Relativity in Geometry and Physics

[Killtech]

This is the bit what i am currently struggling with in physics. Relativity is a concept deeply rooted in the nature of geometry itself because we must always account that space and time are never measured absolutely but in units - and those are always relative to some local and frame dependent reference. For the SI second this is a specific Caesium transition frequency at the observers frame and location. When something is said to be constant it merely means it changes exactly the same as the reference in units of which it is measured. Because all measurement is relative, it is impossible/meaningless to determine if something changes in an absolute sense, hence it is crucial to understand the entire geometric formulation is about relative metrics and distances. But from what i learn from physics, i did get the impression it lacks understanding of the full extend of relativity and its seems stuck in interpreting everything within a singular view. So the baryocentric metric representation discussed here is just one example to show how relative the entire geometry is beyond what is usually used in physics.

To go into detail, let's start with a motivation. In the beginning of the 19th century, time didn't not have the same importance it has in our todays society. Back then each town might have had its own non uniform local time and clocks. Only when rail tracks between the cities were build, time became of higher importance as we needed to organize events in between different locations. The railway time was introduced as an unifying standard and one of the early time standards which effectively became GMT/UTC our clocks still reference today.

In general relativity we have a little bit of a similar scenario with each frame having its very own proper time and clock. The reasons may differ, but the practical problem this creates are the same, hence IAU introduced the TDB coordinate time [https://en.wikipedia.org/wiki/Barycentric_Dynamical_Time] that serves the purpose of 'railway time' for the solar system.

Let's assume i would be an interplanetary traveler and have to buy a clock, then what concept of time would i want it to show? or let's say i want to record the historic accounts of a interstellar civilization - what time should i use?

Proper and coordinate times are very different representations of time for different purposes, both perfectly valid in their own right. But for our intuition, the one is a lot more abstract then the other, so i was wondering what happens when we formulate general relativity in a different time?

Measuring time differently, or rather measuring a different time has a lot of implications. TDB corrects for the effects of gravity fields to proper time - so at the edge of the solar system SI and TDB clocks may yield same numerical results, but on earth TDB clocks will run relatively faster. Consequently, if we were to use such clocks to measure the speed of light in vacuum, we find that light signals appear to travel slower in earths gravity well then at the edge of the solar system. There is no contradiction here because TDB clocks results have to use an own coordinate time unit (i.e. not the SI second), and these units are locally different from SI, allowing no simple comparison.

Mathematically however, we do have the means to adequately predict how physics will look when we keep using such units. Now this is where the discipline of geometry comes into play. First, let's recount what is mathematically the difference between a sphere and an ellipsoid Riemann manifolds? Technically we can take the very same subset of R^3 for both manifolds and use the very same coordinates for either. The actual difference is in fact how we define to *measure* length along a line element. Or more generally, the metric tensor we define on the manifold. Even though geometry is a mathematical discipline the practical concept of measurement is a fundamental part of the theory (as the term "metric" implies) - which the metric tensor represents in a abstract way.

Now going in back to TDB clock question, we need a metric tensor that is correctly able to represent TDB time along a line element. We can do that do that with length as well i.e. using BCRS coordinate lengths [https://en.wikipedia.org/wiki/Barycentric_and_geocentric_celestial_reference_systems] to yield a new metric tensor representing measurements done with the corresponding coordinate units. Admittedly it does not make practical sense with just any coordinates, but for these, the results can be well interpreted. For example the coordinate distance between earth and moon is almost constant since it does not depend on the overserves frame i.e. it is always a specific proper distance. Geometrically speaking, we exchanged the metric tensor g of general relativity for another - and i don't mean its coordinate representation - I really mean to have exchanged the whole geometry. Because we exactly know how these two metric tensors relate to each other at every location, we can write down a transformation procedure to reformulate Einsteins field equations with their original metric tensor to the geometry implied by the TDB-BCRS metric tensor.

Of course that does not change physics in any way - we just change the point of view (i.e. measurement) to yield a different model of physics but with a new interpretation (mapping time and space to different clocks and measurements). If we consistently exchange a physical model along with its interpretation for another, the changes in both cancel each other out in a sense: mathematically speaking we have a commuting diagram [https://en.wikipedia.org/wiki/Commutative_diagram] where the interpretations takes the role of the morphisms while the models and reality are the nodes/vertices.

If we were to look at Maxwells equations in barycentric coordinate units specifically, the transformation prescribed from Einsteins metric tensor to the barycentric geometry will introduce a gravity induced refractive index to the vacuum, stemming from the gravity correction to TDB clocks. As such in the new model (and its units) the speed of light is not a constant and the bending of light by gravity is now described via refractive index instead of the geometry. The actuary predictions however remain ultimately identical.

