Posted: Jul 19, 2010 8:02 am
Post 2 General Relativity and its Canonical Formulation
In this post I want to say a bit about the canonical formulation of GR, since LQG is based on something similar. To do this I’ll start by saying what the mathematical ingredients of GR are – just a flavour no pretence of completeness or rigour.
Manifold
A manifold is a geometric object that looks locally like a piece of Euclidean space RN. RN is just the space of ordered N-tuples of real numbers (x1,...xN). So a (N dimensional) manifold has the property that I can plaster it with patches, and the points in each patch are labelled by the coordinates x1...xN. I may not be able to find one patch to do the job over the whole manifold – for example if I take the 2 dimensional manifold which is the surface of a sphere, I can make much of it look like R2 by using polar coordinates θ and φ. These can’t be used everywhere however, since the poles don’t have a well defined, unique φ coordinate. I can, however, cover the sphere with overlapping patches such that each patch looks like a piece of R2. In GR, the manifold is spacetime, and there are 4 coordinates. Most people number the coordinates from 0 to 3, with 0 being the time coordinate.
Tangent Vector
It’s fairly intuitive what one of these is (but a little more involved to define mathematically). Taking the sphere, I can imagine a tangent vector at a point to be a little arrow stuck onto the sphere at a point P and perpendicular to the line from the origin to P. (If I were to define tangent vector mathematically I wouldn’t need to use the concept of “perpendicular” here).
Metric
The metric (which means “measure”) is a rule which defines how you can take the scalar product of any 2 tangent vectors, so you can use it to compute the angle between the vectors or the length of a vector. Since on an N dimensional manifold I can define N basis tangent vectors, my metric just needs to define how to take the scalar product of any 2 basis vectors, so it has N2 components. However, scalar products are symmetric, so it really only has N(N+1)/2 independent components.
Connection
A connection is a mathematical object which tells you how to parallel transport a vector. Parallel transport is concerned with how the vector changes as you change the point at which it’s attached to the manifold. Suppose it’s attached at point P1 and I want to “move” it to point P2 along a curve C connecting P1 and P2. Conceptually
a) At each intermediate point P along the way, it must become a remain vector at P i.e. it’s not allowed to “dip in” or “dip out” of the manifold – it stays in the tangent space
b) It “preserves the metric”, i.e. keeps the vector the same length
c) It stays “as near as possible to staying parallel to itself whilst respecting (a) and (c)
The mathematical object that tells you how to do this is called a “connection coefficient”, or sometimes “Christoffel symbol”. Γabc
A curve which parallel transports its own tangent vector is called a geodesic and in GR geodesics are the curves traced out in spacetime by particles that are freely falling under gravity.
(For the pedants: When I say "connection" here, I always mean "torsion free Levi Civita connection")
Curvature
Staying with my surface-of-sphere manifold example, if I take a vector at the North pole, parallel transport it along a line of longitude to a point on the equator, parallel transport it a quarter way round the sphere along the equator, and parallel transport it back up to the North pole along a line of longitude. If I do this, it ends up pointing in a different direction from the one it started in, simply because of the transport round the path. The picture of this process is herehttp://en.wikipedia.org/wiki/Connection_%28mathematics%29
If parallel transport round closed circuits like this causes the vector to change relative to its initial state, as it does in the case of the sphere, then the manifold is curved. If I were to do a similar exercise on a cylinder, the vector would end up back where it started from, since the cylinder is not a curved manifold.
Mathematically, I can imagine moving vectors around an infinitesimal quadrilateral circuit bordered by a pair of other vectors and in this way generate the Riemann curvature tensor.The Riemann curvature tensor will have 4 indices – two for the vectors bounding the infinitesimal quadrilateral, and two to make up a matrix with which to transform our test vector after it’s been parallel transported around the infinitesimal quadrilateral. Denote the Riemann tensor by Rabcd . It has several symmetry properties, and you end up with only 20 independent components.
In GR, given the metric, I can compute (by differentiation), the connection coefficients, and by some further differentiation, the Riemann curvature tensor. So the metric lies at the root of the interesting geometrical features – it’s a bit like the potential in electromagnetism – you can get all the physical information you want from it by differentiating. Think of the metric in terms of a potential.
Ricci Tensor and Ricci Scalar
Remember I said that the Riemann curvature tensor had 4 indices Rabcd, .If I sum over the first and third indices, I end up with something with only two indices
Rbd = sum over index a (Rabad)
This is called the Ricci tensor and plays a key part in GR. It has 10 independent components.
