The meaning of constancy of the speed of light

[studiot]

Just now, Killtech said:

Think of it this way: the definition of the SI second defines the Caesium transition frequency as invariant

Does It?

 

I don't think so, nor do I think astronomical observations bear that out.

If it is invariant, why do we need to do corrections for the relativistic shift of the spectral lines ?

Invariant surely means the same in all frames, our observational one and that of some receeding galaxy.

Edited Thursday at 11:10 PM by studiot

[swansont]

1 hour ago, Killtech said:

Think of it this way: the definition of the SI second defines the Caesium transition frequency as invariant (which is entirely electromagnetic in origin).

No, it doesn’t.

Quote

The SI meter defines c as invariant

Also no. It leverages that fact, but defines the meter in terms of the second and the numerical value of c

[KJW]

1 hour ago, swansont said:

Quote

The SI meter defines c as invariant

Also no. It leverages that fact, but defines the meter in terms of the second and the numerical value of c

That's actually an interesting point. Although defining c to have a particular numerical value does seem to be forcing the speed of light in a vacuum to be constant, in fact measuring out a metre of length involves creating the distance travelled by physical light in 1/c of a second. That is, the light will travel at whatever speed it wants to travel and isn't being forced to conform to a constant speed. But the use of physical light to define the metre does rely on the speed of light in a vacuum being able to fulfil the properties of a standard.

 

Edited Friday at 01:52 AM by KJW

[Markus Hanke]

8 hours ago, Killtech said:

yet I do not entirely follow how you use the term metric and what it means to you two of them.

I use it in the formal sense as defined in differential geometry, ie as a structure that allows you to meaningfully define the inner product of tangent vectors at points on the manifold, which in turn gives a meaningful notion of lengths, angles, areas and volumes.

8 hours ago, Killtech said:

We know that the Christoffel symbols of a Levi-Civita connection can be expressed via the metric

Yes.

8 hours ago, Killtech said:

the metric fully specifies all components of this special connection.

You need to be careful here - the Christoffel symbols and the connection are not the same thing. A connection allows you to relate tangent spaces at different points on the manifold to one another, ie it provides a notion of parallel transport. This is quite independent of any metric, which is to say you can meaningfully have a manifold that is endowed with a connection, but not a metric. The Christoffel symbols then give you the connection coefficients, ie they tell you what effects your connection has in a particular coordinate basis. They do this by describing what happens to basis vectors as you transport them between neighbouring points, which is something you can calculate from the metric and its derivatives. Without a metric you can still do parallel transport, but you can’t tell what happens to lengths and angles when you do it.

Long story short - you can have a connection without a metric.

8 hours ago, Killtech said:

That means that if we have two metrics, each produces a distinct LC-connection

See above. Having a different metric changes the Christoffel symbols (they are not tensors!), but not the connection.

8 hours ago, Killtech said:

As for metric equivalents, i used the term they way i know it from metric spaces

Ok, but in the context of physics (SR/GR) the term “metric” is most often used in the differential geometry sense. Physically speaking, equivalence then means a diffeomorphism, so that both metrics describe the same spacetime and thus physical situation.

8 hours ago, Killtech said:

Lets assume the proposition of a alternative clocks and rules that correct for all gravitation effects. Lets g' be a metric associated with these and written in the same coordinates it is:

But here’s the thing - as explained above, you’re still on the same manifold endowed with the Levi-Civita connection. By changing the metric like this, you’re doing one of two things:

1. You’re describing a different spacetime, ie a different physical situation, since the two metrics aren’t related by any valid diffeomorphism; or

2. You’re describing the same physical situation, but the coordinates you are using no longer have the same physical meaning.

I think what you are trying to do is (2). But the thing is that now measurements on your mathematical manifold (ie in the model) no longer correspond to measurements in the real world, so anything you calculate from this - eg the length of a world line - must first be mapped back into suitable physical coordinates to compare them to real-world measurements. Such a mathematical map may or may not exist, depending on the specifics of the setup. This will also change the form of physical laws, so all the various equations etc will be different for each choice of transformation you make.

