Check for geodesically-followed path in a coordinate-free way

[pervect]

Surely mine was, a that level, a pure and simple spatial 2D <-> 3D analogy.Take an ordinary 2-D surface with no metric defined on it (just a 2D smooth manifold with affine connection defined)

Limiting ourselves to it, how can an 'ant' -- from an operational point of view -- actually 'implement' (let me say step-by-step) the parallel transport of its tangent vector along a 'small' closed path in order to detect the geodesic curvature ? I am not sure there exist actually such a way for the ant to do that without an operative procedure to 'implement' the chosen (mathematical) affine connection structure.

The ant starts out at point, and picks out a direction in which to walk. This direction can be represented as a vector. He then proceedes to walk in that direction. While he walks, he also "parallel transports" the vector, representing the direction in which to walk, along with himself. He continues to walk in the direction the vector points. He doesn't need information other than the connection at the points he visits in order to do this - the value of the connection along his path gives him all he needs.

 

[pervect]

A fine point that concerns me a bit, so I'll give a quote from Wald, "General Relativity", pg 34.

The quick summary is to provide a reference and check on my statement that the parallel transport process depends only on the value of the connection coefficients along the curve, and to flesh out the process a bit.

We start by writing the parallel transport equations for a vector $v^a(\tau)$ along a curve with a tangent $t^a$. We will further assume that the manifold is labelled with some coordinates, and an associated coordinate basis for vectors.

In terms of components in the coordinate basis, and the parameter $\tau$ along the curve

$$\frac{dv^\nu}{d\tau} + \sum_{\mu,\lambda}t^\mu \, \Gamma^\nu{}_{\mu\lambda} v^\gamma = 0$$

This shows that the parallel transport of $v^a$ depends only on the values of $v^a$ on the curve, so we may consider the prallel transport properties of a vector defined only along the curve as opposed to a vector fields.

The connection as represented in the coordinate basis are the Christoffel symbols denoted by the symbols $\Gamma$. $\Gamma^a{}_{bc}$ maps two vectors to a number, but it doesn't transform as a tensor, so it's best considered as a map from two vectors to a number, said map defines the connection in the particular coordinates and coordinate basis chosen.

This snippet shows mathematically how to parallel transport a vector along a curve, though it's a bit terse. And it fleshes out how an "ant" can perform the process knowing the connection.

To turn this parallel transport equation for a vector $v^a$ along a curve $t^a$ into the equations for a geodeisc, we only needed to to equate $t^a$ and $v^a$, as a geodesic parallel transports the tangent vector of a curve along the curve. This, for a geodesic, we have:

$$\frac{dv^\nu}{d\tau} + \sum_{\mu,\lambda}v^\mu \, \Gamma^\nu{}_{\mu\lambda} v^\gamma = 0$$

I have assumed that $v^a$ is a unit vector, though I suppose we're considering the case where $v^a$ doesn't necessarily have a length, so this assumption is probably not needed.

For a space or space-time of dimension n, this is just n linear differential equations in the components $v^0, v^1, ...v^{n-1}$

For a flat space-time and cartesian coordinates, the Christoffel symbols are all zero, and you find that the components of the vector $v^a$ are constant.

 

[cianfa72]

The ant starts out at point, and picks out a direction in which to walk. This direction can be represented as a vector. He then proceeds to walk in that direction. While he walks, he also "parallel transports" the vector, representing the direction in which to walk, along with himself. He continues to walk in the direction the vector points. He doesn't need information other than the connection at the points he visits in order to do this - the value of the connection along his path gives him all he needs.

Surely, so to carry out the parallel transport it is enough for the ant to know the connection just at the points he visits. That means that if the manifold has a metric then the ant can derive the connection from it, otherwise 'someone' has to assign it at least at points he visits.

 

[A.T.]

From what we said it is possible for an observer traveling along a path through spacetime to know if its path is actually a geodesic without (locally) looking off its path (basically it suffices to have an accelerometer attached to it).

My takeaway from page 1 was that accelerometers cannot be pointlike, just "local" in the sense of: arbitrarily small, but not zero sized. So an accelerometer "looks off its path" in space time:

My understanding is also that local is not point like but merely at a scale too small for curvature to be noticed. You can certainly make accelerometers that small

 

[vanhees71]

Sure, "local" in this case means a space (or space-time) region across which you can assume the gravitational field being homogeneous, i.e., the tidal forces are negligible.

