Relativity in Geometry and Physics

[studiot]

2 hours ago, Killtech said:

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

Exactly so, except you didn't originally specify a minimising geodesic.

 

But even then you must be careful since the ame piece of geodesic can be minimising or maximising depending upon the final endpoints.

 

For instance London, Brighton, the North nd South Poles are all on the Greenwich Meridian, which is a geodesic.

But is it minimising or maximising ?

Well if I start from Greenwich and travel south I am travelling along a minimising geodesic,

But if I am going to the North Pole it is a maximising geodesic !

[Killtech]

4 hours ago, Markus Hanke said:

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.

Yes, and that connection is always given by the metric of the Riemann manifold via Levi-Civita. This is why the definition skips out out to mention it. In the special case of Riemann geometry, the metric uniquely dictates the connection and gives it a special name, the LC connection.

But don't misunderstand me, i am not insisting that in general a connection requires a metric for its definition. It does not. In that sense it is indeed an entirely independent object around which there is a separate field of study. But we are not discussing the connection itself but geometry, and that is another matter. A some geometric properties have redundant definitions as they can be defined via different concepts, like e.g. geodesics. Besides of what the name geometry already implies, as you can see from the field of pure metric geometry, main geometric definitions don't need to have a connection at all. This is where it is important that when different concepts are available at the same time, their compatibility must be ensured. It is weird to work with a metric and a connection that contradict each other showing two very different geometries. So whenever geometry is concerned specifically, there is a clear link between them.

For the part of physics i want to discuss I assume we have a both a metric and a connection and they have to be compatible so that whenever we talk about geometry in the model, these don't provide contradicting accounts. A change of metric hence requires to find a new connection compatible with the new metric.

4 hours ago, Markus Hanke said:

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

I'm on it.

[Markus Hanke]

2 hours ago, Killtech said:

Yes, and that connection is always given by the metric of the Riemann manifold via Levi-Civita.

You are always free to choose a connection other than Levi-Civita, but if you do that, you will have to adjust your physical laws accordingly, since their form might differ now.

2 hours ago, Killtech said:

But don't misunderstand me, i am not insisting that in general a connection requires a metric for its definition. It does not. In that sense it is indeed an entirely independent object

Ok, so we are in agreement on this.

2 hours ago, Killtech said:

It is weird to work with a metric and a connection that contradict each other showing two very different geometries. So whenever geometry is concerned specifically, there is a clear link between them.

 

The link between them is given by the connection coefficients (Christoffel symbols), which allow one to express the effects of the chosen connection in terms of the metric and its derivatives in a consistent way, should the manifold be endowed with a metric. So there is never any “conflict” between them.

2 hours ago, Killtech said:

A change of metric hence requires to find a new connection compatible with the new metric.

Changing the metric simply changes the connection coefficients, it has no bearing on the connection itself. For example, GR allows for infinitely many different metrics on spacetime, but it always uses the Levi-Civita connection.

2 hours ago, Killtech said:

But we are not discussing the connection itself but geometry, and that is another matter.

Ok, so now you need to define for us just exactly what it is you mean by “geometry”. In standard GR, two given spacetimes are said to have the same geometry if they pass the Cartan-Karlhede algorithm, meaning “geometry” is given by curvature tensors, their curvature invariants, and the functional relationships between them. But I don’t think this is what you have in mind - it sounds more like you wish to model spacetimes without reference to any metric at all, and thus express “geometry” in terms of different dynamical variables. If so, gauge theory gravity might be an example of what you are looking for.

[Killtech]

48 minutes ago, Markus Hanke said:

The link between them is given by the connection coefficients (Christoffel symbols), which allow one to express the effects of the chosen connection in terms of the metric and its derivatives in a consistent way, should the manifold be endowed with a metric. So there is never any “conflict” between them.

Changing the metric simply changes the connection coefficients, it has no bearing on the connection itself. For example, GR allows for infinitely many different metrics on spacetime, but it always uses the Levi-Civita connection.