In either case, this creates a lot of additional work and confusion, for no discernible benefit.

8 hours ago, Killtech said:

What would an experiment be looking for to disprove the isotropic nature of space?

It would look for differences in the outcomes of experiments if you vary direction of relative motion, as mentioned previously. For example, if a uranium atom decays if you move it in one direction, but doesn’t decay if you move it at a 90° angle to that direction (everything else remains the same), then you have anisotropic space. This has nothing to do with conventions.

[swansont]

10 hours ago, KJW said:

That's actually an interesting point. Although defining c to have a particular numerical value does seem to be forcing the speed of light in a vacuum to be constant, in fact measuring out a metre of length involves creating the distance travelled by physical light in 1/c of a second. That is, the light will travel at whatever speed it wants to travel and isn't being forced to conform to a constant speed. But the use of physical light to define the metre does rely on the speed of light in a vacuum being able to fulfil the properties of a standard.

 

Any realization of the meter would have to account for the non-ideal circumstances of the measurement, such as the index of refraction’s effect on the speed of light. It’s also defined on the geoid and at 0 K, because those are conditions for realizing the second, and you have to make adjustments for not being under those conditions.

[studiot]

The derivation of the Lorenz transformations is interesting and I wonder if you have been reading the older material from Voigt, Larmor, Poincare and Lorenz himself ?

 

Quote

https://physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

 

Derivation of the Lorentz Transformation
Lecture Note for PHYS 171H, 270, 374, 411, 601
by Victor M. Yakovenko
http://physics.umd.edu/~yakovenk/teaching/
Department of Physics, University of Maryland, College Park
In most textbooks, the Lorentz transformation is derived from the two postulates: the
equivalence of all inertial reference frames and the invariance of the speed of light. However,
the most general transformation of space and time coordinates can be derived using only the
equivalence of all inertial reference frames and the symmetries of space and time. The general
transformation depends on one free parameter with the dimensionality of speed, which can
be then identified with the speed of light c. This derivation uses the group property of the
Lorentz transformations, which means that a combination of two Lorentz transformations
also belongs to the class Lorentz transformations.
The derivation can be compactly written in matrix form. However, for those not familiar
with matrix notation, I also write it without matrices.

 

[Killtech]

12 hours ago, Markus Hanke said:

Ok, but in the context of physics (SR/GR) the term “metric” is most often used in the differential geometry sense. Physically speaking, equivalence then means a diffeomorphism, so that both metrics describe the same spacetime and thus physical situation.

Diff geo conforms with the general definitions of metric spaces, it's just the we have better means to represent the metric. But just to be clear, a diffeomorphism (see wiki definition) without any further specifications is just a bijective function which is smooth in both directions. A piece of flat space is diffeomorphic to a piece of a spheric surface - and i want to stress those are in no way unique. These spaces will have different connections though. Do you mean a situation where your diffeomorphism pulls back the connection, i.e. the connection of the target space can be expressed via the connection of the source space?

11 hours ago, Markus Hanke said:

But here’s the thing - as explained above, you’re still on the same manifold endowed with the Levi-Civita connection. By changing the metric like this, you’re doing one of two things:

1. You’re describing a different spacetime, ie a different physical situation, since the two metrics aren’t related by any valid diffeomorphism; or

2. You’re describing the same physical situation, but the coordinates you are using no longer have the same physical meaning.

I think what you are trying to do is (2). But the thing is that now measurements on your mathematical manifold (ie in the model) no longer correspond to measurements in the real world, so anything you calculate from this - eg the length of a world line - must first be mapped back into suitable physical coordinates to compare them to real-world measurements. Such a mathematical map may or may not exist, depending on the specifics of the setup. This will also change the form of physical laws, so all the various equations etc will be different for each choice of transformation you make.

In either case, this creates a lot of additional work and confusion, for no discernible benefit.

Hmm, it's really neither of those two, though it is much closer to 1 except that your interpretation how this maps to reality/experiment might differ from mine. In a way you could call it a different physical situation with different laws of physics which despite this produces equivalent outcomes. We need to work on the interpretation of this.