Above I mentioned the idea of a geodesic as a curve traced out in spacetime by something freely falling. If I take a small sphere of test particles and let them go in free fall, then the Ricci tensor is associated with the volume change of this sphere.
I can throw away even more information if I sum over the two indices b,d of the Ricci tensor. The object I end up with is the Ricci scalar, or “curvature scalar”, called R.
Energy-Momentum Tensor
The final ingredient for GR, the energy momentum tensor Tab has two indices. We’ll use the letters a, b, c , d for spacetime indices, i.e. they each run from 0 to 3, and the letters i, j, k, l for space indices, i.e. they run from 1 to 3. Given some matter, component Tij is the i’th component of the momentum flux in the j’th direction. What this means is, imagine a surface whose normal is the j’th direction, then the “flux in the jth direction” is the momentum flowing across this surface. T00 is the energy density etc.
Einstein’s Equations
Ignoring the cosmological constant and the occasional π Einstein’s equations are just
Rab – 1/2Rgab = Tab
Written like this, they look fairly simple, but remember that Rab is a complicated expression involving the Γabc and Γabc is a complicated expression involving the gab. So given a matter distribution Tab, solving for the gab is not an easy matter. Only a handful of exact solutions are known.
In a vacuum, Tab = 0 and Einstein’s equation reduces to Rab = 0. That means 10 curvature components have vanished – however remember that the Riemann tensor has 20 components, the remaining 10 are encapsulated in the Weyl tensor – this is basically the content of the Riemann tensor which isn’t in the Ricci tensor. We said above that the Ricci tensor causes volume change in a sphere of freely falling test particles, well, the effect of the Weyl tensor is to deform the sphere away from its initial spherical shape – it’s a shearing effect, or a tidal force.
Initial Value Formulation
I will hopefully get around to talking about LQG eventually (but not in this post!). LQG theory is constructed in a similar way to an old treatment of GR called the canonical formalism.
The development of the canonical formalism was motivated by a desire to isolate the independent “physical degrees of freedom” of GR. Quantization relies on writing down commutation (or anticommutation for fermions) relations between operators representing these physical degrees of freedom and their conjugates. So essentially by "physical degrees of freedom" I mean a minimal set of functions I can write down, which contain all the physics.
The Riemann curvature tensor has 20 components, but I can get all these from the metric tensor components by differentiating (just like in Maxwell’s theory I can get E and B fields from the electromagnetic potential vector Ab by differentiating). So surely, these metric components are the physical degrees of freedom ? To see why it may not be so simple, we’ll start by taking a look at the easier case of electromagnetism.
In classical Maxwell theory, the electric field vector E has 3 components Ei (i=1,2,3) and the magnetic vector B has 3 components Bi. In trying to find the true physical degrees of freedom, we can do a bit better though, we can use the four-potential Aa = (φ, A), where E=-grad(φ), and B=curl(A). φ is a scalar and A is a three dimensional vector. In fact it's even better than this - I can make changes to the potential of the form Aa ->Aa+daω without altering the physics (ω is just a scalar function). Such changes to Aa are called gauge transformations (another note to pendants - I am leaving out global gauge issues, such as the Aharonov Bohm effect here). In the absence of sources (charges and currents), I can reduce everything to two physical degrees of freedom (which I can express, for example as left and right circularly polarised waves).
That was just the Maxwell case. As you might guess, the same sort of thing applies for the GR case only it's a lot more complicated. People started a long time ago to try to isolate the physical degrees of freedom in GR by asking the question how many functions (and their first time derivatives) would I have to specify at t=0 in order that Einstein's equations would then determine the rest of the spacetime ?
Now in GR what does t=0 mean ? It means that I've introduced a coordinate system and I'm looking at the subset of spacetime I get if I set the time coordinate to zero. Such a subset with a fixed time coordinate is called a spacelike hypersurface, and it might be wiggly because spacetime is wiggly, or even if spacetime is flat it might be wiggly because I happen to have chosen a wiggly coordinate system. Separating these reasons for wiggliness is a large part of the problem. The first systematic attempt to do this that I know of was done by Arnowitt, Deser and Misner, and is called the ADM formalism (although I'm pretty sure I read somewhere that Dirac did something along these lines previously). The ADM reference is a classic.
