

Here is one interesting approach to 1+1 dimensional gravity:

http://arxiv.org/abs/hep-th/9111056

Math is fine, but reality has only 3 dimensions. See "Einstein's most important philosophical error" at http://www.worldsci.org/pdf/abstracts/abstracts_5991.pdf

In the theory of special relativity we considering an observer moving with constant velocity parallel to to one of the spatial coordinates. I look at the theory of general relativity as theory in which the observer is moving with accelerating velocity in arbitrary direction such observer is equivalent to an observer in a gravitational field. So the general relativity is a theory of gravity. According to this understand we can get a theory of general relativity with 1+1 D. which will be like the theory of special relativity, but the observer is moving with accelerating velocity. Such a theory is more general than the theory of special relativity, and less general than the general relativity.

thank you Dr. Borchardt and Muyyad for your valuable suggestions; what impression i am i under is that single dimension will may give more fundamental understanding towards the concept, although i took up the problem not taking this into account.

What i intent to ask is that how far do the dilation theory prove consistent? What are other ways to approach the problem?

Tiime is motion; motion is not a thing. It is what things do. Things can dilate, but their motions cannot.

The dilaton theory of gravity in 1+1 dimensions is mathematically consistent and it captures certain features of gravity in 3+1 d, such as Hawking radiation, for instance. However, it should be noted that there are hundreds of different models one might consider, the question is what features of reality you would like to understand? Is there some particular gravitational phenomenon you would like to understand better? We don't have a final theory, so everything is really provisional. One can assign ranges of validity to various theories, but that is it for now.

There is one feature of 1+1 dimensional theories that is peculiar. Namely, every interaction is renormalizable. In other words, "anything goes" in 1+1. Thus, these theories are not as tightly constrained as higher dimensional theories are. You could basically just write down any lagrangian you like and try to quantize it. (There may be conformal anomalies, but the theory can still be well defined.) Thus, 1+1 d will always have some features which are different than 3+1 d gravity, which is non-renormalizable. It may still be a useful model, one must ask what kind of "insight" you are looking for in such a model.

Dear all

In this case the general theory of relativity identifies with the Newtonian one.practical is the definition of the simple "principle of equivalence"

Rici tensor it identifies with the divergence of the Newtonian gravity acceleration g .

Practically the governing mathematics is the newtons theory in a uniform gravitation

To continue

Practically the governing mathematics is the Newtons theory in a uniformed field . Of course there no tidal forces.

To continue

In the case someone consider the Newtonian aproche as fundamentally wrong we can consider the mathematics of the general theory but for just a puntifom object. An object with no dimensions that that moves on the time truck.

Regards

There are no limits to the dissemination of nonsense via the internet! This feature may

kill serious science, before long.

Your question is very interesting. The general law of gravity collapse there as the curvature tends to zero in this zone. The Riemann tensor should be modified with Dirac delta function to understand the the status of general relativity in space time.

I think that most common use of relativity occur in the global positioning system (GPS) where the clock of the fast moving satellite is to be adjusted by a small amount in 24 hrs. the amount of the time adjustment is calculated by theory of relativity. No adjustment in the satellite clocks for a weak may cause the error of kilometers in the position.

Vibrating or rotating string like objects can be described in this frame work.

In two dimensions, the curvature is a topological invariant of the underlying manifold. This implies that there are only three possibilities to choose the geometry of spacetime. These are euclidean plane, hyperbolic plane and the sphere. So the geometry is highly constrained. If you want to see more diversity, you have to move to higher dimensions, or do some quantum stuff, like causal dynamical triangulation, where the curvature is an observable and different states of the 1+1 dimensional spacetime is labeled by different values of the scalar curvature.

I am sorry; but I feel this discussion ought to stop now! The confusion created by various contributors is increasing just like the entropy of a physical system, meaning that it becomes increasingly impossible to retrieve reliable information!

(To give an example of what I mean: Only the integral of curvature over a surface is a topological invariant! If you only worry about the topology of a surface there are infinitely many inequivalent topologies labelled by the Euler number (# handles, punctures)! The problem is that there isn't any nice possibility of equipping surfaces with Lorentz metrics.)

The experience we are making with Research Gate is that scientific information disseminated via the Internet will become increasingly unreliable, and it will become very dangerous to rely on information downloaded from the Internet - SAD!

Thank you for your comment on my mistakes, Prof Fröhlich. I missed the clear distinction between the local and the global characteristics of spaces - what was the root of my fault.

@Jürg. I never think of of these ResearchGate question and answers as definitive "information," I just think of them as a form of discussion. Some are interesting and some are frustrating. But discussions in real life are like that too :-)

The best answer I've seen here is to look at String Theory, where the perturbation series is generated by a generally-covariant quantum field theory on the 1 + 1 dimensional world-sheet of the propagating string, so String Theory is not just 1 + 1 dimensional general relativity, but 1 + 1 dimensional quantum gravity (with some additional fields). Since all of the local degrees of freedom of the metric can be gauged away in 1 + 1 dimensions, there is not a lot of gravitational dynamics. However, the comments about there being no curvature in this case are false: "space-time" is a Riemann surface, as others have noted, and string theory sums over all of these surfaces. There are even string theories with 1 + 1 dimensional target spaces, so that it is possible to describe black holes in this case. See, for example, the charged black holes with rather interesting Penrose diagrams obtained here: https://www.researchgate.net/publication/222486389_Charged_black_holes_in_two-dimensional_string_theory?ev=prf_pub (Two dimensions here means space + time.)

Article Charged Black Holes in Two-Dimensional String Theory

The Jackiw-Teitelboim model of gravity in one space and one time dimension has been studied extensively in the litterature. It is probably the most popular 2d model of gravity.

1+1 dimensional GR was shown to be trivial by P. Collas in the 70s; see for instance the funny blog post, http://williewong.wordpress.com/2012/02/22/one-and-two-space-time-dimensional-general-relativity/

This being said, there is still major obstacles with the 3+1 dim. theory: it asks for heavy math. machinery (tensors, etc ...), it is terribly difficult to get rigorous math. existence theorems, and it's a complete nightmare from the numerical analyst's point of view.

A simple model, which appears to mimic rather correctly higher dimensional GR was proposed by R.B. Mann (see here on Researchgate) and collaborators during the 90s. The field eqn reads R=T (with zero cosmo. constant), R the Ricci scalar, T the trace of the energy-mass tensor. It matches a special case of a Gowdy-symmetric spacetime.

As a numerical analyst, i tried recently to simulate this precise 1+1 model, without black holes, and mostly in conformal coordinates (R=T yields the Liouville eqn): it is already a nontrivial task to produce reliable approximate solns of a coupled system involving matter and geometry in interaction.

Hope it helps!

I forgot to say something in my previous post: R=T model of 1+1 GR, thanks to the possibility it offers to work in the "conformal gauge", allows to strongly lower the nonlinearity inherent to the coupled system of matter/field equations.

Indeed, in conformal coords, the field eqn is the liouville eqn, which is semilinear only. So only the matter eqns (Euler isothermal for instance) are quasi-linear. This eases one's life quite a bit when it comes to numerical simulation.