Lets go back to how we interpret GR and how we compare it with reality/experiments. More specifically, how does the spacetime show itself in reality? If we agree that the metric is a simple tool in our model to describe the spacetime, then there are a few things we can obtain from it that directly map to objects of reality, like the proper time maps to what clocks measure and proper length what rulers show. And there are also angles.

And does this mapping work the other way around as well? i.e. given the tools like clocks, rulers and goniometers (for angles) is that enough to reconstruct the spacetime with its metric? So is this mapping kind of bijective and in some way smooth? ... can we treat this interpretation mapping between model and reality almost like a diffeomorphism itself, one pulling back a metric?

What would happen to this line of thought if we exchanged the tools we use in reality (clocks & rulers) for a different set of tools wich additionally apply a correction for gravity? Let's assume the type of correction is such that these tools do fulfill all conditions to construct a mathematically valid space and metric. More specifically, lets assume they construct exactly the g' metric from my previous example in the Schwarzschild case.

The benefits are hard to see from the tedious construction, but actually there are multiple reasons i am looking into this. But maybe I could throw in an appetizer for a potential silver lining: the quantization of gravity would greatly benefit if there existed a method to equivalently model GR physics in a flat space where gravity arises as a force. But i do not want to discuss such ideas further for as long as we don't have a common understanding.

19 hours ago, KJW said:

That's actually an interesting point. Although defining c to have a particular numerical value does seem to be forcing the speed of light in a vacuum to be constant, in fact measuring out a metre of length involves creating the distance travelled by physical light in 1/c of a second. That is, the light will travel at whatever speed it wants to travel and isn't being forced to conform to a constant speed. But the use of physical light to define the metre does rely on the speed of light in a vacuum being able to fulfil the properties of a standard.

9 hours ago, swansont said:

Any realization of the meter would have to account for the non-ideal circumstances of the measurement, such as the index of refraction’s effect on the speed of light. It’s also defined on the geoid and at 0 K, because those are conditions for realizing the second, and you have to make adjustments for not being under those conditions.

Lets forego that the definitions of one iteration second and meter were made such that to be consistent with previous ones, so that the original definitions weren't chosen with such a purpose in mind.

But lets go to Einstein. He postulated c to be invariant and with it Maxwell. A postulate is just as much a definition, and so it was set as invariant. The definition of the meter was derived in parts on this idea. Lets not focus how this postulate was obtained but rather what the direct consequence of such a definition is.

Lets just consider what would be if light somewhat tried to travel at different speeds (and for some reason no one catches it). even the prior part of the sentence already causes a lot of contractions with the meter. Because we measure distance in units of light travel time in vacuum. So we practically would try to measure the speed of light in vacuum in units of the speed of light, so we measure it moves at 1 c. It is by all means a tautology! It can neither logically deviate nor create any contradictions with reality. its just how we measure stuff.

If light traveled slower in some region, it would cause Maxwell to look differently, but since the lengths that we use become shorter, plugging in the effect restores Maxwell to its well know invariant form. All effect you will see instead is that space will get a dent around that region - which is no contradiction with GR, since even for the worst case of a missing explanation we have dark matter. What this does it that whatever the reality is, the definition of meter and second guarantee that Maxwell in vacuum is invariant. At the same time there is nothing in this statement capable to produce a conflict with reality.

For the meter, this tautological construction with SR is easy to see, but for the second it is far less obvious.

[swansont]

1 hour ago, Killtech said:

Lets forego that the definitions of one iteration second and meter were made such that to be consistent with previous ones, so that the original definitions weren't chosen with such a purpose in mind.

But lets go to Einstein. He postulated c to be invariant and with it Maxwell. A postulate is just as much a definition, and so it was set as invariant. The definition of the meter was derived in parts on this idea. Lets not focus how this postulate was obtained but rather what the direct consequence of such a definition is.