I should also mention that the barycentric nature of these coordinates means the new model has chooses a preferred frame. and since it measures time intervals and length in coordinates differences i.e. entirely invariant of the observers frame, Lorentz transformation will now distort those. Instead it is more natural to go back to Galilean transformations which preserve these. But looking at Maxwell in vacuum, this now means that this equations looks like a classical wave equation in a medium. Furthermore gravity itself will also appear back as a force field (and some other additional fields). An increase in complexity which is compensated by trivializing the geometry. Conceptually this is similar to a reversed Newton-Cartan approach, where we take general relativity and translate it back to a Newtonian-like "absolute time" (which TDB is).

[Genady]

The following theoretical construction perhaps contains the answer to your question:

(let me know if you want to read the rest of this Box - it does not add anything to the construction itself)

[Killtech]

1 hour ago, Genady said:

The following theoretical construction perhaps contains the answer to your question:

[...]

(let me know if you want to read the rest of this Box - it does not add anything to the construction itself)

Thanks. I understand that construction. Note that it is called the "geodesic clock" and as the name suggests it is constructed via geodesics. A geodesic is defined via the geometry and more precisely the Christoffel symbols derived from the metric tensor of said geometry. As i tried to highlight, the definition metric tensor already implies a what clocks and length measurements must produce.

The theory of relativity never formally defines what we should count as a clock even though Einstein uses the term a lot. But we can't just use any oscillators because each may be impacted differently by the environment at a given location and frame (unless we properly correct for the deviances). The geodesic clock is actually a theoretic derivation what clocks we must use to be to consistent with Einsteins theory and atomic clocks are constructed/defined and corrected in just such a way to be as close as possible to that ideal. This is why what you posted is so very significant - it gives us the final details for a most accurate interpretation of the theory. 

I am however looking at using a time standard instead of proper time which is a major difference. I should maybe again highlight that TDB is normally not thought in terms of clocks for the good reason to avoid confusion. In fact TDB time has to be calculated via an atomic clock (which is the most accurate implementation of an ideal clock of general relativity) and correcting it by the influence of gravity (which requires additional information) so it is not just more complex method, but also explicitly deviates from what a theoretical clock should yield.

Note that most clocks we use in daily life are invalid in the context of relativity because they don't measure proper time but also give time according to a time standard. So if you were to climb Mount Everest with two clocks, one radio-synched watch and an atomic clock we will find a discrepancy: the UTC time standard simply ignores our actual location whereas an atomic clock doesn't, hence general relativity will make them run apart over time - yet neither is going off. They just measure different things! It is important to understand that.

What i am looking for is the question if physics skipped the part where the mathematical discipline of Riemann geometry is so flexible that it allows us to treat TDB as if it was a time measure we could base clocks on for physical experiments. Of course it has practically a limited use since measuring time in SI-seconds allows for a much greater accuracy the measurement in TDB which inherit a big uncertainty from the gravitational correction which is hard to measure and calculate precisely. Then again, if we want to calculate whether two celestial bodies will eventually collide, we have that error one way or another.

Edited September 24, 2023 by Killtech

[Genady]

1 hour ago, Killtech said:

A geodesic is defined via the geometry and more precisely the Christoffel symbols derived from the metric tensor of said geometry.

Geodesic is a physical phenomenon. It is a worldline of a free-falling particle and thus independent on clocks and length measurements. The latter are defined by the former.

The following paragraph agrees with your

1 hour ago, Killtech said:

The geodesic clock is actually a theoretic derivation what clocks we must use to be consistent with Einsteins theory and atomic clocks are constructed/defined and corrected in just such a way to be as close as possible to that ideal. 

 

 

I don't know about aspects of using standard time in general and TDB in particular. Hopefully, other knowledgeable members will contribute to this regard.

Edited September 24, 2023 by Genady

[Markus Hanke]

9 hours ago, Killtech said:

A geodesic is defined via the geometry and more precisely the Christoffel symbols derived from the metric tensor of said geometry.

The actual definition of a geodesic is a curve that parallel-transports its own tangent vector. This requires a connection, but not necessarily a metric - IOW, a manifold that is endowed with a connection but not a metric, will exhibit a geodesic structure. Of course, if you have both a connection and a metric (as is the case in GR), then the geodesic equation can be written in terms of derivatives of the metric tensor. But that is not its fundamental definition.

14 hours ago, Killtech said:

Geometrically speaking, we exchanged the metric tensor g of general relativity for another - and i don't mean its coordinate representation - I really mean to have exchanged the whole geometry.