 

[pervect]

Surely, so to carry out the parallel transport it is enough for the ant to know the connection just at the points he visit. That means that if the manifold has a metric then the ant can derive the connection from it, otherwise 'someone' has to assign it at least at points he visit.

Knowing the metric on the path is not in general sufficient for the ant to navigate a geodesic. Knowing the metric, and the first partial derivatives of the metric with respect to the coordinates is sufficient, given that one is assuming the Levi-Civita connection. Knowing the connection is sufficient, and given a metric, the Levi-Civita connection can be calculated from the (inverse) metric and it's partial derivatives. And the inverse metric can be calculated from the metric.

The issue here the value of a function f(x) at a point is not sufficient to know it's derivative at the point, i.e f'(x). But if you know the value of f(x) in some neighborhood, and not just at a point, you can calculate it's derivative from f(x).

The issue with the connections is a multi-dimensional case of the same issue. Knowing the value of the metric at a point doesn't give the partial derivatives or the values of the connection coefficients/Christoffel symbols. But knowing it in a neighborhood does.

Note that it's possible and sometimes useful to use some other connection than the Levi-Civita connection, though it's unnecessary to use any other connection in GR. It's a bit off topic, so I won't go into it more unless asked.

 

[pervect]

Something else that might be useful to point out. In an orthonormal basis representing the "proper reference frame" in GR (not a coordinate basis), the reading of an accelerometer at a point with constant coordinates is given by the values of certain of the connection coefficients / Christoffel symbols at that point.

To be specific, $\Gamma^\hat{0}{}_{\hat{j}\hat{0}} = a^j$. See for instance MTW, pg 330. Here it is assumed that the coordinate $x^{\hat{0}} = \tau$, $\tau$ being the proper time, and that $a^1, a^2, a^3$ are the readings of the accelerometers measuring the components of the proper acceleration in the three spatial directions.

Thus, if we know the value of the connection coefficients in an orthonormal basis, we know the accelerometer readings of an accelerometer with constant coordinates. Further, we can see , as I argued previously, that in general we need to know the metric and it's partial derivatives to calculate these readings.

Unsurprisingly, in an inertial frame, or for a particle following a geodesic, the accelerometer readings above will all be zero.

Some of the other connection coefficients can be processed by the Levi-Civita symbol to give the rotation of the basis, $\omega^i$ as well, though I won't give the details.

 

[PeterDonis]

My takeaway from page 1 was that accelerometers cannot be pointlike, just "local" in the sense of: arbitrarily small, but not zero sized. So an accelerometer "looks off its path" in space time

When we describe an object (more precisely, a test object, an object whose presence does not affect the spacetime geometry) using a worldline in a spacetime model, we are treating it as being "pointlike" even though no actual object is pointlike. An accelerometer is treated as "pointlike" in the same sense in the model, even though no actual accelerometer is pointlike. So in the model, an accelerometer does not need to "look off the path" in spacetime.

In actual spacetime, any accelerometer, like any object at all, has finite size; but in actual spacetime, no real object is a "test object" in the strict sense--every object has some finite amount of stress-energy and has some finite effect on the spacetime geometry, even if it's too small to matter. So the model we use is idealized in these respects.

 

[Ibix]

So in the model, an accelerometer does not need to "look off the path" in spacetime.

Yes - but isn't examining that idealisation relevant to this question? As pervect points out, velocity and acceleration are perfectly well defined at a point in a classical theory, but actually measuring them requires you to look at a small region. At least, I can't think of an accelerometer that can work at one event. For a concrete example, imagine an accelerometer consisting of a mass attached by six springs to the walls of a box. I can't take a snapshot of the state of the accelerometer and deduce anything about my acceleration. For example, if I've just turned off my rockets the mass will be out of equilibrium for a small time, so the mere fact that the springs are stretched doesn't tell me that I'm accelerating.

 

[PeterDonis]

isn't examining that idealisation relevant to this question?

It depends on what kind of answer the OP wants.

