Rods, Clocks & Free Fall: Metric & Connections in GR

[shahbaznihal]

In Classical GR the metric tensor $g_{\mu\nu}$ determines the length of rods and ticking of clocks while the connection $\Gamma^{\alpha}_{\mu\nu}$ determine the equation of geodesic (the free fall motion of particle). Furthermore, in GR the Levi-Civita connection is uniquely determined by the metric tensor as,$$
\Gamma_{\mu\nu}^{\alpha}=\frac{1}{2}g^{\alpha\delta}(g_{\mu\delta,\nu}+g_{\nu\delta,\mu}-g_{\mu\nu,\delta})$$
In certain theories of gravity and Quantum gravity, the connection and the metric tensor are taken as independent quantities. It appears to me that when this happens the free fall geodesic equation, describing the motion of particle in a manifold, becomes independent of the structure of space-time, the metric tensor (rods and clocks). This makes, to me at least, very little intuitive sense (may be because the picture painted by classical GR is too strongly imprinted on my mind).

So, how is this (making metric and connection independent) physically motivated? Or it just a mathematical curiosity?

 

[Demystifier]

Classically, the momentum
$$p(t)=mv(t)=m\frac{dx(t)}{dt}$$
is uniquely determined by $x(t)$. But in quantum mechanics, $x$ and $p$ are independent. In particular, if you know the former than you cannot know the latter. And that's very non-intuitive if classical picture of $x$ is painted in your mind. I think this is quite analogous to the quantum part of the problem that bothers you.

How about the classical part, where concepts should be more intuitive? Can position and momentum be treated as independent quantities in classical mechanics? Yes, in the Hamiltonian formalism they are treated as independent quantities before the equations of motion are solved. In Hamiltonian formalism, the equation above is not a definition, but a solution of one of the equations of motion. So in classical physics it's just a mathematical trick. As you may guess, the independent treatment of metric and connection in classical gravity is something very similar.

 

[shahbaznihal]

Thanks for your reply. My literature review of the question got me to the same conclusion.

 

[haushofer]

In certain theories of gravity and Quantum gravity, the connection and the metric tensor are taken as independent quantities.

Again, which theories are you referring to?

The Palatini formalism is a mathematical "reformulation" of GR. Normally, one has the Riemann tensor depending on the connection and its derivatives. And these connections depend on the metric and its derivatives. So, the Einstein equations are second order differential equations for the metric. But from the theory of differential equations, we know that a second order diff.eqn. can be recasted as a set of two first order diff.eqns. (if you're not convinced: try Newton's second law for the position of a particle for instance!)

This trick also works for the Einstein equations. We can recast this second order diff.eqn. for the metric as two first (!) order diff.eqns. This recasting is the Palatini equation. Basically, in this formalism you take the Einstein Hilbert action and treat a priori the metric and connection as independent fields. So you vary the action with respect to the metric and to the connection, giving you two first order differential equations. The equation of motion for the connection then gives you the usual relation between metric and connection. See e.g. Samtleben's notes on Supergravity, page 9 onward.

So, to make the example of the Newtonian point particle a bit explicit: say, we have Newton's second law

$m\ddot{x}=F$

This is a second order diff.eqn. for x. But we can recast it as

$\dot{x}=v , \ \ \ \ \ \ \ \ \dot{v}=\frac{F}{m}$

using that

$\dot{x}=v$

is the velocity of the particle. Now we have turned a second order diff.eqn for x into two first order eqns. for x and v. Mathematically, one can often use techniques from linear algebra to solve this system for a given F, and physically one can directly draw conclusions for the corresponding phase space.

Likewise, the Palatini formalism can make calculations involving variations of the action sometimes easier, as Samtleben also explains on page 9. I'm not sure if it is used to draw conclusions for the corresponding phase space; I've never seen this.

I'm not aware of theories of gravity in which the connection is treated as being independent of the metric in all of the dynamics, as in having it's separate degrees of freedom in phase space. That's why I'm asking you: which theories do you refer to?

 

[Demystifier]

The main reason for using a first order formalism is the fact that $R$ contains second time derivatives of the metric, while the usual Lagrangian formalism assumes that Lagrangian depends only on canonical positions and its first time derivatives. One has to eliminate the second time derivatives in the Lagrangian, and the first order formalism is one way to do it.

To see how that works, instead of gravity let us study a simple toy model with similar properties. Consider a single degree of freedom $x(t)$ described by the Lagrangian
$$L=-\frac{\dot{x}^2}{2}-x\ddot{x} \;\;\;\;\; (1)$$

a) One way to eliminate the second derivative $\ddot{x}$ is to use the identity
$$x\ddot{x}=\frac{d}{dt}(x\dot{x})-\dot{x}^2$$
so the Lagrangian is
$$L=\frac{\dot{x}^2}{2}+{\rm total \; derivative} \;\;\;\;\; (1')$$
where the total derivative term can be ignored because it does not contribute to the variation of the action $\int dt\, L$. Hence the physics is determined by the first term in (1'), which gives the equation of motion
$$\ddot{x}=0 \;\;\;\;\; (2) $$

b) Another way to eliminate the second derivative $\ddot{x}$ is the first order formalism. One first introduces the velocity $v=\dot{x}$ to write (1) as
$$L=-\frac{v^2}{2}-x\dot{v} \;\;\;\;\; (3)$$
and then treats $x$ and $v$ in (3) as independent quantities. Hence there are two equations of motion, one for $x$ and another for $v$. The equation for $x$ is
$$\frac{d}{dt}\frac{\partial L}{\partial\dot{x}}=\frac{\partial L}{\partial x}$$
which gives
$$0=\dot{v} \;\;\;\;\; (4) $$
The equation for $v$ is
$$\frac{d}{dt}\frac{\partial L}{\partial\dot{v}}=\frac{\partial L}{\partial v}$$
which gives
$$\dot{x}=v \;\;\;\;\; (5) $$
Clearly, Eqs. (4) and (5) are equivalent to Eq. (2), showing that the two formalism a) and b) are equivalent. Note, however, that (5) is not a definition but an equation of motion derived from the Lagrangian (3).