I don’t know what you mean by this. A metric is a structure on a manifold that has a very precise definition, which has to do with fiber bundles and tangent spaces, but not with any particular coordinate choices. What, exactly, do you want to replace here? 

15 hours ago, Killtech said:

we can write down a transformation procedure to reformulate Einsteins field equations with their original metric tensor to the geometry implied by the TDB-BCRS metric tensor.

The Einstein equations are a covariant tensor equation, so its form is the same irrespective of what geometry the manifold has. That’s the entire point of general covariance. IOW, you are quite free to use a different concept of time (coordinate basis) to describe your scenarios, but that doesn’t change the laws of physics, and thus all tensor equations remain unaffected. 

15 hours ago, Killtech said:

Furthermore gravity itself will also appear back as a force field

In that case you will obtain incorrect predictions for the polarisation states of gravitational waves, which cannot be modelled by any rank-1 model. You need at least a rank-2 model to correctly account for all relevant degrees of freedom, so gravity cannot be a force in the Newtonian sense.

15 hours ago, Killtech said:

Newtonian-like "absolute time" (which TDB is).

It’s not absolute, it’s just a convenient choice of coordinate basis that makes certain astrophysical calculations easier. You are always free to choose your coordinates and units as is convenient - that doesn’t change anything about the laws of physics, in particular not their form when written as tensor equations.

[Markus Hanke]

16 hours ago, Killtech said:

But from what i learn from physics, i did get the impression it lacks understanding of the full extend of relativity

I disagree. Relativity is just a particular application of (much more general) differential geometry, which is extremely well understood and worked out.

[studiot]

4 hours ago, Markus Hanke said:

The actual definition of a geodesic is a curve that parallel-transports its own tangent vector. This requires a connection, but not necessarily a metric - IOW, a manifold that is endowed with a connection but not a metric, will exhibit a geodesic structure. Of course, if you have both a connection and a metric (as is the case in GR), then the geodesic equation can be written in terms of derivatives of the metric tensor. But that is not its fundamental definition.

I don’t know what you mean by this. A metric is a structure on a manifold that has a very precise definition, which has to do with fiber bundles and tangent spaces, but not with any particular coordinate choices. What, exactly, do you want to replace here? 

The Einstein equations are a covariant tensor equation, so its form is the same irrespective of what geometry the manifold has. That’s the entire point of general covariance. IOW, you are quite free to use a different concept of time (coordinate basis) to describe your scenarios, but that doesn’t change the laws of physics, and thus all tensor equations remain unaffected. 

In that case you will obtain incorrect predictions for the polarisation states of gravitational waves, which cannot be modelled by any rank-1 model. You need at least a rank-2 model to correctly account for all relevant degrees of freedom, so gravity cannot be a force in the Newtonian sense.

It’s not absolute, it’s just a convenient choice of coordinate basis that makes certain astrophysical calculations easier. You are always free to choose your coordinates and units as is convenient - that doesn’t change anything about the laws of physics, in particular not their form when written as tensor equations.

+1 to Markus.

 

The actual definition of a geodesic is lost somewhere in the Terra Incognita of algebraic geometry  - I have several all pretty impenetrable jungle.

 

Here is the beginning of a good introduction for Relativists from this book.

[Killtech]

4 hours ago, Markus Hanke said:

I don’t know what you mean by this. A metric is a structure on a manifold that has a very precise definition, which has to do with fiber bundles and tangent spaces, but not with any particular coordinate choices. What, exactly, do you want to replace here? 

Okay, from your responses i see we are talking past each other. All you say is true, but it is also not exactly related to what i was writing. 

Let's go back to the example of a single differentiable manifold that we can make to be either an ellipsoid or a sphere depending to what metric and connection we define on it. Because both are build on the very same manifold however means that any covariant tensor equation in the ellipsoid geometry can be translated into one in the sphere geometry like this:

 

(btw. how do you use latex here around these forums when in need of writing down some formulas?)

There is also an interesting discussion of this by Henry Poincaré you can find here in section XII: https://en.wikisource.org/wiki/The_Foundations_of_Science/The_Value_of_Science/Chapter_2

It discusses the difficult relation we have in physics where in order to conduct any measurement at all we need to start with some assumptions. These kind of assumptions must be distinguierend from other physical laws as these assumptions are not testable in an experiment as Poincare remarks. For example: you will find that we cannot simply assume the speed of light to be non-constant as this - given how we define the measurement of time and distance - will produce contradictions whenever we try to model these measurements with such an assumption. 

 

Anyhow, once you have made a transition to another geometry, you have to be very careful with the interpretation. if you use the wrong one, the new model will of course make wrong predictions. But same as for general relativity, you first have to deduct the correct clocks and rods that are represented by the new metric tensor and its connection. And only then the combined model and interpretation will yield identical prediction to your starting theory. 

