What is gauge in General Relativity?

[oldman]

What is "gauge" in General Relativity?

I am trying to concoct a dumbed-down explanation of the significance of gauge in physics (and especially in General Relativity (GR)) that my limited intellect can cope with. I need some serious correcting about how to unravel the differences between “gauge” in mathematics and in physics:

To mathematicians (see for instance Terence Tao’s explanation) gauge seems to be associated with
using coordinates to quantitatively describe and model a geometrical object. He writes:

Classical mathematics, such as practiced by the ancient Greeks, could be loosely divided into two disciplines, geometry and number theory, where I use the latter term very broadly, to encompass all sorts of mathematics dealing with any sort of number. The two disciplines are unified by the concept of a coordinate system, which allows one to convert geometric objects to numeric ones or vice versa.

(e.g. to convert lengths into numbers associated with units).

The utility of such conversion is that the tools of mathematical analysis may then be used, say to compare objects when coordinate systems vary with

“location” with respect to some “base space” or“parameter space

say Differential Geometry when the base space being treated is non-Euclidean. Or, in ordinary space, integration, Fourier analysis etc.

To use coordinates one must choose items like: a scale, units, an origin, an orientation and maybechirality (direction of circulation, clockwise or anticlockwise). This is called fixing a gauge. If the same fixed gauge is chosen across the space spanned by the object or objects being described,the gauge is said to be global. If the choice varies from one location to another it is said to be local. Terence Tao writes:

By fixing a gauge (thus breaking or spending the gauge symmetry), the model becomes something easier to analyse mathematically, such as a system of partial differential equations (in classical gauge theories) or a perturbative quantum field theory (in quantum gauge theories),though the tractability of the resulting problem can be heavily dependent on the choice of gauge that one fixed. Deciding exactly how to fix a gauge (or whether one should spend the gauge symmetry at all) is a key question in the analysis of gauge theories, and one that often requires the input of geometric ideas and intuition into that analysis.

In physics (which is much younger than mathematics) quantitative mathematical models ofphysical objects and their behaviours depend on parameters --- quantities that vary from case to case. A model may be said to span a “parameter space”, where, I suppose, it forms a geometrical object complete with a numerical character that can be modeled and analysed mathematically.

In situations where a parameter cannot be measured physically, as with the vector potential in the theory of electromagnetism, the arbitrarily fixed “gauge” of the parameter space is global (and explains electric charge conservation). Here the gauge-invariant parameter space is Euclidean so that differential geometry need not be invoked. A similar situation prevails when the parameter space is circular, modulo 2 pi, as with the phases of wave functions in quantum mechanics.

The non-Euclidean geometry of the four-dimensional spacetime of General Relativity, on the other hand, is rendered dynamic by the presence of mass and energy, which means that differential geometry, or tensor analysis with coordinates, is used numerically to describe geometrical objects. (Marcus has pungently described this as: “... Coordinates are "gauge" meaning physically meaningless redundant trash”...) But in this case the gauge used to convert geometrical objects in spacetime into numeric objects, although invariant, is just the S.I.: confusingly even when "dynamic" means the “expansion” of the isotropic everywhere-and-everywhen FLRW model of cosmology. Its invariance across spacetime corresponds to the conservation of momentum and energy (at least on a local scale).

Is this the sense in which GR is considered to be a gauge theory?

Or is it only when GR is treated as a quantum field theory with excitations called gravitons that GR is classed as a gauge theory?

Or why (without too much jargon, please) is GR classed as a gauge theory?

 

[atyy]

What is "S.I."?

 

[George Jones]

Or why (without too much jargon, please) is GR classed as a gauge theory?

I haven't thought much about them, and they might not meet your criterion, but see posts *3 and #4 in

https://www.physicsforums.com/showthread.php?p=1294659#post1294659.

 

[Phrak]

Simple language; the way I like it. A gauge symmetry is a continuous symmetry.

What is a symmetry? A symmetry means you change something (call it A) and something else (call it B) doesn't change. B is symmetrical under changes of A.

As mundane as it sounds, this is what a symmetry means.

What's a continuous symmetry? 'A' changes continuously, over the real numbers for instance, rather than in jumps like the integers.

 

[oldman]

What is "S.I."?

Apology for this bit of jargon, especially since I asked others not to use too much of it!

S.I. stands for Systeme Internationale, the name of the system of units (Metre, Kilogram Second, etc.), that physicists and others outside the United States of America (U.S.A.) use.

 

[oldman]

Simple language; the way I like it. A gauge symmetry is a continuous symmetry.