Ok, so now you need to define for us just exactly what it is you mean by “geometry”. In standard GR, two given spacetimes are said to have the same geometry if they pass the Cartan-Karlhede algorithm, meaning “geometry” is given by curvature tensors, their curvature invariants, and the functional relationships between them. But I don’t think this is what you have in mind - it sounds more like you wish to model spacetimes without reference to any metric at all, and thus express “geometry” in terms of different dynamical variables. If so, gauge theory gravity might be an example of what you are looking for.

Okay, i totally failed explaining what i mean by "changing the metric". In my defense i don't know any appropriate terminology for that particular procedure and googling didn't help... so i turn to the forums.

Maybe let me try to rephrase it: Given a Riemann manifold (X,g) we can also consider it a simple metric space (ignoring its differential structure for the start). Now let's consider the identity map id of X to itself. I want to introduce an alternative metric structure on X to make it a different metric space (X,f). In that scenario id also becomes a map between two metric spaces and the intention of the choice of f is that id won't be an isometry. Now accounting that X is a smooth manifold we have two distinct Riemann manifolds, each with its own LC connection and they must consequently fail the Cartan-Kalhede test.

I started reading on teleparallelism and it goes quite along what i am interested in. The tetrad field in my case would be build from the unit vectors of the TDB and BCRS coordinates. I am just not sure i understand the choice of metric and connection in that case yet. gimme some time. But choice of metric indeed also tries to study a case of a flat geometry, but i intend to stay within the context Riemann geometry. The major difference is that i do not want to postulate any new physical laws on my own but rather would like to deduct the laws in the new geometry from the starting theory using a transformation like Steven posted. In particular, i want to move all influence of gravity from the geometry and torsion (rendering it trivial) and instead separate it out into its own fields: in terms of the transition to the new equation of motion of a particle, the remaining difference between the new and the old Christoffel symbols needs to be interpreted as physical fields representing gravity. 

 

 

[Markus Hanke]

14 hours ago, Killtech said:

Now accounting that X is a smooth manifold we have two distinct Riemann manifolds, each with its own LC connection and they must consequently fail the Cartan-Kalhede test.

 

14 hours ago, Killtech said:

I want to introduce an alternative metric structure on X to make it a different metric space (X,f).

This is the bit I don’t get - what exactly do you mean by “alternative metric structure”? You can always just pick a new metric tensor that isn’t related to the old one by any diffeomorphism, which gives you a new spacetime that is not isomorphic to the old one. But that’s probably not what you have in mind? And why would you want to do this at all - what is the advantage?

14 hours ago, Killtech said:

In particular, i want to move all influence of gravity from the geometry and torsion (rendering it trivial) and instead separate it out into its own fields:

In that case you need to go away from metric structures altogether, and consider non-metric approaches. The aforementioned Gauge Theory Gravity is an example for this. More fundamentally, it is probably gauge theory in general that you should take a closer look at.

[Killtech]

On 10/2/2023 at 8:47 AM, Markus Hanke said:

This is the bit I don’t get - what exactly do you mean by “alternative metric structure”? You can always just pick a new metric tensor that isn’t related to the old one by any diffeomorphism, which gives you a new spacetime that is not isomorphic to the old one. But that’s probably not what you have in mind? And why would you want to do this at all - what is the advantage?

Yes, this is indeed where i want to start from. As I understand it, the metric is the important bridge between model and experiment and one finds almost all measurements contain the units of lengths and time. The interpretation isn't actually trivial, because looking deeper at the definitions, one has to make quite a few implicit assumptions in order to define any kind of unit. So the question arises what happens, if we changed some of those assumptions/definitions? We could for example assume a real physical oscillator, on which frequency we base our unit on, is influenced by certain local conditions and consequently we want to apply location and frame specific correction factors to counter these effects. 

But that interests me for another reason: In science we want to test the assumptions of our models against experiments and in the process it's sometimes easier to formulate counter-hypothesis and check those instead. For some postulates that runs into logical problems: e.g. the isotropy of the speed of light. If we assume a model where it isn't a constant, we run into contradictions with how measurement in experiments works which still implicitly assumes otherwise. If we however account those new assumptions in measurement, the required corrections will yield different measurements as we practically use a different metric (implicitly also geometry). In that case, we may end up doing what i want to discuss. If physics can be reformulated into a different geometry with different laws of physics such that it yields identical predictions, then a counter-hypothesis may prove physically equivalent to the base postulate. In that case i would consider such postulates untestable and treat them as conventions.