[Genady]

5 hours ago, Markus Hanke said:

The actual definition of a geodesic is a curve that parallel-transports its own tangent vector. This requires a connection, but not necessarily a metric - IOW, a manifold that is endowed with a connection but not a metric, will exhibit a geodesic structure.

And the other way around: geodesics define the parallel transport, the covariant derivative, and the connection. But not necessarily a metric.

[Markus Hanke]

On 9/25/2023 at 11:58 AM, Killtech said:

Because both are build on the very same manifold however means that any covariant tensor equation in the ellipsoid geometry can be translated into one in the sphere geometry like this

I’m having trouble following you here. A covariant tensor equation is precisely this - covariant. This means that its form does not change at all when you choose a different metric.

In GR in particular, you are free to choose any suitable system of coordinates you like to describe your spacetime, and this has no consequences for what form the Einstein equations take - they will always look the same.

What does have real consequences is choosing a different connection on your manifold. GR assumes the Levi-Civita connection, and if you choose a different one, you will in general get a different model that might make different predictions. Is this what you mean, perhaps?

PS. To reiterate again, connection and metric are very different things. You need both to meaningfully speak of lengths, angles, volumes etc, but general tensors and their relationships (though not all index manipulations) are defined without recourse to a metric, and you can still make some statements about the topology and geometry of your manifold (“affine geometry”) even without any metric at all.

[Killtech]

On 9/26/2023 at 7:36 PM, Markus Hanke said:

I’m having trouble following you here. A covariant tensor equation is precisely this - covariant. This means that its form does not change at all when you choose a different metric.

In GR in particular, you are free to choose any suitable system of coordinates you like to describe your spacetime, and this has no consequences for what form the Einstein equations take - they will always look the same.

Yes, a covariant form is as you say independent of the choice of coordinates - but i do not intend to change the coordinates. The issue is that "changing the metric" has an ambiguous meaning, because in physics it's used differently - because the actual metric is never considered to be changable. Physics don't consider the possibility that the very same physical situation can be described using different geometries. Analog to how we can freely choose coordinates, mathematics does actually allow to change the geometry as well, for as long as the topology is shared.

Consider that Newton's classical theory of gravity and Newton-Cartan theory represent the same physics, yet they achieve the same predictions by the use of different geometries.

On 9/26/2023 at 7:36 PM, Markus Hanke said:

What does have real consequences is choosing a different connection on your manifold. GR assumes the Levi-Civita connection, and if you choose a different one, you will in general get a different model that might make different predictions. Is this what you mean, perhaps?

PS. To reiterate again, connection and metric are very different things. You need both to meaningfully speak of lengths, angles, volumes etc, but general tensors and their relationships (though not all index manipulations) are defined without recourse to a metric, and you can still make some statements about the topology and geometry of your manifold (“affine geometry”) even without any metric at all.

I do assume to always use a Levi-Civita connection as you can see from the formulas in my last post. I left out the connection mostly because i prefer to have it compatible with the metric and therefore implied by the known formula for the Christoffel symbols. But in my case it is applied to a different tensor, hence the resulting connection is different as well. I haven't said that explicitly but it's in the formulas i have posted. If we agree to restrict to LC-connections, it is enough to specify only the metric tensor.

If the formulas in my example weren't enough or the idea too unfamiliar, note that a Weyl transform is a special case of this, however it has a different purpose and i don't intend to restrict to rescaling of the metric but rather any local transformation that preserves the rank of the tensor. 

The idea is indeed to redefine the meaning of lengths, angles and what is parallel: what one metric tensor (and its connection) may consider orthogonal, another may not! In mathematics we know that we can work with different definitions of orthogonality all the same - it is just not trivial to find the proper interpretation of what that means. Same as using different coordinates requires using a different interpretation for each point is (i.e. (0,0,1) is a very different location in polar then in cartesian coordinates), we must adapt our interpretation when using a different geometry similarly (lengths, angles etc.).

You are right, that if we change the connection and leave the interpretation unchanged, we will get a different model that will make wrong predictions. However if we change both the model and its interpretation accordingly, we can leave all predictions untouched.

[Markus Hanke]

5 hours ago, Killtech said:

The issue is that "changing the metric" has an ambiguous meaning

No, it has a very precise meaning, but clearly it doesn’t correspond to whatever it is you have in mind here.

5 hours ago, Killtech said:

because the actual metric is never considered to be changable

Can you explain precisely what you actually mean by “changeable”? You seem to be using the term in a non-standard way, so far as tensor calculus is concerned.

Or are you referring to the covariant derivative, which of course vanishes for the metric tensor?