What is a symmetry? A symmetry means you change something (call it A) and something else (call it B) doesn't change. B is symmetrical under changes of A.

As mundane as it sounds, this is what a symmetry means.

What's a continuous symmetry? 'A' changes continuously, over the real numbers for instance, rather than in jumps like the integers.

Thanks --- I understand this, but I'm still puzzled as to why exactly General Relativity is known as a gauge theory. Sure, the way it is presented in texts is with coordinates, which require a gauge, and that gauge is invariant (has continuous symmetry) --- even when applied to cosmology. But this seems too trivial a reason for making a fuss about the importance of gauge, in this case at least. Gauge invariance is usually associated with conservation laws, but ordinary classical mechanics also has gauge invariance (again the Systeme Internationale) which leads to energy and momentum conservation: so in this respect: what's new about General Relativity?

 

[oldman]

I haven't thought much about them, and they might not meet your criterion, but see posts *3 and #4 in

https://www.physicsforums.com/showthread.php?p=1294659#post1294659.

Thanks, George, for directing me to this thread. When folk like John Baez and MeJennifer get to puzzling, I'm a bit out of my league. But I'll mull in a doggy sort of way at what they say, and there's just a chance that I'll see the light.

 

[Fredrik]

Can someone explain to me what coalquay404 is doing here?

Everyone knows that a necessary and sufficient condition for the action to be stationary are that the Euler-Lagrange equations are satisfied:

$$L_i = -\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}^i}\right) + \frac{\partial L}{\partial q^i} = 0$$

What is rarely mentioned, however, is that a much more enlightening way to write the Euler-Lagrange equations is

$$L_i = -W_{ij}(q,\dot{q})\ddot{q}^j + V_i = 0,$$

where

$$W_{ij}(q,\dot{q}) = -\frac{\partial^2L}{\partial\dot{q}^i\partial\dot{q}^j}$$
$$V_i = -\frac{\partial^2L}{\partial\dot{q}^i\partial q^j}$$

From this you can see that the $\ddot{q}^i$s at a given time are uniquely determined by $(q,\dot{q})$ at that time if and only if $W_{ij}$ is invertible.

This is what I get:

$$0=-\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}^i}\right) + \frac{\partial L}{\partial q^i}=-\frac{\partial^2 L}{\partial \dot q^i \partial q^j}\dot q^j-\frac{\partial^2 L}{\partial \dot q^i \partial \dot q^j}\ddot q^j+\frac{\partial L}{\partial q^i}=W_{ij}(q,\dot q)\ddot q^j-\frac{\partial^2 L}{\partial \dot q^i \partial q^j}\dot q^j+\frac{\partial L}{\partial q^i}$$

 

[xepma]

The notion of gauge invariance in General Relativity has to do with the principle of diffeomorphism invariance. The "action" of a diffeomorphism results in the same thing as a coordinate transformation, yet there is a subtle difference.

As Carroll in his book puts it: diffeomorphisms are "active" (coordinate) transformations, while traditional [sic] coordinate transformations are "passive".

To be more precise: a passive transformations corresponds to simply a new choice of coordinates. You have some manifold $$M$$, and some coordinate system $$x^{\mu}: M\rightarrow\mathbf{R}^n$$. But we can also choose some other coordinate system, $$y^{\mu}: M\rightarrow\mathbf{R}^n$$. Standard rules tell you how components of tensors etc change. This is the passive point of view, since we only change the way we choose to describe the system.

In the active point of view we make use of diffeomorphisms, pullbacks and pushforwards. We choose one particular (global) coordinate system, $$x^{\mu}: M\rightarrow\mathbf{R}^n$$. Suppose we now have a diffeomorphism of the manifold M to itself: $$\phi: M\rightarrow M$$, i.e. some smooth mapping which is invertible. This mapping is an active mapping: it maps point on the manifold M to other points on the manifold. We can use this diffeomorphism to also map any tensors, specifally, the metric $$g_{\mu\nu}$$. Note the similarity with the passive point of view: the pushforward of the coordinate mapping defines a new coordinate system! $$(\phi^\star x)^\mu:M\rightarrow \mathbf{R}^n$$. This is why these two concepts are so similar.

But back to diffeomorphism: what does it mean in the context of general relativity? Suppose we have a manifold $$M$$, with some curvature $$g_{\mu\nu}$$ caused by some matter distributions represented by the fields $$\psi$$, i.e. the set $$(M,g_{\mu\nu},\psi)$$. We can use some diffeomorphism $$\phi: M\rightarrow M$$ and construct a new physical system on M, defined by the triple $$(M,\phi^{\star}g_{\mu\nu},\phi^{\star}\psi)$$. Diffeomorphism invariance means that these two systems represent the same physical situation.