We only had ~50 years of testing the validity of relativity before adopting the definition of the second, and another ~15 before defining the meter in terms of c (which includes the Hafele-Keating and Vessot rocket clock confirmations)

1 hour ago, Killtech said:

Lets just consider what would be if light somewhat tried to travel at different speeds (and for some reason no one catches it). even the prior part of the sentence already causes a lot of contractions with the meter. Because we measure distance in units of light travel time in vacuum. So we practically would try to measure the speed of light in vacuum in units of the speed of light, so we measure it moves at 1 c. It is by all means a tautology! It can neither logically deviate nor create any contradictions with reality. its just how we measure stuff.

If light traveled slower in some region, it would cause Maxwell to look differently, but since the lengths that we use become shorter, plugging in the effect restores Maxwell to its well know invariant form. All effect you will see instead is that space will get a dent around that region - which is no contradiction with GR, since even for the worst case of a missing explanation we have dark matter. What this does it that whatever the reality is, the definition of meter and second guarantee that Maxwell in vacuum is invariant. At the same time there is nothing in this statement capable to produce a conflict with reality.

For the meter, this tautological construction with SR is easy to see, but for the second it is far less obvious.

You’re missing the point about Maxwell’s equation. It’s not a wave equation anymore if c isn’t invariant. It has nothing to do with measuring distances or times. You’re focusing on that but ignoring that it’s not how the experiments are done.

 

[KJW]

25 minutes ago, Killtech said:

What would happen to this line of thought if we exchanged the tools we use in reality (clocks & rulers) for a different set of tools wich additionally apply a correction for gravity? Let's assume the type of correction is such that these tools do fulfill all conditions to construct a mathematically valid space and metric. More specifically, lets assume they construct exactly the g' metric from my previous example in the Schwarzschild case.

I believe that this violates the principle of general relativity. The principle of general relativity requires that clocks and rulers be allowed to behave naturally as clocks and rulers, whereas you are applying corrections to the clocks and rulers based on gravitation in violation of the principle.

That the application of the corrections leads to flat spacetime means that the corrections are destroying information about the spacetime being measured. All the information about the measured spacetime is contained in the applied corrections and not at all in the flat spacetime. So, unless you somehow retain the information contained within the corrections and use that information in the description of the measured spacetime, the flat spacetime will not be a valid description of the measured spacetime.

You appear to be constructing the following:

[math]g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}[/math]

where:

[math]g_{\mu\nu}[/math] is the measured metric tensor field using uncorrected clocks and rulers

[math]\eta_{\mu\nu}[/math] is the flat spacetime metric tensor field

[math]h_{\mu\nu}[/math] is the corrections field

One difficulty of the above worth noting is that [math]\eta_{\mu\nu}[/math] and hence [math]h_{\mu\nu}[/math] can be mathematically chosen independently from [math]g_{\mu\nu}[/math], whereas given [math]g_{\mu\nu}[/math], there would seem to be a natural choice of [math]\eta_{\mu\nu}[/math] and hence [math]h_{\mu\nu}[/math].

In obtaining the curvature tensor fields, you would substitute the above expression for [math]g_{\mu\nu}[/math] into the expression of the curvature tensor fields in terms of [math]g_{\mu\nu}[/math] to obtain the expression of the curvature tensor fields in terms of [math]\eta_{\mu\nu}[/math] and [math]h_{\mu\nu}[/math].

 

[Killtech]

1 hour ago, KJW said:

I believe that this violates the principle of general relativity. The principle of general relativity requires that clocks and rulers be allowed to behave naturally as clocks and rulers, whereas you are applying corrections to the clocks and rulers based on gravitation in violation of the principle.

But what does that even means "behave naturally as clocks"? Naturally i could just interpret the situation another way: Lets say I don't know much about modern physics and naively start my exploration of the world with radio clocks instead. those don't need any corrections and in a sense they do tick at the same rate regardless if they are in deep space or close to a black hole. If i compare their time with that of SI atomic clocks, i notice that those run inconsistently - and i can figure out a physical reason to attribute that effect to: gravity. Furthermore i can make the atomic clocks consistent to my radio clocks, if i were to correct them for that effect.