5 hours ago, Killtech said:

Physics don't consider the possibility that the very same physical situation can be described using different geometries.

I don’t know what you mean by this - the Einstein equations relate energy-momentum to average curvature, so they place a local constraint on the metric. Along with boundary conditions they provide a unique geometry for any given distribution of energy-momentum. If you change that geometry, you have to also change that distribution, so you are no longer dealing with the same physical situation. If you want to keep your physical situation, but use a different geometry to describe it, then you have to change the form of your physical laws. Steven mentions this too in the screenshot you posted.

5 hours ago, Killtech said:

Consider that Newton's classical theory of gravity and Newton-Cartan theory represent the same physics, yet they achieve the same predictions by the use of different geometries.

NC gravity is much more general than standard Newtonian gravity - they are equivalent only if you impose certain external constraints on NC that don’t follow from the theory itself. Otherwise you get a large collection of different models that aren’t necessarily equivalent to Newton. Here is a good overview. 

5 hours ago, Killtech said:

I do assume to always use a Levi-Civita connection as you can see from the formulas in my last post. I left out the connection

Do you mean the screenshot of Steven’s post from PhysicsForums?

6 hours ago, Killtech said:

But in my case it is applied to a different tensor, hence the resulting connection is different as well. I haven't said that explicitly but it's in the formulas i have posted. If we agree to restrict to LC-connections, it is enough to specify only the metric tensor.

I’m sorry, I can’t make any sense of what you are saying here - first you say the connection will be different, then you seem to say we are using LC?

So if I understand all this correctly, you want to reformulate GR by using a geometry different from a semi-Riemannian manifold endowed with LC connection and metric to model gravity, right? Note that many of such alternatives already exist.

[Killtech]

16 hours ago, Markus Hanke said:

I don’t know what you mean by this - the Einstein equations relate energy-momentum to average curvature, so they place a local constraint on the metric. Along with boundary conditions they provide a unique geometry for any given distribution of energy-momentum. If you change that geometry, you have to also change that distribution, so you are no longer dealing with the same physical situation. If you want to keep your physical situation, but use a different geometry to describe it, then you have to change the form of your physical laws. Steven mentions this too in the screenshot you posted.

Steven's post was the best and shortest summary of what i intend to discuss - and yes, just as Poincaré implies and Steven writes, the change of geometry entails the change of laws physics. Einsteins field equations are linked with a very specific interpretation and geometry. Changing the latter requires to update the prior.

The starting point is that i want to use different devices as clocks which will produce time measurements that will disagree with the proper time general relativity expects. Instead of discarding these devices as false, i intend to find a model that fits them and therefore I need a metric tensor that is able to reproduce their time measurements. Measurements with such clocks naturally will also show a disagreement when testing various laws of physics as we know them, hence we do indeed need different laws to make the new clocks work.

16 hours ago, Markus Hanke said:

NC gravity is much more general than standard Newtonian gravity - they are equivalent only if you impose certain external constraints on NC that don’t follow from the theory itself. Otherwise you get a large collection of different models that aren’t necessarily equivalent to Newton. Here is a good overview. 

Your link is a bit general. can you tell me which chapter i should be looking up in more detail?

when comparing Newton's old theory to the relativistic case, we find that it is itself a collection of different proxy models, because we have to choose the frame in which gravity acts instantaneous. depending on the choice/definition what we consider simultaneous in reality, we get each a model that will produce slightly different predictions.

Furthermore, we need to interpret Newton's model and in particular each time measurement. Do we use an interpretation where any SI-clock is a valid measure of the absolute time or do we identify Newton's time with a coordinate time like TDB? while that does not change the working of the physical model itself, this has big impact on translating measurements into initial conditions and later back into predictions we can compare with experiments.

Newton's gravity is by no means a uniquely determined theory either. 

16 hours ago, Markus Hanke said:

Do you mean the screenshot of Steven’s post from PhysicsForums?

I’m sorry, I can’t make any sense of what you are saying here - first you say the connection will be different, then you seem to say we are using LC?

Hmm, not sure what you have trouble with here. As you can see from Steve's post we have two connections Gamma and Gamma' wich are both derived via the formula from their corresponding metric tensor g or g'. That formula is what makes each into an LC connection of the corresponding geometry. Since both manifolds use the very same set and coordinate map, we can evaluate all those terms in those coordinates at each location and find they are simply different matrices.

Let's go back to the simple case of a single differentiable manifold, which in one case we equip with a Riemann metric to make it into a sphere and in the other we pick a another metric to make it into an ellipsoid. Both cases have each a metric tensor and an associated LC connection. But the LC connection on a sphere cannot be the same as on an elipsoid. A LC connection is only unique per each Riemann metric - therefore on a single differentiable manifold we have as many LC connections as we have valid tensor fields that fulfil the requirements of a Riemann metric (bilinear, symmetric, non-degenerate everywhere).