I've tried to show their approach in this diagram:
Spacetime is 4 dimensional, but I will not draw 2 of the space dimensions (because I can't draw them) and just draw 1 space and 1 time dimension. I have a spacelike hypersurface at time x0 (which will be a 3 dimensional surface, represented as a wiggly line in my diagram), and another one at a slightly later time x0+dx0. What happens if you look at a fixed space coordinate, x (as in the Maxwell discussion above, bold symbols are 3 dimensional, and so have 3 components) ? You get a wiggly vertical line (labelled by what I call a "3-coordinate" in the diagram because it coordinatizes the 3 space dimensions). I've also shown the nearby fixed space coordinate x+dx.
DA is the infinitesimal vector, at D, normal (according to the spacetime metric gab) to the surface, with its tip on the second surface at A.
DA = Ndx0 for some scalar function N
If, instead of going in the normal direction along DA, I were to keep the space coordinate x constant, I would have travelled up the integral curve of x0 and ended up at point B. If I initially went up DA I can get back to B by going along a 3-vector AB, tangent to the 3 surface. The coordinates of this tangent vector AB are traditionally denoted Ni, i=1..3.
N and Ni are called respectively the lapse function and shift vector (or as Paul Tod used to say, lips and shaft). If I want to change the point in the 3 surface a bit as well, I get to point C. Working like this I can decompose my metric as
gab = [ -N2+sum over i,j(NiNjgij).......Nj ]
.......[ Ni..................................gij ]
Here gij is just the metric "induced" in the 3 surface from the metric gab on spacetime.
N and Ni aren't physical degrees of freedom - they're just objects which tell you how to evolve in time. The physical degrees of freedom are contained in gij.
In order to attempt to quantize the system, a set of conjugate variables πij are required (think of the p's in elementary quantum mechanics, being conjugate to the q's). There are four equations that the gij and πij components have to satisfy among themselves which can be derived from the ADM formalism, called the constraint equations.
So you can see that in GR, the question of what is fundamentally physical and what is not is quite difficult.
LQG does not actually use the ADM decomposition, it uses a slightly less obvious description of GR, invented by Abhay Ashtekar, which I might describe in a later post ( when I've sorted out how it works !). I just mentioned the ADM formalism to give a flavour of what’s involved in trying to isolate the physical degrees of freedom.
In the next post, I’ll give some overview of how spacetime arises in LQG from spin networks.
Edited for formatting
In this post I want to say a bit about the canonical formulation of GR, since LQG is based on something similar. To do this I’ll start by saying what the mathematical ingredients of GR are – just a flavour no pretence of completeness or rigour.
Manifold
A manifold is a geometric object that looks locally like a piece of Euclidean space RN. RN is just the space of ordered N-tuples of real numbers (x1,...xN). So a (N dimensional) manifold has the property that I can plaster it with patches, and the points in each patch are labelled by the coordinates x1...xN. I may not be able to find one patch to do the job over the whole manifold – for example if I take the 2 dimensional manifold which is the surface of a sphere, I can make much of it look like R2 by using polar coordinates θ and φ. These can’t be used everywhere however, since the poles don’t have a well defined, unique φ coordinate. I can, however, cover the sphere with overlapping patches such that each patch looks like a piece of R2. In GR, the manifold is spacetime, and there are 4 coordinates. Most people number the coordinates from 0 to 3, with 0 being the time coordinate.
Tangent Vector
It’s fairly intuitive what one of these is (but a little more involved to define mathematically). Taking the sphere, I can imagine a tangent vector at a point to be a little arrow stuck onto the sphere at a point P and perpendicular to the line from the origin to P. (If I were to define tangent vector mathematically I wouldn’t need to use the concept of “perpendicular” here).
Metric
The metric (which means “measure”) is a rule which defines how you can take the scalar product of any 2 tangent vectors, so you can use it to compute the angle between the vectors or the length of a vector. Since on an N dimensional manifold I can define N basis tangent vectors, my metric just needs to define how to take the scalar product of any 2 basis vectors, so it has N2 components. However, scalar products are symmetric, so it really only has N(N+1)/2 independent components.
Connection
A connection is a mathematical object which tells you how to parallel transport a vector. Parallel transport is concerned with how the vector changes as you change the point at which it’s attached to the manifold. Suppose it’s attached at point P1 and I want to “move” it to point P2 along a curve C connecting P1 and P2. Conceptually
a) At each intermediate point P along the way, it must become a remain vector at P i.e. it’s not allowed to “dip in” or “dip out” of the manifold – it stays in the tangent space
b) It “preserves the metric”, i.e. keeps the vector the same length
c) It stays “as near as possible to staying parallel to itself whilst respecting (a) and (c)
The mathematical object that tells you how to do this is called a “connection coefficient”, or sometimes “Christoffel symbol”. Γabc
A curve which parallel transports its own tangent vector is called a geodesic and in GR geodesics are the curves traced out in spacetime by particles that are freely falling under gravity.