Now if i have two clock types, ones that give me different times at different regions and another that show consistent results, which clock seems more like the natural choice for time keeping? If i was to run space ports around the universe, which kind of clocks would i announce my flight schedule with? Just to throw that in reminiscence of how connecting towns with the rail and trains required to make clocks at different towns to run consistently, i.e. introducing the railway time.

It really depends on your perspective which clocks you declare to behave naturally. But yes, radio clocks have a significant drawback: they always have a preferred frame and that is that of their signal source. In empty space there is no natural choice of a preferred frame - and as you say eta and h there is some arbitrarily there. I am not sure if the presence of gravity may induce some kind of preference though. If you think of Poincarés corrected LET but now expand it with gravity deducted from GR - which is more or less what this approach leads to - it would require addition of density + current for the aether along with a flow equation. The choice of h would impact its form since it mostly would describe the aether, thus you would look for what makes its time evolution equation most natural.

1 hour ago, KJW said:

gμν=ημν+hμν

wait, we can use latex here!?! how?

Edited Saturday at 01:09 AM by Killtech

[swansont]

26 minutes ago, Killtech said:

But what does that even means "behave naturally as clocks"? Naturally i could just interpret the situation another way: Lets say I don't know much about modern physics and naively start my exploration of the world with radio clocks instead. those don't need any corrections and in a sense they do tick at the same rate regardless if they are in deep space or close to a black hole.

Radio clock? In that you are synchronizing to a remote source? You can't say that they tick at the same rate regardless of where they are, since you aren't relying on that clock, you're relying on the remote one. And you will notice that they get out of synch faster depending on where you are in a gravity well or how fast you move. If they didn't, you wouldn't have to continually reset them

 

26 minutes ago, Killtech said:

If i compare their time with that of SI atomic clocks, i notice that those run inconsistently - and i can figure out a physical reason to attribute that effect to: gravity. Furthermore i can make the atomic clocks consistent to my radio clocks, if i were to correct them for that effect.

Now if i have two clock types, ones that give me different times at different regions and another that show consistent results, which clock seems more like the natural choice for time keeping? If i was to run space ports around the universe, which kind of clocks would i announce my flight schedule with? Just to throw that in reminiscence of how connecting towns with the rail and trains required to make clocks at different towns to run consistently, i.e. introducing the railway time.

It really depends on your perspective which clocks you declare to behave naturally. But yes, radio clocks have a significant drawback: they always have a preferred frame and that is that of their signal source.

Exactly. It's not the local clock that's telling you the time, it's the remote one. Your notion that it doesn't vary with the location is a misrepresentation.

 

[Markus Hanke]

7 hours ago, Killtech said:

These spaces will have different connections though.

No they won’t. Same manifold, same connection, same metric even - just expressed in a different coordinate system.

7 hours ago, Killtech said:

Do you mean a situation where your diffeomorphism pulls back the connection, i.e. the connection of the target space can be expressed via the connection of the source space?

No, I’m not talking about connections. I’m talking about a situation where you express the same physical situation (ie spacetime) in a different coordinate system, so that all laws of physics and all tensorial quantities remain the same - including the metric tensor. This is just the usual diffeomorphism invariance of GR.

7 hours ago, Killtech said:

And does this mapping work the other way around as well? i.e. given the tools like clocks, rulers and goniometers (for angles) is that enough to reconstruct the spacetime with its metric?

Yes, it’s enough.

7 hours ago, Killtech said:

What would happen to this line of thought if we exchanged the tools we use in reality (clocks & rulers) for a different set of tools wich additionally apply a correction for gravity?

You end up with a set of parameters that don’t correspond to what clocks and rulers actually read, so for every quantity you calculate from such a model you need to apply a mapping that takes it back to real-world measurements - and that map is just precisely the inverse of the “correction” you applied in the first place. Like I said, lots of extra work for no discernible benefit.

7 hours ago, Killtech said:

the quantization of gravity would greatly benefit if there existed a method to equivalently model GR physics in a flat space where gravity arises as a force.

Gravity isn’t a force, and can’t be modelled as one - this is precisely the difference between Newtonian gravity and GR. A rank-1 theory such as a vector field model cannot capture all relevant degrees of freedom of gravity. For example, the polarisation modes of gravitational radiation in any force-based model will be inclined by 90°, whereas in reality these modes are at 45°. 