16 hours ago, Markus Hanke said:

So if I understand all this correctly, you want to reformulate GR by using a geometry different from a semi-Riemannian manifold endowed with LC connection and metric to model gravity, right? Note that many of such alternatives already exist.

I would be very interested in looking these up. It's quite possible i was looking (googling) for the wrong thing and my results just came up empty.

[Markus Hanke]

4 hours ago, Killtech said:

the change of geometry entails the change of laws physics. Einsteins field equations are linked with a very specific interpretation and geometry. Changing the latter requires to update the prior.

Yes, that’s fine, you can always do this. Several such reformulations already exist.

4 hours ago, Killtech said:

The starting point is that i want to use different devices as clocks which will produce time measurements that will disagree with the proper time general relativity expects.

Proper time is defined to equal the geometric length of the clock’s world line through spacetime - this is very convenient, and greatly simplifies much of the maths. While it is certainly possible to make other choices (affine parametrisation), I don’t know why you would.

But so far as I can see this doesn’t lead to any different geometries - you’re still dealing with the same geodesics on the same manifold, you’re just parametrising them differently.

4 hours ago, Killtech said:

wich are both derived via the formula from their corresponding metric tensor g or g'

Connections are not derived from metrics, they are separate and more fundamental structures. The LC connection is a very specific one, namely that for which torsion vanishes.

4 hours ago, Killtech said:

Both cases have each a metric tensor and an associated LC connection. But the LC connection on a sphere cannot be the same as on an elipsoid. A LC connection is only unique per each Riemann metric - therefore on a single differentiable manifold we have as many LC connections as we have valid tensor fields that fulfil the requirements of a Riemann metric (bilinear, symmetric, non-degenerate everywhere).

Without intending any disrespect, but it seems like there’s some confusion here about the meaning of “metric” and “connection” in differential geometry, because what you write above doesn’t make any sense.

The connection exists quite independently from metrics and coordinates; its purpose is to relate tangent spaces at different points of the manifold to one another, so that a covariant derivative can be defined. A metric provides a way to define an inner product for vectors and forms, and thus a notion of lengths, angles, volumes etc. These are different things, and you can have a connection without a metric on your manifold - this allows you to do a certain amount of topology, define parallel transport, as well as tensor fields and some operations between them (excepting index raising/lowering, which requires a metric). Choosing a different metric thus has no bearing on your connection at all, it only changes the measurements of lengths and angles.

4 hours ago, Killtech said:

I would be very interested in looking these up.

Standard GR uses curvature on a semi-Riemannian manifold to model gravity. An example of an alternative approach is teleparallel gravity - here you use a parallelizable manifold and endow it with a Weizenböck connection, which yields a situation where you have no curvature at all, but only torsion. So gravity here is described solely through torsion on parallel geodesics, with the field equations adapted accordingly. 

If I understand you correctly, that’s an example of what you mean by “different geometry”.

A second example would be Einstein-Cartan gravity - here you choose a connection that allows both curvature and torsion, and adapt your field equations accordingly.

A third example is the ADM formalism - you replace your manifold with a foliation of 3D hypersurfaces, and wrap all your dynamics into how these surfaces are related to one another, using the Hamiltonian formulation.

And there are many more such formalisms. Do note that these are all specific examples of gauge theories - which is kind of the overarching framework when it comes to “different geometries”.

Is this helpful?

[joigus]

Great summary. I agree with Genady and Markus.

Connections are quite independent of metric in general. It's one of the hallmarks of Einstein's GR that the connection is a metric one. Thereby the words "metric connection".

Rods haven't been a standard for quite a while.

A gauge fibre bundle is an example of a metric-less connection. The gauge field A provides the parallel transport along the manifold, while the gradient of A gives you the parallel tranport on the fibres \( \Psi \), the whole structure is (locally) a product MxF (M=manifold, F=fibres), but with no metric for the \( \Psi \)'s.

Sometimes I have a problem understanding what the OP sets out to do. This is one of those times. Before one starts thinking about physics, one should get a clear picture of what needs to be solved.

Going back to metric connections in order to try to solve a problem GR doesn't have doesn't look promising.

Things that are considered solved are considered solved for a reason or, should I say, for a bundle* of reasons.

*  

[Markus Hanke]

17 hours ago, joigus said:

Things that are considered solved are considered solved for a reason or, should I say, for a bundle* of reasons.