If the OP wants to know how the models we use work, which will be a pretty good for practical purposes, though simplified, description of how the actual world works, then we want to look at the models as they are used, idealizations and all.

If the OP wants to know how the actual world works in all details, then we need to examine how the models are idealized and what aspects of the actual world those idealizations leave out.

 

[PeterDonis]

I can't think of an accelerometer that can work at one event.

Not if "one event" literally means "one point in the spacetime of the actual world", where we place no limit on our measurement resolution.

But if "one event" means "a region small enough that, in our simplified model, which is good enough for almost all practical purposes, we can treat it as a point", then an accelerometer certainly can work at one event. It just needs to have a response time that is very short compared to the characteristic time of changes in the system. For example, if the springs in your accelerometer with springs are stiff enough, they will readjust to a new equilibrium much faster than your rocket's engine can go from, say, 1 g acceleration to zero acceleration, so you will never be able to see your accelerometer show a mismatch with the actual acceleration, because the accelerometer is adjusting faster than the acceleration itself is changing.

 

[cianfa72]

If the OP wants to know how the models we use work, which will be a pretty good for practical purposes, though simplified, description of how the actual world works, then we want to look at the models as they are used, idealizations and all.

If the OP wants to know how the actual world works in all details, then we need to examine how the models are idealized and what aspects of the actual world those idealizations leave out.

Both. The point seems to me quite clear though.
From the model point of view we can consider a 'small' region as a 'point' making sense to how an accelerometer actually works. Otherwise we need to "deep in" examining all the details

 

[cianfa72]

Knowing the metric on the path is not in general sufficient for the ant to navigate a geodesic. Knowing the metric, and the first partial derivatives of the metric with respect to the coordinates is sufficient, given that one is assuming the Levi-Civita connection. Knowing the connection is sufficient, and given a metric, the Levi-Civita connection can be calculated from the (inverse) metric and it's partial derivatives. And the inverse metric can be calculated from the metric.

A point I would like to stress is that, in case a metric is not defined on a manifold, the ant should not be able to navigate a geodesic. In that case, as said before, I believe 'someone' has to 'inform' the ant about the chosen connection that is how to carry out the parallel transport of its tangent vector along the points on its path.

 

[pervect]

A point I would like to stress is that, in case a metric is not defined on a manifold, the ant should not be able to navigate a geodesic. In that case, as said before, I believe 'someone' has to 'inform' the ant about the chosen connection that is how to carry out the parallel transport of its tangent vector along the points on its path.

If you are saying that a connection defines parallel transport, in the presence or absence of a metric, I'd agree - but I'm not sure that's what you are saying.

When you have a metric, in general you still have to specify the connection. In GR the Levi-Civita connection is the implied choice of connection. But it's possible to make other choices.

Given that you have a connection, something does have to "inform" the ant about it, I suppose. But the connection is what the ant needs to know to follow a geodesic.

A case of point - there is a choice of connection that makes lines of constant lattitude geodesics on a 2-sphere. This connection is useful, but it's not the Levi-Civita connection. So, an ant crawling along a circle of constant latitude (for instance, by using a compass and always heading east) may or may not be following a geodesic. It depending on the choice of connection.

 

[cianfa72]

If you are saying that a connection defines parallel transport, in the presence or absence of a metric, I'd agree - but I'm not sure that's what you are saying.

Surely, that actually was my point.

When you have a metric, in general you still have to specify the connection. In GR the Levi-Civita connection is the implied choice of connection. But it's possible to make other choices.

yes, definitely true.

Given that you have a connection, something does have to "inform" the ant about it, I suppose. But the connection is what the ant needs to know to follow a geodesic.

A case of point - there is a choice of connection that makes lines of constant latitude geodesics on a 2-sphere. This connection is useful, but it's not the Levi-Civita connection. So, an ant crawling along a circle of constant latitude (for instance, by using a compass and always heading east) may or may not be following a geodesic. It depending on the choice of connection.

Right, those (bold is mine) are a sort of 'instructions' to give to the ant to carry out the parallel transport of its tangent vector. Basically the rule is : starting from a point A look at the direction shown by a compass there and move heading est respect to it arriving at the neighboring point B. The tangent vector in A parallel transported in B is --by the very definition of the connection chosen-- the direction heading est respect to the direction shown here locally by the compass...and so on

This kind of 'operations' basically define a geodesic path for the ant (in the chosen connection).

