# Einstein-Hilbert Action -> GR is a saddle point?

• JustinLevy
In summary, the Einstein-Hilbert action leads to the Einstein Field Equations of General Relativity, but the action itself is not bounded from below and can also be a saddle point. This does not affect our ability to use the action principle to describe the equations of GR, and it is not related to the issues with quantizing gravity. Additionally, the principle of least action is more accurately described as the principle of stationary action. On a curved manifold, geodesics serve as the locally shortest curves connecting two points, but only in a small neighborhood. The concept of "shortest" becomes more complicated on a curved manifold, as seen with the example of an ellipsoid.
JustinLevy
Einstein-Hilbert Action --> GR is a saddle point?

[Clarification: I'm not claiming GR is wrong or anything. I'm asking if there are issues in interpreting GR as an action principle, and since I assume not, why not?]

We can get the equations of GR from the Einstein-Hilbert action by finding the extremal action in variations of the metric. However, the action does not appear to be bounded below as the curvature could just go arbitrarily negative for a finite period of time. Similarly it doesn't seem to be bounded from above. So are the equations we find when solving for the extremal action actually some kind of "saddle point"?

Doesn't this prevent us from considering GR in terms of a least action principle?

Is this issue of being unbounded from below in anyway related to the problems of trying to quantize gravity?

JustinLevy said:
the action does not appear to be bounded below as the curvature could just go arbitrarily negative for a finite period of time. Similarly it doesn't seem to be bounded from above.

Remember that the action is a functional of the metric; both the Ricci scalar ##R## and the factor ##\sqrt{-g}## depend on the metric. Finding the extremal action means varying the action with respect to the metric, and both factors will vary. So you can't just look at the Ricci scalar; you have to look at ##\sqrt{-g}## as well, and ask how that will vary if you vary the metric to make ##R## arbitrarily negative or positive. My understanding is that when you take all that into account, the extremal metric, i.e., the one that solves the Einstein Field Equation, is in fact a minimum of the action.

Taylor & Gray's paper, "When action is not least", <<link>> might also be of interest. I'll quote from the abstract. Perhaps I'm missing something, but basically my impression is that the action being a saddle point is not at all that unusual or anything to be concerned about.

We examine the nature of the stationary character of the Hamilton action for a space-time trajectory worldline tof a single particle moving in one dimension with a general time-dependent potential energy function U(x,t). We show that the action is a local minimum for sufficiently short worldlines for all potentials and for worldlines of any length in some potentials. For long enough worldlines in most time-independent potentials U, the action is a saddle point,that is, a minimum with respect to some nearby alternative curves and a maximum with respect toothers. The action is never a true maximum, that is, it is never greater along the actual worldline thanalong every nearby alternative curve.

vanhees71
Even in classical mechanics is it the ”principle of least action”, although it is a common misnomer. The correct nomenclature would be the ”principle of stationary action”.

Demystifier, romsofia and dextercioby
Less technically than the paper I mentioned, but hopefully easier to communicate in an post, consider whether or not a straight line is the shortest path between two points. Specifically, would we be better off saying that a "straight line" extremizes the distance between two points.. We are basically asking, can we use the principle of minimizing distance of a path to define a straight line, or is "extremal distance" better.

On a plane the answer is that straight lines are the curves of shortest distance between two points. On a curved manifold, we will need to be a bit more precise about our terminiology. We will use the concept of "geodesic" rather than straight lines, and we will also assume that we use the Levi-civiti connection.

Then these geodesics on a curved manifold are locally the shortest curves connecting two points, but only in a sufficiently small neighborhood of the original point. The paper I cited uses great circles on a sphere to illustrate this point, but I find it more compelling to consider geodesics on an ellipsoid. Stretch a sphere in one direction, so that it becomes ellipsoids. A curve that minimizes the distance on the sphere also minimizes it on the ellipsoid, but the great circles on the sphere that minimize the distance are "stretched".

On the sphere, there are sevaral different geodesics connecting the antipodes, points on the opposite side of the sphere.

On the ellipsoid, here are now several geodesics connecting the antipodes on what used to be the sphere. But now, some of them will be longer than others,. THus we see that there is no guarantee that a particular geodesic path is the shortest one, though it's always a local minimum.

## 1. What is the Einstein-Hilbert Action?

The Einstein-Hilbert Action is a mathematical expression used in General Relativity (GR) to describe the dynamics of spacetime. It is a function of the metric tensor, which represents the curvature of spacetime, and its derivatives.

## 2. How does the Einstein-Hilbert Action relate to GR?

The Einstein-Hilbert Action is the mathematical formulation of Einstein's field equations, which are the cornerstone of General Relativity. It describes how matter and energy interact with spacetime, and how this interaction produces the curvature of spacetime.

## 3. What does it mean for GR to be a saddle point?

In mathematics, a saddle point is a critical point on a surface that is neither a maximum nor a minimum. In the context of GR, this means that the Einstein-Hilbert Action is a critical point of the functional that describes the behavior of spacetime. This critical point represents the equations of motion of GR, which govern the dynamics of spacetime.

## 4. Why is the saddle point property important for GR?

The saddle point property is important for GR because it ensures that the field equations have a unique solution. This means that the equations of motion will always lead to a well-defined, self-consistent description of the dynamics of spacetime.

## 5. Can you provide an example of the saddle point property in action in GR?

One example of the saddle point property in action in GR is the Schwarzschild solution, which describes the gravitational field around a non-rotating, spherically symmetric mass. This solution arises from the Einstein-Hilbert Action and has a unique, well-defined behavior at all points in spacetime.

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