Before i can go deeper exploring various approaches for physical models, i want to first get a good understanding of the the fundamental relations between a model, its interpretation, measurement and experimental testing. Furthermore, what exact role does the metric plays in this and is the way i think about it correct?

On 10/2/2023 at 8:47 AM, Markus Hanke said:

In that case you need to go away from metric structures altogether, and consider non-metric approaches. The aforementioned Gauge Theory Gravity is an example for this. More fundamentally, it is probably gauge theory in general that you should take a closer look at.

I have looked tiny bit into GTG and it does sound quite interesting, though i have not yet understood how its interpretation works. I will have to look better into the techniques applied, though the idea to start a theory from the action and deduct the model from there is inconvenient for my case, because my starting point is indeed the metric and i know too little about what the resulting model may be. Also, i'm not sure if a flat Minkowski space is a good basis for a formulation where c(x) is deliberately made non constant.

During the week i usually don't have too much time to focus on such topics. For now you gave me plenty of stuff to read

[Markus Hanke]

8 hours ago, Killtech said:

We could for example assume a real physical oscillator, on which frequency we base our unit on, is influenced by certain local conditions and consequently we want to apply location and frame specific correction factors to counter these effects.

That’s just a particular choice of coordinate basis, which you are always free to make. You can even choose coordinates that don’t correspond to any physical clocks and/or rulers at all (eg Kruskal-Szekeres coordinates), and still obtain useful solutions.

8 hours ago, Killtech said:

Furthermore, what exact role does the metric plays in this and is the way i think about it correct?

The metric in GR essentially represents your chosen way of how you label physical events in your spacetime, and allows you to define measurements (angles, distances, volumes etc) in terms of those labels. It basically tells you how events are related, given a particular choice of coordinate basis. It also relates vectors to 1-forms and vice versa, and allows for the definition of certain important operations, like the Hodge dual. If you picture a street map of your local city, the metric would be the scheme by which you assign street names, as well as define distances and routes between addresses. Just like you are free to re-name streets without affecting the physical layout of your city, so you can choose different coordinates on the same spacetime without affecting its geometry.

8 hours ago, Killtech said:

Also, i'm not sure if a flat Minkowski space is a good basis for a formulation where c(x) is deliberately made non constant.

To be honest, I don’t see how such a formulation would be helpful, given that local Lorentz invariance is an extremely well established experimental result. c simply isn’t seen to locally vary in any meaningful way, so I don’t see why one would want to write a model based on this.

Is your basic idea to try and replace the rank-2 tensor description of GR with some kind of scalar field theory?

[Killtech]

10 hours ago, Markus Hanke said:

That’s just a particular choice of coordinate basis, which you are always free to make. You can even choose coordinates that don’t correspond to any physical clocks and/or rulers at all (eg Kruskal-Szekeres coordinates), and still obtain useful solutions.

yes and no. you are right that i can do a lot of things already with the coordinates but not all of it.

but coordinates are one thing, units another. energy for example does not depend on the choice of coordinates (apart from the frame), yet its unit is made up of length and time units. using a locally different time unit as a basis for energy defines a very different physical entity that actually belongs to a different geometry. Noether guarantees us that as long as we can find the symmetries in the new geometry, there will be a alternative concept of energy which will be preserved.

Coordinates cannot help with the energy question. Consider Euclidean clocks which behave differently between frames compared to proper time, particularly lacking the singularity at c - if we insist on those to measure an alternative energy, we get diverging results. It requires very different laws of physics to make that new energy (and its action) produce the same outcomes as the relativistic Minkowski geometry does. That is what i am aiming to look for by changing of the metric and particularly hope it can provide a direct translation mechanism in between these different concepts of energy and geometry. 