Nicely put +1

[joigus]

23 hours ago, joigus said:

The gauge field A provides the parallel transport along the manifold

Here I should've said the Levi-Civita connection, sorry. Anyway, even the manifold connection can be introduced independently from the metric.

[Killtech]

On 9/29/2023 at 6:52 AM, Markus Hanke said:

Connections are not derived from metrics, they are separate and more fundamental structures. The LC connection is a very specific one, namely that for which torsion vanishes.

Without intending any disrespect, but it seems like there’s some confusion here about the meaning of “metric” and “connection” in differential geometry, because what you write above doesn’t make any sense.

I have to admit that i am a bit rusty in the field and had to look up a few things again. I mean no disrespect either but what you said is at least partially at odds with literature on geometry - because you leave important things out. 

While the connection is a fundamental tool in geometry, it is not actually used in most geometric definitions because it is only available under special circumstances. And whenever you talk about a LC connection, you forget that is is explicitly defined as a metric connection, that is via its compatibility with the metric (please check that if you don't believe me). So maybe let's start with clarifying a few things then.

The concept of a geodesic as a generalization of a straight line in a curved space, is roughly defined as the shortest local path between two points (skipping some details). That is amongst the most general definition for which only a metric space is needed but no differentiable structure (hence no connection). Note that also the setup of Riemann geometry, the Riemann manifold, is defined minimalistically in terms of only a smooth manifold and a metric with no mention of a connection because being a metric space, the entire geometry along with the geodesic structure is fully specified this way already.

However, for any practical use it is incredibly cumbersome to construct geodesics from that alone and since we are in a special case, we can make use of the additional tools available. This is where the LC connection comes into play. Aside from being torsion-free, its other defining property is that it must be compatible with the metric on the Riemann manifold. Only the combination of both conditions makes the connection unique. The second condition is crucial because it assures that the geodesics constructed via the connection agree with their metric definition - and frankly speaking this is the core motivation for that definition. The torsion-free is chosen for simplicity and more importantly to make it unique.

In case of Riemann geometry, the definition of geodesics via parallel transport is equivalent to their metric definition through that link. But that of course requires that we cannot treat the connection as independent from the metric at all.

On 9/29/2023 at 6:52 AM, Markus Hanke said:

The connection exists quite independently from metrics and coordinates; its purpose is to relate tangent spaces at different points of the manifold to one another, so that a covariant derivative can be defined. A metric provides a way to define an inner product for vectors and forms, and thus a notion of lengths, angles, volumes etc. These are different things, and you can have a connection without a metric on your manifold - this allows you to do a certain amount of topology, define parallel transport, as well as tensor fields and some operations between them (excepting index raising/lowering, which requires a metric). Choosing a different metric thus has no bearing on your connection at all, it only changes the measurements of lengths and angles.

What you write sounds as like lengths, angles and volumes are effectively independent of the geometry. but you are aware that angles are a way to express whether two vectors are orthogonal or parallel or in between? So your independent metric may get into an argument with your connection about its idea of a parallel transport.

On 9/29/2023 at 6:52 AM, Markus Hanke said:

Standard GR uses curvature on a semi-Riemannian manifold to model gravity. An example of an alternative approach is teleparallel gravity - here you use a parallelizable manifold and endow it with a Weizenböck connection, which yields a situation where you have no curvature at all, but only torsion. So gravity here is described solely through torsion on parallel geodesics, with the field equations adapted accordingly. 

If I understand you correctly, that’s an example of what you mean by “different geometry”.

A second example would be Einstein-Cartan gravity - here you choose a connection that allows both curvature and torsion, and adapt your field equations accordingly.

A third example is the ADM formalism - you replace your manifold with a foliation of 3D hypersurfaces, and wrap all your dynamics into how these surfaces are related to one another, using the Hamiltonian formulation.

And there are many more such formalisms. Do note that these are all specific examples of gauge theories - which is kind of the overarching framework when it comes to “different geometries”.

Is this helpful?

Yes, this is very helpful! Albeit i will have to do some reading before i can sort out how these relate to what i want to look into.

Give me a few days. I'm sure i'll back full of questions

 

Edited September 30, 2023 by Killtech

[studiot]

15 minutes ago, Killtech said:

The concept of a geodesic as a generalization of a straight line in a curved space, is roughly defined as the shortest local path between two points (skipping some details).

Isn't it also the longest path in some cases?

 

Also what about connectivity ?

[Killtech]

2 hours ago, studiot said:

Isn't it also the longest path in some cases?

I very roughly summarized the definition. technically, it is merely minimizing the distance locally. Like we know it from optimization problems, finding a local minimum doesn't guarantee at all it's also a global one. This can happen when there is more then one geodesic connecting two points with each other.