You really do not at least a rank-2 tensor model, such as GR.

7 hours ago, Killtech said:

its just how we measure stuff.

No, it’s a lot more than that. As other posters here have correctly pointed out, for there to be radiation at all, the second derivatives wrt time and space of your “waving quantity” need to be related via a very specific form:

\[\frac{\partial^{2}}{\partial t^{2}} =c^{2}\frac{\partial^{2}}{\partial x^{2}}\]

In the case of light, the relevant quantity is the electromagnetic 4-potential, and the equation thus becomes (in Lorentz gauge)

\[\square A^{\mu}=0\]

If c isn’t a constant, this relationship is violated - there is no electromagnetic radiation in such a universe, at least not of the form we see in the real world. This has nothing to do with measurements or conventions.

 

[Killtech]

7 hours ago, Markus Hanke said:

No they won’t. Same manifold, same connection, same metric even - just expressed in a different coordinate system.

You misread my statement or i do not follow you here. A piece of a sphere is diffeomorphic to a flat space of R^2 (calling the former a sphere and the latter flat does imply that i talk about more then there mere differential manifold). The coordinate maps themselves provide such a diffeomorphism (at least for piece of the sphere there is no problem with map singularities at the poles). So with the regular connection you have on the sphere you get a parallel transport like this: https://en.wikipedia.org/wiki/Parallel_transport#/media/File:Parallel_Transport.svg. On the flat R^2 you will not find a path where such a twisting will occur because it comes with a different connection. But of course you can pullback your connection from the sphere piece via the coordinate map to get a connection on the flat space which will also twists your parallels - and basically reinterpret the flat space as curved.

Think of a balloon painted as a globus. Also paint the coordinate grid on it. Usually a balloon is spherically shaped. But we can cut a piece out of it, stretch it into a flat surface and pin it on the wall. All points of interesting like Greenwich, the Cape Town and Katmandu will remain at their locations as described via coordinates. But let's try a parallel transport between Greenwich, the Cape Town, Katmandu and back to Greenwich - in the original state and after the flattening. Apparently it now uses a different connection. Yet the piece of the balloon is the same smooth manifold regardless how it is shaped - it remains the same set of points and how they are linked with each other.

Diffeomorphism, or more generally homeomorphisms is what we use in diff geo to reshape a geometry without changing its topology. You know, like a coffee mug is homeomorph to a torus (https://en.wikipedia.org/wiki/Homeomorphism#/media/File:Mug_and_Torus_morph.gif). The diffeomorphism will additionally make sure the reshaping does not produce any kinks or folds that would cause issues for differentials. But a diffeomorphism does not preserve the geometry, for that you need stronger things (https://en.wikipedia.org/wiki/Isometry#Manifold).

7 hours ago, Markus Hanke said:

No, I’m not talking about connections. I’m talking about a situation where you express the same physical situation (ie spacetime) in a different coordinate system, so that all laws of physics and all tensorial quantities remain the same - including the metric tensor. This is just the usual diffeomorphism invariance of GR.

In my case i have only one set of coordinates and never change it, because coordinates and equations purely expressed in terms of them are one of the few thing that remain unaffected under a reinterpretation of the geometry.

Now imagine the balloon not to be made out of a simple fabric but its surface being a futuristic display that displays live physics interactions that happen on the globus (think of a live meteorological satellite stream). The physics displayed on that surface will have no idea about what kind of stretching and reshaping we did with it and act entirely independently from that. All the reshaping may however let the same physical interactions playing out on the surface appear to to subject to different laws of physics with things moving faster in the stretched regions and slowed in the contracted ones. Also our coordinates as originally painted as a grid onto the surface do not notice any kind of reshaping we do - they only look different for someone outside their world looking at the manifold.

7 hours ago, Markus Hanke said:

A rank-1 theory such as a vector field model cannot capture all relevant degrees of freedom of gravity. For example, the polarisation modes of gravitational radiation in any force-based model will be inclined by 90°, whereas in reality these modes are at 45°. 