10 hours ago, Markus Hanke said:

The metric in GR essentially represents your chosen way of how you label physical events in your spacetime, and allows you to define measurements (angles, distances, volumes etc) in terms of those labels. It basically tells you how events are related, given a particular choice of coordinate basis. It also relates vectors to 1-forms and vice versa, and allows for the definition of certain important operations, like the Hodge dual. If you picture a street map of your local city, the metric would be the scheme by which you assign street names, as well as define distances and routes between addresses. Just like you are free to re-name streets without affecting the physical layout of your city, so you can choose different coordinates on the same spacetime without affecting its geometry.

I think it is also the metric which carries units while coordinates are usually treated as dimensionless. of course we often use conventions like c=1 to hide and simplify the equations.

10 hours ago, Markus Hanke said:

To be honest, I don’t see how such a formulation would be helpful, given that local Lorentz invariance is an extremely well established experimental result. c simply isn’t seen to locally vary in any meaningful way, so I don’t see why one would want to write a model based on this.

It is worthwhile to spend some time reading Henry Poincaré's notes on measuring time and what that means for the speed of light. So how would be even notice c to vary, if even Poincaré's corrected Lorentz Aether Theory concludes the same result for the Michelson interferometer? Any attempt to measure c requires us to be able to measure time and length or at least assure we can maintain intervals of constant lengths for the measurement. Yet all the definitions of length and time we used were purely electromagnetic in origin. And it is precisely there where we go the full circle. if our concepts of length and time are implicitly based on c and use it as a reference rendering it constant, then all our measurement will show exactly this and none will be able to record any deviations unless it breaks from the specifications of the SI system.

Look at the definition of a geodesic clock @Genady posted earlier and consider how it is affected by a locally varying c(x). it demonstrates how the assumptions on the speed of light is tight to definition of clocks and that it will work with any assumption you put into it. Now consider using that clock in reverse to measure c - those are two side of the same coin.

We do know from experiments that clocks run out of synch depending how close they are to a gravity well. We can interpret it as usual, or we can assume that our reference oscillator for time is affected by some local effect and needs correction - same as we had to correct for thermal expansion of the original meter bar and same as the official SI definition of second via Caesium lists required corrections singling out gravity as the only local influence that must not be corrected. If we do that however, we speed up time locally at the immediate consequence that c(x) gets a local dependence. isotropy of light is incompatible with isotropy of clocks.

10 hours ago, Markus Hanke said:

Is your basic idea to try and replace the rank-2 tensor description of GR with some kind of scalar field theory?

 

I am open to the result of whatever the metric transition will require. i highly doubt it will be however a scalar field theory, after all the existing degrees of freedom gravity has in GR embedded into the geometry have to go somewhere. I do however think there will be at least one dominant scalar field, especially in the Maxwell equation: a refractive index c(x). But if gravity travels at c and has transversal waves, it likely needs quite a few field equations... actually i would think it might look analogue to Maxwell with two force fields, a vector current and a scalar density. each dimension reflecting one degree of freedom of the GR metric tensor.

actually i stumbled on this recently: https://arxiv.org/pdf/gr-qc/0205035.pd . haven't yet time to go through it, but it goes in a similar direction albeit with a different starting point.

[Markus Hanke]

10 hours ago, Killtech said:

i highly doubt it will be however a scalar field theory, after all the existing degrees of freedom gravity has in GR embedded into the geometry have to go somewhere.

Yes, that’s right.

10 hours ago, Killtech said:

But if gravity travels at c and has transversal waves, it likely needs quite a few field equations... actually i would think it might look analogue to Maxwell with two force fields, a vector current and a scalar density.

You need at least a rank-2 tensor in there as well, in order to capture all relevant degrees of freedom.

But regardless, I agree that the result - if it exists at all - would end up being mathematically very complicated and require extra fields, whereas standard GR has pretty simple field equations. So the question naturally arises: why bother? What actual problem in the existing models are you attempting to address?

10 hours ago, Killtech said:

So how would be even notice c to vary, if even Poincaré's corrected Lorentz Aether Theory concludes the same result for the Michelson interferometer?