 

[shahbaznihal]

Again, which theories are you referring to?

The Palatini formalism is a mathematical "reformulation" of GR. Normally, one has the Riemann tensor depending on the connection and its derivatives. And these connections depend on the metric and its derivatives. So, the Einstein equations are second order differential equations for the metric. But from the theory of differential equations, we know that a second order diff.eqn. can be recasted as a set of two first order diff.eqns. (if you're not convinced: try Newton's second law for the position of a particle for instance!)

This trick also works for the Einstein equations. We can recast this second order diff.eqn. for the metric as two first (!) order diff.eqns. This recasting is the Palatini equation. Basically, in this formalism you take the Einstein Hilbert action and treat a priori the metric and connection as independent fields. So you vary the action with respect to the metric and to the connection, giving you two first order differential equations. The equation of motion for the connection then gives you the usual relation between metric and connection. See e.g. Samtleben's notes on Supergravity, page 9 onward.

So, to make the example of the Newtonian point particle a bit explicit: say, we have Newton's second law

$m\ddot{x}=F$

This is a second order diff.eqn. for x. But we can recast it as

$\dot{x}=v , \ \ \ \ \ \ \ \ \dot{v}=\frac{F}{m}$

using that

$\dot{x}=v$

is the velocity of the particle. Now we have turned a second order diff.eqn for x into two first order eqns. for x and v. Mathematically, one can often use techniques from linear algebra to solve this system for a given F, and physically one can directly draw conclusions for the corresponding phase space.

Likewise, the Palatini formalism can make calculations involving variations of the action sometimes easier, as Samtleben also explains on page 9. I'm not sure if it is used to draw conclusions for the corresponding phase space; I've never seen this.

I'm not aware of theories of gravity in which the connection is treated as being independent of the metric in all of the dynamics, as in having it's separate degrees of freedom in phase space. That's why I'm asking you: which theories do you refer to?

I have already gone through the math of the process. The question is not about how different formulations do the math. The question is if there is any physical importance of treating them separately or is it just a mathematical curiosity. All I get from papers is that it is just a generalization and no paper, neither in your reply, do I see any physical motivation for doing it.

 

[martinbn]

All I get from papers is that it is just a generalization and no paper, neither in your reply, do I see any physical motivation for doing it.

Which papers?

 

[stevendaryl]

In the Einstein-Cartan theory, which is a generalization of GR, the connection is not determined by the metric. But in the formulation described in Wikipedia, the independent variables are not the metric tensor and the connection, but the metric tensor and the torsion (which is the anti-symmetric part of the connection coefficients: $T^k_{ij} = \Gamma^k_{ij} - \Gamma^k_{ji}$

 

[stevendaryl]

In the Einstein-Cartan theory, which is a generalization of GR, the connection is not determined by the metric. But in the formulation described in Wikipedia, the independent variables are not the metric tensor and the connection, but the metric tensor and the torsion (which is the anti-symmetric part of the connection coefficients: $T^k_{ij} = \Gamma^k_{ij} - \Gamma^k_{ji}$

https://en.wikipedia.org/wiki/Einstein–Cartan_theory

 

[haushofer]

I have already gone through the math of the process. The question is not about how different formulations do the math. The question is if there is any physical importance of treating them separately or is it just a mathematical curiosity. All I get from papers is that it is just a generalization and no paper, neither in your reply, do I see any physical motivation for doing it.

I'm not aware of any physical motivation, but if you refuse to be more specific it's hard to see what kind of motivation you want in the first place. E.g., it's still not clear to me whether you're referring to standard Palatini formalism or a dynamical modification of GR.

Maybe this insight helps:

https://www.physicsforums.com/insights/general-relativity-gauge-theory/

 

[robphy]

Possibly useful:
https://www.google.com/search?q=ehlers+pirani+schild
https://www.google.com/search?q=ehlers+pirani+schild+palatini

https://scholar.google.com/scholar?q=ehlers+pirani+schild
https://scholar.google.com/scholar?q=ehlers+pirani+schild+palatini

 

[haushofer]

I'm not aware of any physical motivation, but if you refuse to be more specific it's hard to see what kind of motivation you want in the first place. E.g., it's still not clear to me whether you're referring to standard Palatini formalism or a dynamical modification of GR.

Maybe this insight helps:

https://www.physicsforums.com/insights/general-relativity-gauge-theory/

Just to add: the reason why I link to this approach, describing GR as the gauge theory of the Poincare algebra, is because in this approach too you consider the connection and metric to be independent to start with. The dependency of the connection on the metric can be derived both from the Palatini formalism and the curvature constraint, eqn.(7) in the Insight. If you don't impose this constraint, you end up with local translations in your theory which are not part of the usual formulation of GR. See the Insight for the algebraic details (and references).