 

[pervect]

Surely, that actually was my point.yes, definitely true.Right, those (bold is mine) are a sort of 'instructions' to give to the ant to carry out the parallel transport of its tangent vector. Basically the rule is : starting from a point A look at the direction shown by a compass there and move heading est respect to it arriving at the neighboring point B. The tangent vector in A parallel transported in B is --by the very definition of the connection chosen-- the direction heading est respect to the direction shown here locally by the compass...and so on

This kind of 'operations' basically define a geodesic path for the ant (in the chosen connection).

That sounds right. In the general case, I suppose , you can think of the ant having some way of knowing where it's at, so it knows it's coordinates.

It also knows the basis vectors at any point.

The connection, which underlies the parallel transport operator, tells you how to "connect" the basis vectors at one point, to the basis vectors at another point, given a path between the points.

Another piece of the puzzle is what we mean by basis vectors. Given coordinates, mathematically, the coordinate basis vectors are identified with partial derivative operators. This may not be the way you intuitively think of basis vectors, mainly because the coordinate basis vectors are NOT necessarily of unit length.

Consider an ant on a plane. In cartesian coordinates, the Levi-Civita connection is simple. $\partial / \partial x$ and $\partial / \partial y$ are of unit length, and , with our assumption of the the Levi-Civita connection, these basis vectors always point in the same direciton, so no matter what path you take, so that $\partial / \partial x$ connects to itself.

Then try and think about the same ant, on the same plane, but the ant is given polar coordinates, instead. And it's also given the basis vectors associated with it's polar coordinates.

Then the ant that wishes to follow a Levi-Civita geodesic has to comensate for the fact that the basis vectors , using the Levi-Civita connection, change, as the ant moves around, in order to be able to travel in a "straight line". By design, the formalism is coordinate independent, so using the Levi-Civita connection the ant using cartesian coordinates follows the same 'straight line' as the ant using polar coordinates. To accomplish this, the ant needs a different "program", the values of the Christoffel symbols. I'd tend to envision the ant as computing these from $(r, \theta)$ on the fly, but that gets into the internal idea of the structure of the ant, which people are free to imagine differently.

 

[cianfa72]

That sounds right. In the general case, I suppose , you can think of the ant having some way of knowing where it's at, so it knows it's coordinates.

It also knows the basis vectors at any point.

Then the ant that wishes to follow a Levi-Civita geodesic has to compensate for the fact that the basis vectors , using the Levi-Civita connection, change, as the ant moves around, in order to be able to travel in a "straight line". By design, the formalism is coordinate independent, so using the Levi-Civita connection the ant using cartesian coordinates follows the same 'straight line' as the ant using polar coordinates. To accomplish this, the ant needs a different "program", the values of the Christoffel symbols. I'd tend to envision the ant as computing these from $(r, \theta)$ on the fly, but that gets into the internal idea of the structure of the ant, which people are free to imagine differently.

Good point. Nevertheless from a geometrical point of view (i.e. in a coordinate-free way) in case of flat plane and using Levi-Civita connection the ant can be 'programmed' to carry out parallel transport of tangent vectors from a point A to a point B based on the following: given A e B consider the path locally minimizing the distance (metric) between them (actually a geodesic segment). The direction of that path at B (its tangent vector at B) is -- by definition of such connection -- the parallel transport on B of that path's tangent vector in A.

This way the ant has a 'rule' that 'connect' two tangent vectors each living in a different tangent space (A and B tangent spaces respectively). Starting from that the ant is actually able to 'connect' other tangent vectors, I believe.

 

[Dale]

As pervect points out, velocity and acceleration are perfectly well defined at a point in a classical theory, but actually measuring them requires you to look at a small region.

I am not sure that the distinction is physically meaningful. I mean, yes, you do need to have information from points in a neighborhood, but that neighborhood is mathematically infinitesimal. Since an infinitesimal is smaller than any real number any measurement device with any degree of accuracy will measure that to be zero. So mathematically the distinction between an infinitesimal neighborhood and a point is valid but physically I am not sure it matters much.