Is this a powerful statement? Is this an eye-opener? As Carroll puts it: it actually conveys very little information. To quote him: the theory is free of "prior geometry", and there is no preferred coordinate system for spacetime. What's the consequence of this? Well, it's the idea of some "hidden" gauge structure - a redundancy present in the way we choose to describe the physics. In this case we choose some geometric structure, with a metric, matter fields and a coordinate system. But another configuration, with a different metric, matter fields, etc, might describe the same physical system - because of diffeomorphism invariance. So these two systems are not physically distinguishable.

One situation where this comes into play is with the treatment of perturbed spacetimes. We might have some background metric $$g_{\mu\nu}$$ with some small perturbation on this metric, $$h_{\mu\nu}$$. The existence of diffeomorphism invariance allows us to construct a whole family of perturbations, $$h_{\mu\nu}^{\epsilon}$$, all small and all with respect to the same background metric. They are related through (infinitesimal) diffeomorphisms and we can switch from one to the other via a gauge transformation. Under such tranformations the perturbation $$h_{\mu\nu}$$ changes, but the Riemann tensor does not. (Note that there also exist diffeomorphisms which blow up the perturbation, such that it isn't small anymore). The presence of this gauge influences, for instance, the number of physical degrees of freedom present in the perturbation.

But this comes down to the same idea that we can choose a different coordinate system to describe the perturbation! But it is not completely the same.

I'll be glad to here some comments on my remark :)

 

[xepma]

Can someone explain to me what coalquay404 is doing here?

This is what I get:

$$0=-\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}^i}\right) + \frac{\partial L}{\partial q^i}=-\frac{\partial^2 L}{\partial \dot q^i \partial q^j}\dot q^j-\frac{\partial^2 L}{\partial \dot q^i \partial \dot q^j}\ddot q^j+\frac{\partial L}{\partial q^i}=W_{ij}(q,\dot q)\ddot q^j-\frac{\partial^2 L}{\partial \dot q^i \partial q^j}\dot q^j+\frac{\partial L}{\partial q^i}$$

Something is missing. The definition of $$V_i$$ cannot be correct - it has too many components (one the l.h.s., 2 on the r.h.s.).

 

[atyy]

A gauge is a redundancy in our description of a physical system.

In electrostatics, the electric potential is meaningless, only potential differences are meaningful - hence when we use the potential, we are using a non-unique, but convenient description of describing a physical system.

In electrodynamics, the vector potential is similarly meaningless, because you can add the derivative of another field to it, and still describe the same physical situation.

In GR, coordinates in which we use to write the metric are also non-unique - as more than one set of coordinates describe the same physical situation.

The electric potential, vector potential, metric are all specified by differential equations in electrostatics, electrodynamics and GR respectively, and in this sense are "dynamical". So if we restrict ourselves to talking about how multiple solutions of the dynamical equations describe the same physical situation, then change of coordinates is the "gauge" of GR.

However, just because a field is not "dynamical" does not mean it is not "physical". In all theories with a metric, including special relativity, we can always change to arbitrary coordinates, as long as we change the coordinates for the metric also - after all the metric is a physical object like a ruler. The difference is that in special relativity, the metric is fixed regardless of the matter content, and the metric has the global special form diag(-1,1,1,1), and the Lorentz transformations are especially useful. In comparison, the metric is not independent of the matter content in GR, and the metric does not have that form globally, and all coordinate transformations become equally useful - because the Lorentz transformations are no longer especially useful.

 

[oldman]

The notion of gauge invariance in General Relativity has to do with the principle of diffeomorphism invariance. The "action" of a diffeomorphism results in the same thing as a coordinate transformation, yet there is a subtle difference.

As Carroll in his book puts it: diffeomorphisms are "active" (coordinate) transformations, while traditional [sic] coordinate transformations are "passive"...



...Is this a powerful statement? Is this an eye-opener? As Carroll puts it: it actually conveys very little information. To quote him: the theory is free of "prior geometry", and there is no preferred coordinate system for spacetime. What's the consequence of this? Well, it's the idea of some "hidden" gauge structure - a redundancy present in the way we choose to describe the physics. In this case we choose some geometric structure, with a metric, matter fields and a coordinate system. But another configuration, with a different metric, matter fields, etc, might describe the same physical system - because of diffeomorphism invariance. So these two systems are not physically distinguishable...