(For the pedants: When I say "connection" here, I always mean "torsion free Levi Civita connection")
Curvature
Staying with my surface-of-sphere manifold example, if I take a vector at the North pole, parallel transport it along a line of longitude to a point on the equator, parallel transport it a quarter way round the sphere along the equator, and parallel transport it back up to the North pole along a line of longitude. If I do this, it ends up pointing in a different direction from the one it started in, simply because of the transport round the path. The picture of this process is herehttp://en.wikipedia.org/wiki/Connection_%28mathematics%29
If parallel transport round closed circuits like this causes the vector to change relative to its initial state, as it does in the case of the sphere, then the manifold is curved. If I were to do a similar exercise on a cylinder, the vector would end up back where it started from, since the cylinder is not a curved manifold.
Mathematically, I can imagine moving vectors around an infinitesimal quadrilateral circuit bordered by a pair of other vectors and in this way generate the Riemann curvature tensor.The Riemann curvature tensor will have 4 indices – two for the vectors bounding the infinitesimal quadrilateral, and two to make up a matrix with which to transform our test vector after it’s been parallel transported around the infinitesimal quadrilateral. Denote the Riemann tensor by Rabcd . It has several symmetry properties, and you end up with only 20 independent components.
In GR, given the metric, I can compute (by differentiation), the connection coefficients, and by some further differentiation, the Riemann curvature tensor. So the metric lies at the root of the interesting geometrical features – it’s a bit like the potential in electromagnetism – you can get all the physical information you want from it by differentiating. Think of the metric in terms of a potential.
Ricci Tensor and Ricci Scalar
Remember I said that the Riemann curvature tensor had 4 indices Rabcd, .If I sum over the first and third indices, I end up with something with only two indices
Rbd = sum over index a (Rabad)
This is called the Ricci tensor and plays a key part in GR. It has 10 independent components.
Above I mentioned the idea of a geodesic as a curve traced out in spacetime by something freely falling. If I take a small sphere of test particles and let them go in free fall, then the Ricci tensor is associated with the volume change of this sphere.
I can throw away even more information if I sum over the two indices b,d of the Ricci tensor. The object I end up with is the Ricci scalar, or “curvature scalar”, called R.
Energy-Momentum Tensor
The final ingredient for GR, the energy momentum tensor Tab has two indices. We’ll use the letters a, b, c , d for spacetime indices, i.e. they each run from 0 to 3, and the letters i, j, k, l for space indices, i.e. they run from 1 to 3. Given some matter, component Tij is the i’th component of the momentum flux in the j’th direction. What this means is, imagine a surface whose normal is the j’th direction, then the “flux in the jth direction” is the momentum flowing across this surface. T00 is the energy density etc.
Einstein’s Equations
Ignoring the cosmological constant and the occasional π Einstein’s equations are just
Rab – 1/2Rgab = Tab
Written like this, they look fairly simple, but remember that Rab is a complicated expression involving the Γabc and Γabc is a complicated expression involving the gab. So given a matter distribution Tab, solving for the gab is not an easy matter. Only a handful of exact solutions are known.
In a vacuum, Tab = 0 and Einstein’s equation reduces to Rab = 0. That means 10 curvature components have vanished – however remember that the Riemann tensor has 20 components, the remaining 10 are encapsulated in the Weyl tensor – this is basically the content of the Riemann tensor which isn’t in the Ricci tensor. We said above that the Ricci tensor causes volume change in a sphere of freely falling test particles, well, the effect of the Weyl tensor is to deform the sphere away from its initial spherical shape – it’s a shearing effect, or a tidal force.
Initial Value Formulation
I will hopefully get around to talking about LQG eventually (but not in this post!). LQG theory is constructed in a similar way to an old treatment of GR called the canonical formalism.
The development of the canonical formalism was motivated by a desire to isolate the independent “physical degrees of freedom” of GR. Quantization relies on writing down commutation (or anticommutation for fermions) relations between operators representing these physical degrees of freedom and their conjugates. So essentially by "physical degrees of freedom" I mean a minimal set of functions I can write down, which contain all the physics.