 

[PAllen]

Good point. Nevertheless from a geometrical point of view (i.e. in a coordinate-free way) in case of flat plane and using Levi-Civita connection the ant can be 'programmed' to carry out parallel transport of tangent vectors from a point A to a point B based on the following: given A e B consider the path locally minimizing the distance (metric) between them (actually a geodesic segment). The direction of that path at B (its tangent vector at B) is -- by definition of such connection -- the parallel transport on B of that path's tangent vector in A.

This way the ant has a 'rule' that 'connect' two tangent vectors each living in a different tangent space (A and B tangent spaces respectively). Starting from that the ant is actually able to 'connect' other tangent vectors, I believe.

For a mathematical model assumed to represent physics, mathematical elements like metric or connection are assumed to have physical analogs. Thus, the connection (parallel transport) definition of geodesic is locally implemented by always 'walking the direction I am facing'. If the connection is physical, this must have local meaning.The metric definition is locally implemented effectively using the triangle inequality. Consider a string going from where I was a tiny bit before to my current position, and then also extending (taut) to various nearby points; and also a string from just before directly to one of the possible path extension points. When the two strings are equal length, that is the correct path extension.

 

[cianfa72]

For a mathematical model assumed to represent physics, mathematical elements like metric or connection are assumed to have physical analogs. Thus, the connection (parallel transport) definition of geodesic is locally implemented by always 'walking the direction I am facing'. If the connection is physical, this must have local meaning.The metric definition is locally implemented effectively using the triangle inequality. Consider a string going from where I was a tiny bit before to my current position, and then also extending (taut) to various nearby points; and also a string from just before directly to one of the possible path extension points. When the two strings are equal length, that is the correct path extension.

With metric definition (in bold) do you mean the metric derived connection (Levi-Civita) ?

 

[PAllen]

With metric definition (in bold) do you mean the metric derived connection (Levi-Civita) ?

No. I’m saying the metric definition of geodesics is about local length minimization (in Riemannian, rather than pseudo-Riemannian geometry), while parallel transport definition is about not changing direction, and is based directly on a connection. The metric definition need not involve the connection at all. When one talks about a physical theory using connection without a metric, one assumes change of direction is a locally determinate physical question.

As an example, Einstein-Cartan theory has the feature that the connection is not the unique torsion free metric compatible connection, but is determined by matter field equations, and may include torsion. As a result, the parallel transport and interval extremal definitions of geodesic are different, and both types are physically determinate in the theory.

 

[cianfa72]

Thus, the connection (parallel transport) definition of geodesic is locally implemented by always 'walking the direction I am facing'. If the connection is physical, this must have local meaning.

When one talks about a physical theory using connection without a metric, one assumes change of direction is a locally determinate physical question.

Sorry, maybe I still fail to quite grasp it. As pointed out the connection-based definition of geodesic (parallel transport) is physically implemented by 'walking the direction I am facing'. Let's go back to the example of the ant walking on an ordinary 2-D spherical surface and consider how the ant 'implements' the concept of 'walking the direction I am facing'.

Now if that just means walking the direction I am facing locally relatively to myself , I believe the answer --regardless the connection chosen-- can be just one: walk following the 'great circle' passing through the starting point of the ant journey (in other words the answer is actually the same as the metric-based geodesic definition)

Not sure if above actually makes sense or not Thanks in advance

 

[cianfa72]

Sorry, maybe I still fail to quite grasp it. As pointed out the connection-based definition of geodesic (parallel transport) is physically implemented by 'walking the direction I am facing'. Let's go back to the example of the ant walking on an ordinary 2-D spherical surface and consider how the ant 'implements' the concept of 'walking the direction I am facing'.