But this comes down to the same idea that we can choose a different coordinate system to describe the perturbation! But it is not completely the same.

I'll be glad to here some comments on my remark :)

This post, taken together with that of Coalquay404 (referred to by ] George Jones [/url] here) clarifies the situation for me. It seems that whether one regards General Relativity as a gauge theory (in the sense that electromagnetism and other field theories are gauge theories)--- or not --- hinges on the distinction between isomorphism and diffeomorphism. This distinction looks to me (no mathematician I!) rather like an abstruse technical distinction with, in this case, small physical content. It can be made much of, as Steven Weinstein has done (pointed to by MeJennifer via George Jones, as above) but this may well be much ado about nothing. In contrast, others (like Lee Smolin) think that this distinction lies at the very heart of Quantum Gravity.

Is the Texas saying: "All hat and no cattle" appropriate here? I can't decide.

 

[Mentz114]

GR can be deduced from scratch, as it were, by gauging the translation group. This means that the global translational invariance of the Lagrangian ( which is necessary to ensure global energy and momentum conservation ) is made local. The global symmetry is thus broken and to ensure conservation of E&P a gauge field must be introduced. This turns out to be gravity encoded in the torsion of a Riemann-Cartan space. It transforms to ordinary GR quite simply. There is a consensus that the gauge group of gravity is the translation group. This ties in with the fact that the Noether current of the translation group is also the source of gravity.

A good reference is Gronwald and Hehl (1996) arXiv:gr-qc\9602013.

 

[Rebel]

... It seems that whether one regards General Relativity as a gauge theory (in the sense that electromagnetism and other field theories are gauge theories)--- or not --- hinges on the distinction between isomorphism and diffeomorphism. This distinction looks to me (no mathematician I!) rather like an abstruse technical distinction with, in this case, small physical content. ...

How is that about isomorphism and diffeomorphism? Can you please deep a little bit more in this?

 

[oldman]

I am still trying to concoct a dumbed-down explanation of the significance of gauge in physics (and especially in General Relativity (GR)) that my limited intellect can cope with. For the record, this is what I've come up with. It's not a theoretical physicists view, but it may be distantly relation to one:

Suppose an electrician wants to investigate and then describe the state of an electrical
circuit. To do this he might use a voltmeter as a “gauge” to measure a parameter that
provides a quantitative description of the state of the circuit — in this case voltage.
Suppose the circuit consists only of a battery with a condenser connected across its
terminals. When the condenser is fully charged the static equilibrium state of the circuit
can be described by plotting a graph of voltage versus position along the circuit.

It turns out that the shape of this graph doesn’t depend on where the electrician
chooses to connect one of the voltmeter’s two probes while he uses the other to
measure voltages, or on whether the circuit is connected (at anyone point) to an
earthed water pipe or (very carefully) to an 11 kilovolt power line.

The electrician’s description doesn’t depend on the zero he chooses for voltage. He
might say that the circuit can “float” at any voltage.

In physics one might say that the circuit has a global symmetry – because the same
(global) change can be made to the parameter “voltage” at all points in the circuit
without affecting the system. Or the size of the unit used to gauge voltage, the volt,
could be changed (redefined) for all measurements.

When you do something to a system and it doesn’t change, you’ve got a symmetry.

The situation can be simplified by imagining the circuit sans battery and condenser —
an “empty” circuit whose state is described by drawing a straight-line graph, with zero
slope, of voltage (aka electrical potential) against position. This empty circuit is also globally
symmetric.

The distinction between these two circuits is the presence in the first circuit of isolated
electrical charges (charges that are macroscopically displaced from their microscopic
positions by the electromotive force of the battery) and static electrical fields
(which may be regarded as gradients of potential) inside the battery and between the
plates of the condenser.

The common global symmetry of these circuits is an invariance associated with the
conservation of net charge, which is the difference between total positive and negative
displaced charge. In these circuits this net charge is zero because the circuits are
closed loops.

The concept of symmetry here described as “global” can be extended to “local” scales
if it is recognised that variations of potential along the circuit can be regarded as
variations from place to place in the choice of zero for potential, or of the unit for
gauging potential. This feature in the description of the state of the circuit is called a
“gauge symmetry” or “local symmetry”. It is associated with the twin presences of
electric “charges” and electric “fields”, in terms of which an alternative and equivalent
description of the circuit’s state can be given. A description using charges and fields is
equivalent to one using potential — electric fields are gradients of potential and the
forces fields exert on charges are known as “gauge forces”.