The Riemann curvature tensor has 20 components, but I can get all these from the metric tensor components by differentiating (just like in Maxwell’s theory I can get E and B fields from the electromagnetic potential vector Ab by differentiating). So surely, these metric components are the physical degrees of freedom ? To see why it may not be so simple, we’ll start by taking a look at the easier case of electromagnetism.
In classical Maxwell theory, the electric field vector E has 3 components Ei (i=1,2,3) and the magnetic vector B has 3 components Bi. In trying to find the true physical degrees of freedom, we can do a bit better though, we can use the four-potential Aa = (φ, A), where E=-grad(φ), and B=curl(A). φ is a scalar and A is a three dimensional vector. In fact it's even better than this - I can make changes to the potential of the form Aa ->Aa+daω without altering the physics (ω is just a scalar function). Such changes to Aa are called gauge transformations (another note to pendants - I am leaving out global gauge issues, such as the Aharonov Bohm effect here). In the absence of sources (charges and currents), I can reduce everything to two physical degrees of freedom (which I can express, for example as left and right circularly polarised waves).
That was just the Maxwell case. As you might guess, the same sort of thing applies for the GR case only it's a lot more complicated. People started a long time ago to try to isolate the physical degrees of freedom in GR by asking the question how many functions (and their first time derivatives) would I have to specify at t=0 in order that Einstein's equations would then determine the rest of the spacetime ?
Now in GR what does t=0 mean ? It means that I've introduced a coordinate system and I'm looking at the subset of spacetime I get if I set the time coordinate to zero. Such a subset with a fixed time coordinate is called a spacelike hypersurface, and it might be wiggly because spacetime is wiggly, or even if spacetime is flat it might be wiggly because I happen to have chosen a wiggly coordinate system. Separating these reasons for wiggliness is a large part of the problem. The first systematic attempt to do this that I know of was done by Arnowitt, Deser and Misner, and is called the ADM formalism (although I'm pretty sure I read somewhere that Dirac did something along these lines previously). The ADM reference is a classic.
I've tried to show their approach in this diagram:
- ADM.jpg (26.54 KiB) Viewed 11028 times
Spacetime is 4 dimensional, but I will not draw 2 of the space dimensions (because I can't draw them) and just draw 1 space and 1 time dimension. I have a spacelike hypersurface at time x0 (which will be a 3 dimensional surface, represented as a wiggly line in my diagram), and another one at a slightly later time x0+dx0. What happens if you look at a fixed space coordinate, x (as in the Maxwell discussion above, bold symbols are 3 dimensional, and so have 3 components) ? You get a wiggly vertical line (labelled by what I call a "3-coordinate" in the diagram because it coordinatizes the 3 space dimensions). I've also shown the nearby fixed space coordinate x+dx.
DA is the infinitesimal vector, at D, normal (according to the spacetime metric gab) to the surface, with its tip on the second surface at A.
DA = Ndx0 for some scalar function N
If, instead of going in the normal direction along DA, I were to keep the space coordinate x constant, I would have travelled up the integral curve of x0 and ended up at point B. If I initially went up DA I can get back to B by going along a 3-vector AB, tangent to the 3 surface. The coordinates of this tangent vector AB are traditionally denoted Ni, i=1..3.
N and Ni are called respectively the lapse function and shift vector (or as Paul Tod used to say, lips and shaft). If I want to change the point in the 3 surface a bit as well, I get to point C. Working like this I can decompose my metric as
gab = [ -N2+sum over i,j(NiNjgij).......Nj ]
.......[ Ni..................................gij ]
Here gij is just the metric "induced" in the 3 surface from the metric gab on spacetime.
N and Ni aren't physical degrees of freedom - they're just objects which tell you how to evolve in time. The physical degrees of freedom are contained in gij.
In order to attempt to quantize the system, a set of conjugate variables πij are required (think of the p's in elementary quantum mechanics, being conjugate to the q's). There are four equations that the gij and πij components have to satisfy among themselves which can be derived from the ADM formalism, called the constraint equations.
So you can see that in GR, the question of what is fundamentally physical and what is not is quite difficult.
LQG does not actually use the ADM decomposition, it uses a slightly less obvious description of GR, invented by Abhay Ashtekar, which I might describe in a later post ( when I've sorted out how it works !). I just mentioned the ADM formalism to give a flavour of what’s involved in trying to isolate the physical degrees of freedom.
In the next post, I’ll give some overview of how spacetime arises in LQG from spin networks.
Edited for formatting