Now if that just means walking the direction I am facing locally relatively to myself , I believe the answer --regardless the connection chosen-- can be just one: walk following the 'great circle' passing through the starting point of the ant journey (in other words the answer is actually the same as the metric-based geodesic definition)

Not sure if above actually makes sense or not Thanks in advance

I believe the main confusing point for me about the 'general case' connection-based definition of geodesic --locally implemented for a given generic connection (not Levi-Civita in case a metric exists) as 'walking the direction I am facing'-- is that along the 'geodesic journey' the ant shouldn't transport only the path's tangent vector from where it was a tiny bit before to its current position but actually some "twist" (locally respect to its 'feet') is actually required before continuing to walk in that direction the ant now is facing to

 

[PAllen]

Maybe this will clarify things. Suppose you want to keep walking the same direction on the surface of the earth. One way is to use a compass and follow a constant bearing. This will certainly feel like you are going straight = not turning. Of course, you will be turning compared length minimizing geodesic, and to its metric compatible notion of not turning. The compass allows you to locally, physically, implement a non metric compatible connection. On the surface of the earth, it is actually harder to implement a local procedure for the metric great circle geodesic.

 

[Dale]

One way is to use a compass and follow a constant bearing. This will certainly feel like you are going straight = not turning.

No, I don't think it will. Of course real compasses are not so accurate, but suppose we had a hypothetical super compass and that the Earth's magnetic field is a perfect dipole. Then, if you are near the magnetic north pole, keeping a constant bearing will definitely feel like you are turning. Further away from the pole you will still be turning but it will be less obvious.

I agree that this is a physically implemented non metric-compatible connection, but the fact that it is non metric-compatible means that it will not "feel like you are going straight".

 

[cianfa72]

No, I don't think it will. Of course real compasses are not so accurate, but suppose we had a hypothetical super compass and that the Earth's magnetic field is a perfect dipole. Then, if you are near the magnetic north pole, keeping a constant bearing will definitely feel like you are turning. Further away from the pole you will still be turning but it will be less obvious.

I agree that this is a physically implemented non metric-compatible connection, but the fact that it is non metric-compatible means that it will not "feel like you are going straight".

That was actually my understanding. Starting from a point on the Earth in a given direction the command 'walk the direction you are facing' for the ant does actually mean: follow the geodesic path as given by (metric compatible) Levi-Civita connection. The non metric-compatible connection we were talking about does imply a local 'change of direction' at each point along the journey in order to keep constant the angle respect to the compass needle direction (constant bearing).

 

[PAllen]

No, I don't think it will. Of course real compasses are not so accurate, but suppose we had a hypothetical super compass and that the Earth's magnetic field is a perfect dipole. Then, if you are near the magnetic north pole, keeping a constant bearing will definitely feel like you are turning. Further away from the pole you will still be turning but it will be less obvious.

I agree that this is a physically implemented non metric-compatible connection, but the fact that it is non metric-compatible means that it will not "feel like you are going straight".

I agree; forgot about the case when the difference would be really significant. But "feel like going straight" really just means sensing a different physical process that couple to metric and its compatible connection. Literally, one learns the difference in force application through ones legs to go straight, as say defined by a laser pointer, which is probably the most convenient local implementation of metric straightness.

Anyway, we agree on my main point: Two different connections can be locally detectable as long as each has some physical realization. I thought of the compass as the simplest plausible example of this. Straight per compass versus straight per laser pointer.

 

[PAllen]

That was actually my understanding. Starting from a point on the Earth in a given direction the command 'walk the direction you are facing' for the ant does actually mean: follow the geodesic path as given by (metric compatible) Levi-Civita connection. The non metric-compatible connection we were talking about does imply a local 'change of direction' at each point along the journey in order to keep constant the angle respect to the compass needle direction (constant bearing).

It's all a matter of what you define as straight versus turning. One definition of straight is associated with minimizing distance, the other is not - it is defined by constant compass bearing.

 

[Dale]

Anyway, we agree on my main point: Two different connections can be locally detectable as long as each has some physical realization. I thought of the compass as the simplest plausible example of this. Straight per compass versus straight per laser pointer.

Yes, I agree. It is a valid connection and you can define "straight" accordingly. I don't know which theorems carry over and which do not when you use a non metric-compatible connection.

 

[cianfa72]

It's all a matter of what you define as straight versus turning. One definition of straight is associated with minimizing distance, the other is not - it is defined by constant compass bearing.

Thus, sticking at that given non-metric connection, the sentence "walking in the direction you are facing" actually means "keep walking forward step-by-step in the direction at a given fixed angle to the direction shown locally by the compass needle"