You can look at ratios (!) of radioactive decay products over long periods. This has been done using natural fission reactors, with the consensus being that the relevant constants of nature involved in these processes have not changed over at least the last ~2 billion years.

[Killtech]

On 10/5/2023 at 8:31 AM, Markus Hanke said:

You can look at ratios (!) of radioactive decay products over long periods. This has been done using natural fission reactors, with the consensus being that the relevant constants of nature involved in these processes have not changed over at least the last ~2 billion years.

You have to be careful with such statements. c is a very different constant from all else given its connection to the definitions of time and length. As Poincaré nicely demonstrates by the example of how Astronomers determined that c was constant, that in fact they had to assume how light moves through the vacuum beforehand in order to make measurements at all, hence they showed that if c is const then c ist const. Therefore if you account how the interpretation ties into the model, the experiment actually measured the function c(c). His example is a good point of study for the general unresolvable interdependence. 

Besides, with the current definition of the SI meter, it is logically impossible that c can vary in any way. i am aware that this definition was chosen much later and for a reason. But let's view it the other way around: if we simply ignore what c "really" does, define its behavior ourself instead and make all experiments maintain that convention (SI system), would we be able to find out that we are "wrong"? For as long as a meter defined liked this provides a well defined measure for length (a rod that has some hysteresis when moved around won't suffice the axioms of a length measure), then no, because all experiments will just provide some results and we can always will find some model that reproduces them. The question is a chicken or the egg causality dilemma between definitions of units and laws of physics. if one assumes a constancy the other inherits it.

In reality we can only observe how physical entities change relatively to each other but never how they change absolutely. So what we can do is compare two physical processes where c is involved against each other and compare that c obtained from one is same as the other. A deviation would be interpreted as our model/understanding of one the processes is wrong. I know experiments were conducted to check how stable constant were, but we have to be a lot more careful interpreting the results.

On 10/5/2023 at 8:31 AM, Markus Hanke said:

You need at least a rank-2 tensor in there as well, in order to capture all relevant degrees of freedom.

But regardless, I agree that the result - if it exists at all - would end up being mathematically very complicated and require extra fields, whereas standard GR has pretty simple field equations. So the question naturally arises: why bother? What actual problem in the existing models are you attempting to address?

Well, one can package all components of Maxwell into a rank-2 energy-stress tensor and the field equation of GR provide the its time evolution. The analogue could work for gravity... maybe it would just effectively replace the Einstein Tensor with another and reshape the remaining Maxwell energy stress tensor into a trivial geometry. With the metric trivialized to globally Euclidean and the two tensors becoming analogue in interpretation, one could combine them into one thingy. So i am not convinced this must lead to a less simple formalism.

There are a lot of reasons to ask questions. We still haven't solved the issue of quantum gravity. Looking on a problem from another perspective may help, specifically in a flat Euclidean geometry quantization might be easier. Also reshaping spacetime like this allows comparison to familiar classical fluids and their equations. I would be interested to study how light inside a warp-bubble solution compares to the situation of sound waves inside a cockpit of a supersonic jet. Maybe if we can bring sound and light into a comparable metric, an analogy which might help us know where to look to find solutions to circumvent certain speed limits.

There is also an unresolved issue with galaxy rotation, which i have a hypothesis i an interested to test experimentally. I wanted post on that later, but since it has connection to this concept here, i decided to post this first.

[studiot]

10 hours ago, Killtech said:

You have to be careful with such statements.

 

On 10/4/2023 at 8:55 PM, Killtech said:

but coordinates are one thing, units another. energy for example does not depend on the choice of coordinates (apart from the frame), yet its unit is made up of length and time units. using a locally different time unit as a basis for energy defines a very different physical entity that actually belongs to a different geometry.

You have omitted one of the dimensions of energy.

 

Actually one thing Markus has not stressed is that

On 10/4/2023 at 8:55 PM, Killtech said:

yes and no. you are right that i can do a lot of things already with the coordinates but not all of it.

but coordinates are one thing, units another. energy for example does not depend on the choice of coordinates (apart from the frame),

Using coordinates directly does not satisfy the Principle of Relativity.