Of course electricity is not in practice a static phenomenon. The accepted analysis and
description of systems in which charges move and fields change — the theory of
electrodynamics — was formulated by Maxwell some 150 years ago. It incorporates
the complicating concept of a magnetic field (later shown to be a relativistic
phenomenon) but retains the features of global symmetry, charge conservation and
gauge or local symmetry described above. Electromagnetism is now classed as a
gauge theory that involves a generalised form of potential (the vector potential) which
determines the observable electric and magnetic fields of a system of moving charges.
Electromagnetism is now considered to be the original gauge theory which can be
described in terms of a parameter that can be differently defined or scaled everywhere.

It has turned out that other interactions of nature are also described by “gauge”
theories in which physically measurable quantities (like electromagnetic interactions
and the “weak” interactions of particle physics) are linked to local changes in
parameters that are globally invariant and locally gauge symmetric. Indeed the various
kinds of known particles (Fermions like electrons, neutrinos and quarks) are in gauge
theories such as Quantum Electrodynamics and Quantum Chromodynamics
themselves regarded as “matter fields”, together with their Bosons (photons in
electromagnetism), which are called “force carrier fields”. In “gauge theories” systems
are unaffected by global changes in the potential energy.

The best description we have of gravity, Einstein’s General Relativity (GR), is usually
regarded as a gauge theory because more than one set of coordinates can describe
the same physical situation.

But is it possible that GR could be regarded as a gauge theory in a quite different
sense?

The sometimes odd choice of the word “gauge” in modern physics goes back to a
discarded idea of Weyl’s, namely that the metre — the unit we use to define distance;
the “gauge” of distance — could be arbitrarily rescaled without affecting a physical
system. Weyl was then attempting (unsuccessfully) to unify electromagnetism with GR.
Consider revisiting this original concept of "gauge".

Our modern definition of the gauge of distance — the metre — is the distance
traveled by light in a certain time. If distance were to be rescaled "á le Weyl" , modern
physics would attribute this to the rescaling of either c, or the defined second, or both.

However the speed of light c is a universal constant, always and everywhere measured
locally to be the same. And the laws of physics, which incorporate c intimately in
various ways (as in the fine structure constant) are observed to operate in the same
way always and everywhere. Could c nevertheless be globally or locally rescaled
without affecting physical systems, and without affecting its locally measured
invariance? I don’t know — it seems unlikely.

But starlight is deviated by the sun and gravitational lensing of remote galaxies is
observed, which is reminiscent of the ordinary refraction and lensing of light, where
light moves slower in an optical medium than in vacuum.

On the other hand the Pound-Rebka experiment shows that the gravitational time
dilation of GR causes light from systems in a gravitational well to appear red shifted to
outside observers. And it is accepted that on a black hole event horizon outside-
observer-time appears to stop, while infalling observers experience invariant local
physics.

But could not these phenomena are viewed as a local rescaling of c (or time) that
leaves local physics invariant? Then the observed local invariance of c we are so
familiar with tells us that such rescaling is unobservable, as is the rescaling of potential
in electromagnetism. And the twin entities mass/energy and gravitational force could
be regarded as analogies of the matter fields and force carrier fields of other gauge
theories. Crazy stuff.

 

[Haelfix]

This is another example of a topic where language can get you into trouble and where not everyone agrees on definitions. Things become a little bit context dependant and technical.

For something more along the lines of what Coalquay mentioned (which is more how I tend to think of things), I suggest reading Henneaux and Teitelboim. Here gauge symmetry is elucidated in the hamiltonian formalism (mostly), and were you talk about first and second class constraints-- eg first class primary constraints generate gauge transformations. Chapter 4 on General covariance then gets into the topic.

In general you probably want to fully understand the difference between active and passive diffeomorphisms, general covariance, the equivalence principle and general invariance. At that point, you learn gauge theory (say in the Yang Mills context), notice that there is an analogy with GR (but with subtle differences), and then try to make GR into a bonafide gauge theory from scratch (see the arxiv paper linked by Mentz for one interpretation).

However you have to be careful here, b/c different people do things differently. Is the *global* gauge group of gravity really Diff(M) or perhaps something different? We know that the Poincare group is a subgroup of the diffeomorphism group, and so how do we want that to enter the picture. Surely that is the 'prefered' local symmetry? Then you start asking questions like, well shouldn't we be modding out by these gauge or diffeomorphism groups to get the real physical objects of interest, or at least an equivalence class thereof.

Mathematically, there are different approaches to these questions and unfortunately the same words have subtle differences (though presumably the physics remains the same).

I apologize if this post is confused, but the intention is to tantalize, b/c the literature is vast and interesting.