The definition requires the existence of a continuous curve between the start and the end, so it only concerns a connected subsets of the space.

You can find the definition here: https://arxiv.org/pdf/2007.09846.pdf
and this is how this general concept translates into the special case of Riemann geometry: https://www.cis.upenn.edu/~cis6100/cis61008geodesics.pdf

Edited September 30, 2023 by Killtech

[studiot]

2 hours ago, Killtech said:

I very roughly summarized the definition. technically, it is merely minimizing the distance locally. Like we know it from optimization problems, finding a local minimum doesn't guarantee at all it's also a global one. This can happen when there is more then one geodesic connecting two points with each other.

The definition requires the existence of a continuous curve between the start and the end, so it only concerns a connected subsets of the space.

You can find the definition here: https://arxiv.org/pdf/2007.09846.pdf
and this is how this general concept translates into the special case of Riemann geometry: https://www.cis.upenn.edu/~cis6100/cis61008geodesics.pdf

My queries were meant more as a prompt for you to look more carefully at the mathematics you are flinging about, rather than a request for information.

 

You are wrong here

2 hours ago, Killtech said:

very roughly summarized the definition. technically, it is merely minimizing the distance locally. Like we know it from optimization problems, finding a local minimum doesn't guarantee at all it's also a global one. This can happen when there is more then one geodesic connecting two points with each other.

local v global ?

 

The great circle distance London to New York is 3451 (minimum) or 21404 (maximum) miles, depending upon which way you go round.

Interesting there spherical geometry is an example of your second statement where all pairs of points are connected by an infinity of great circles.

[Killtech]

14 minutes ago, studiot said:

You are wrong here

local v global ?

 

the general definition only requires to minimize distance locally, the extension of that condition to hold on the entire interval makes it hold globally. 

 

[studiot]

2 hours ago, Killtech said:

 

the general definition only requires to minimize distance locally, the extension of that condition to hold on the entire interval makes it hold globally. 

 

 

Are you tryng to tell me that if I travel the 21,404 mile route from London to New York I am not following a geodesic ?

 

In that case how would you describe that path ?

[Markus Hanke]

13 hours ago, Killtech said:

And whenever you talk about a LC connection, you forget that is is explicitly defined as a metric connection

Yes, it’s a connection for which torsion vanishes - this is what I wrote above. It also makes the covariant derivative of the metric identically vanish, which I likewise mentioned in a previous post.

13 hours ago, Killtech said:

The concept of a geodesic as a generalization of a straight line in a curved space, is roughly defined as the shortest local path between two points (skipping some details).

As I explained already, this is not its fundamental definition, because this applies only to those manifolds that are endowed with a metric. The actual definition is a curve that parallel-transports its own tangent vector - you don’t need a metric for this.

If you do have a metric (as in GR), then geodesics extremize (either minimise or maximise) the separation of events, but doing this extremisation procedure requires a metric.

13 hours ago, Killtech said:

Note that also the setup of Riemann geometry, the Riemann manifold, is defined minimalistically in terms of only a smooth manifold and a metric with no mention of a connection because being a metric space, the entire geometry along with the geodesic structure is fully specified this way already.

Spacetime manifolds in GR are pseudo-Riemannian (ie locally Lorentzian), so they are always endowed with a connection. Without that there wouldn’t be a notion of parallel transport, and thus curvature, and so it would be useless for the purposes of the model.

13 hours ago, Killtech said:

But that of course requires that we cannot treat the connection as independent from the metric at all.

Of course we can - this is the basis of differential topology and affine geometry. You use a connection for these, but no metric.

13 hours ago, Killtech said:

What you write sounds as like lengths, angles and volumes are effectively independent of the geometry.

It’s rather the other way around. There are some aspects of geometry and topology that are entirely independent of any metric. Parallel transport is one of them.

13 hours ago, Killtech said:

So your independent metric may get into an argument with your connection about its idea of a parallel transport.

Parallel transport doesn’t have anything to do with angles, it is defined purely in terms of the connection. However, if you do have a metric as well, then the two are related via the connection coefficients and the metric compatibility condition, so of course there is never any conflict.

13 hours ago, Killtech said:

Yes, this is very helpful! Albeit i will have to do some reading before i can sort out how these relate to what i want to look into.

Give me a few days. I'm sure i'll back full of questions

I am still not sure if I understand what you are actually trying to do. But I’ll wait for further comments first.

[Killtech]

9 hours ago, studiot said:

Are you tryng to tell me that if I travel the 21,404 mile route from London to New York I am not following a geodesic ?

In that case how would you describe that path ?

When you stick to that definition, your 21404 mile route is a geodesic, just not a mimimizing geodesic.