You need to use coordinate differences for that.

It is also worth realising that n dimensional coordinate rquire n+1 pieces of information (unlike geodesics for instance )

 

10 hours ago, Killtech said:

There are a lot of reasons to ask questions. We still haven't solved the issue of quantum gravity. Looking on a problem from another perspective may help, specifically in a flat Euclidean geometry quantization might be easier. Also reshaping spacetime like this allows comparison to familiar classical fluids and their equations. I would be interested to study how light inside a warp-bubble solution compares to the situation of sound waves inside a cockpit of a supersonic jet. Maybe if we can bring sound and light into a comparable metric, an analogy which might help us know where to look to find solutions to circumvent certain speed limits.

I totallty agree. Relativity cannot cope with singularities.

It is interesting to note that, as you say, such singularities arise in Fluid Mechanics and Nature has its own resolution, not predicted in the mathematics on eithe side of a singularity.

Shock waves and the Hydraulic jump are specific real world examples, and energy is involved in the analysis.

[Markus Hanke]

On 10/7/2023 at 4:18 AM, Killtech said:

Well, one can package all components of Maxwell into a rank-2 energy-stress tensor and the field equation of GR provide the its time evolution. The analogue could work for gravity... maybe it would just effectively replace the Einstein Tensor with another and reshape the remaining Maxwell energy stress tensor into a trivial geometry. With the metric trivialized to globally Euclidean and the two tensors becoming analogue in interpretation, one could combine them into one thingy. So i am not convinced this must lead to a less simple formalism.

Well, there’s nothing wrong with trying this, so long as the resulting model can replicate already known (and well-tested) results. I would be interested to see what this would look like. 

On 10/7/2023 at 4:18 AM, Killtech said:

Looking on a problem from another perspective may help, specifically in a flat Euclidean geometry quantization might be easier.

I don’t know how you would recover local Lorentz invariance from a Euclidean metric, but again, there’s nothing wrong with trying, if you can show that it provides correct predictions.

[Killtech]

On 10/8/2023 at 9:30 AM, Markus Hanke said:

Well, there’s nothing wrong with trying this, so long as the resulting model can replicate already known (and well-tested) results. I would be interested to see what this would look like. 

Indeed me too. I am still figuring if such an approach is viable in general and if there are already similar concepts people worked on that may help here. As for the details, i am still figuring out how the correction function applied to clocks in TDB looks like when generalized to any possible theoretical situation, i.e. what the metric that TDB units imply looks like relative to the usual metric of GR - because that's what mostly defines the transition between geometries.

On 10/8/2023 at 9:30 AM, Markus Hanke said:

I don’t know how you would recover local Lorentz invariance from a Euclidean metric, but again, there’s nothing wrong with trying, if you can show that it provides correct predictions.

It doesn't work like that. A such a big change in geometry also is accompanied by a change of some symmetries. Particularly having a locally dependent speed of light does not work too well with Lorentz invariance. Other geometries means other laws of physics, and in this special case the laws of physics using a time (metric) that is shared between all frames and locations (effectively is an absolute time by Newtons definition), their invariance can at best be only Galilean. This is the form where the equations can start to resemble those of fluid dynamics.

[Markus Hanke]

7 hours ago, Killtech said:

Other geometries means other laws of physics, and in this special case the laws of physics using a time (metric) that is shared between all frames and locations (effectively is an absolute time by Newtons definition), their invariance can at best be only Galilean.

Well, this is going to be a problem then. I don’t think you will get anywhere useful if you plan to abandon local Lorentz invariance, and thus necessarily also CPT invariance, as well as some fundamental properties such as spin. Clearly, the outcome of particle accelerator experiments are what they are (in terms of how different types of particles behave in specific circumstances etc), and whatever form you choose to write your laws of physics in, they must be able to reproduce these outcomes in some way, or else these formalisms will be of no use at all, because they wouldn’t relate to what we actually see and measure in the real world.

But why don’t you keep working on it a bit more, and once you have an actual formalism to present, people will be happy to take a look at it.

 

[KJW]

On 9/29/2023 at 9:38 AM, 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. 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.

In what way are these different devices clocks? In what way are these measurements that disagree with the proper time of general relativity time measurements? I'm not discarding these devices as false, only your assertion that they measure time.

It is my belief that it is the most fundamental principle of reality that the laws of physics are invariant. For without invariance, there are no laws of physics.

[studiot]

1 hour ago, KJW said:

It is my belief that it is the most fundamental principle of reality that the laws of physics are invariant. For without invariance, there are no laws of physics.

That is a very strong statement.

Doesn't it depend upon what you mean by invariant ?

For instance the Principle of Relativity requires the same form in any inertial system or that these laws are independent of the coordinate system.

[Killtech]

 

16 minutes ago, KJW said:

It is my belief that it is the most fundamental principle of reality that the laws of physics are invariant. For without invariance, there are no laws of physics.

And you are not entirely wrong here. However, math is a mean bastard when you go deep into some seeming trivial details. There is just no one singular way to represent and model physics because it turns out you will requiring quite a few of additional assumptions that cannot be experimentally verified. Those are technically conventions. The choice of those will however have an impact on the resulting invariances and therefore laws of physics. 

16 minutes ago, KJW said:

In what way are these different devices clocks? In what way are these measurements that disagree with the proper time of general relativity time measurements? I'm not discarding these devices as false, only your assertion that they measure time.

Uff, that isn't so easy to answer.

But how do you define a clock or a time measurement in general? let's say we are in a different universe then this one with other laws an all (or just in a computer simulated reality like in the matrix films). How do we define in a general abstract case? Usually it helps by asking what do we need time measurements for to solidify which axioms those measurements have to adhere to in order to fulfill that purpose. This is maybe a mathematical approach physicist don't often consider.

A key aspect of measurements is their ability to compare results and translate real world relations into numeric values we can do calculus on. The mathematical concept of a metric very accurately reflect this fundamental of measurements. But given one metrizible topology, we know from mathematics there is way more then one possible metric.

Note that nature does not actually need any numbers to work. But we do to model nature. So for nature a smooth topology is enough and it is us that adds a numeric structure of a metric with its comparison relation. in doing so we introduce a lot of untestable assumptions (that implicitly define the metric we use) that mix with the laws of nature into their familiar representation.

As a somewhat analog example consider how different symmetries look from the perspective of different coordinates. The same is true for geometries / metrics.

But your initial believe holds in a sense as long as the model suffices to Noethers prerequisites: if we choose a suitable metric, it will still have invariances and well handable laws of physics - those will depend on chosen metric / geometry though. Technically you could however work with violated Noether assumptions... but that will be very annoying to handle a system where the total energy isn't conserved but evolves by a deterministic function which means the laws of physics will have some nasty absolute time dependence.

I did pick the TDB time coordinate as a time metric explicitly because it guarantees a Galilean invariance with a corresponding energy conservation but requires altered laws of physics that fit those while still reproducing the same relativistic physics.

[KJW]

41 minutes ago, studiot said:

That is a very strong statement.

Doesn't it depend upon what you mean by invariant ?

For instance the Principle of Relativity requires the same form in any inertial system or that these laws are independent of the coordinate system.

You are correct. There is the question of "invariant to what?". For example, Special Relativity is invariant to Poincaré transformations, whereas General Relativity is invariant to all coordinate transformations, a much bigger group. But if the group of transformations for which the laws of physics are invariant is limited, and that the laws of physics are not invariant to transformations outside this group, then it becomes impossible to state what these varying laws of physics are because one doesn't have a reference against which one can state which laws of physics apply to which location in the space to which the transformations apply. But if the laws of physics are invariant, then they can be stated unequivocally due to that invariance.

For example, if the laws of physics are expressed entirely in terms of partial derivatives, then they will vary according to the coordinate system. But the coordinate system is not actually known, so one can't know which laws of physics apply. But if the laws of physics are expressed entirely in terms of covariant derivatives, then they will be invariant to coordinate transformations, and therefore one no longer needs to know the coordinate system because the same laws of physics apply to all of them.
 

23 minutes ago, Killtech said:

But how do you define a clock or a time measurement in general?

I often see the question "What is time?". The answer I give is "Time is what a clock measures.". But what is a clock? A clock is defined by the instructions that are used to construct it. While this may not be particularly helpful in understanding the nature of time, time is nevertheless well defined by the instructions.

[Markus Hanke]

@KJW Well, mark my words…I remember you from the good old days back on The Science Forum…great to see you again

[KJW]

5 hours ago, Markus Hanke said:

@KJW Well, mark my words…I remember you from the good old days back on The Science Forum…great to see you again

Good day, Markus. Yes, it is me from The Science Forum. It seems that The Science Forum has finally died. It did seem like an inevitability for quite a long time. But I did stick with it till the very end. Anyway, I joined this site earlier today, posted some stuff.  I notice you are still using the generalised Stokes theorem as your signature. 😉

[Killtech]

 

  Hmm, what an inkonvenient technical pause to the conversation.

On 10/17/2023 at 12:54 AM, KJW said:

I often see the question "What is time?". The answer I give is "Time is what a clock measures.". But what is a clock? A clock is defined by the instructions that are used to construct it. While this may not be particularly helpful in understanding the nature of time, time is nevertheless well defined by the instructions.

Sure, so you reduce time to the instructions to construct it and in the case of SI, we have very specific instructions. That definition is well defined, sure, but it is decided upon by a committee of people and not actually nature. If you look into the details, you will find a lot of instructions how to correct the Caesium atoms readings for specific effects and you will additionally find a passage explicitly stating that effects of gravity must not be corrected. These seemingly arbitrary specifications make it clear that it is a convention we come up with, same as Einstein's synchronization is and that raises further questions.

A mathematicians first natural reflex here is to ask, what other choices of such instructions could we use instead that lead us to a well defined time? are all of those equivalent? And really, thinking a bit about it, it turns out that geometry rises such question and has figured out the answers long ago. It turns out that a metrizable topological space with a differential structure allows for way more one Riemann manifold to construct on it.

So we know that there is a large set of possible alternative concepts of time that are not isometric to the one we use.  We can deduct how clocks of the different definitions of time relate to each other, and we can formulate how the laws of physics and their symmetries look from the perspective of other alternative clocks. We cannot go wrong when we change conventions our theories work with, can we?

[geordief]

On 10/17/2023 at 11:36 AM, KJW said:

 It seems that The Science Forum has finally died. It did seem like an inevitability for quite a long time. But I did stick with it till the very end

How did you know it had died?You are right .The domain is for sale. (224  Dollars or whatever currency they are displaying on Godaddy)

Without the archive I can't see any point in buying it (Skinwalker ,the owner of thescienceforum.org once said he was keeping an eye on the domain -about 4 or 5 years ago)

 

[Markus Hanke]

On 10/17/2023 at 12:36 PM, KJW said:

It seems that The Science Forum has finally died. It did seem like an inevitability for quite a long time.

It’s a real shame, I have many fond memories of my years on TSF. But that’s how it goes sometimes. This here is a good place though.

On 10/17/2023 at 12:36 PM, KJW said:

I notice you are still using the generalised Stokes theorem as your signature. 😉

Lol yes, I copied this across when I first came here. I still think it is one of the most beautiful (and important) results in all of maths :)

[exchemist]

On 10/17/2023 at 11:36 AM, KJW said:

Good day, Markus. Yes, it is me from The Science Forum. It seems that The Science Forum has finally died. It did seem like an inevitability for quite a long time. But I did stick with it till the very end. Anyway, I joined this site earlier today, posted some stuff.  I notice you are still using the generalised Stokes theorem as your signature. 😉

Good to see you here. I wonder if we will get any more refugees.

[Phi for All]

4 hours ago, exchemist said:

Good to see you here. I wonder if we will get any more refugees.

Our hearts go out to those who've been intellectually abused by the loss of their favorite forums. Cheese Nips and coffee are over there on the periodic table, please help yourselves. So glad momentum brought y'all here!