You really do not at least a rank-2 tensor model, such as GR.

Yes, this you are absolutely right. When we stretch the spacetime geometry onto the flat, gravity must at least become similar to electromatic field which is a 2nd degree tensor. So unlike Newtons gravity there must be at least something like a gravity analoge of the magnetic field when represented in a flat spacetime. Probably even more then that.

 

7 hours ago, Markus Hanke said:

You end up with a set of parameters that don’t correspond to what clocks and rulers actually read, so for every quantity you calculate from such a model you need to apply a mapping that takes it back to real-world measurements - and that map is just precisely the inverse of the “correction” you applied in the first place. Like I said, lots of extra work for no discernible benefit.

We have exchanged the clocks and rules and use those for our real-world measurements instead. Think as if you were forbidden from even using SI conforming clocks and rules and are forced to rely solely on the new ones for all experiments - i want to you understand what this situation mathematically means (not practically). In that case i don't have to do any additional mappings: because as you said, the correction we did coupled with the mapping back (which is the inverse) combined together yields f ° f^-1 = id. No additional work needed if i simply forget the old word entirely, both in terms of the the model and measurements in the real world.

Edited Saturday at 12:43 PM by Killtech

[Markus Hanke]

19 hours ago, Killtech said:

Yet the piece of the balloon is the same smooth manifold regardless how it is shaped - it remains the same set of points and how they are linked with each other.

Not it isn’t the same set of points, because in order to flatten it, you had to cut a piece out of the surface (as you correctly stated yourself), so you have lost information in the process.

19 hours ago, Killtech said:

But a diffeomorphism does not preserve the geometry

This depends on what exactly you mean by “geometry”. In the context of GR, this means the various tensor fields that describe the distribution of energy-momentum and the associated effects this has on the world lines of test particles. And these very much are invariant under diffeomorphisms, which is to say you can label the same physical events in different ways.

19 hours ago, Killtech said:

because coordinates and equations purely expressed in terms of them are one of the few thing that remain unaffected under a reinterpretation of the geometry.

If you change neither the physical meaning of coordinates nor the form of equations expressed with them, then you can’t have a different geometry, unless you describe a different physical situation. You can’t have it both ways.

19 hours ago, Killtech said:

When we stretch the spacetime geometry onto the flat

If that is what you want to do, you should take a look a teleparallel gravity. This model has no curvature (ie all geodesics remain parallel even globally), and all the information about gravity is contained in the form of torsion along world lines. Einstein himself investigated this in some detail.

19 hours ago, Killtech said:

No additional work needed if i simply forget the old word entirely

But then this model is entirely useless, because you cannot compare anything calculated from it against quantities physically measured with real-world instruments. You might as well be talking about invisible pink unicorns, for all the use it has in modelling the real world.

Physics makes models that describe aspects of the physical world around us - as such we must be able to extract predictions from those models and compare them to real-world measurements. If you ask us to just forget about the real world, then I’m sorry to say we’re not interested, because it’s of no use to us when solving practical issues like eg calculating the orbit of a satellite around a gravitating body, and knowing how a clock on that satellite relates to a clock on the surface. In standard GR, this comes right out of the model, because what clocks physically read is always identical to the geometric (mathematical) length of the world line it traces out in spacetime, so the problem is rather straightforward, if not always simple. In what you propose that is not so, and you’re asking us to just ignore this…?

Im sorry, but I’m still completely failing to see the actual point in all of this.

[Killtech]

 

4 hours ago, Markus Hanke said:

If you ask us to just forget about the real world, then I’m sorry to say we’re not interested

Arg, no, that is not what i was saying at all. You misunderstand me.

I am interested in exploring the definition of SI second and meter and how they affect measurements made by experiments, the laws of physics those experiments perceive when using those standards and finally how that shapes the entire physical model that we use. Also Einstein uses in his gedankenexperiments a lot of clocks, yet this object is never actually defined or meaningfully explored. Implicitly all his considerations assume there is only one concept of time, which all real world clocks have to measure.

But there is more then one concept of time and of course we can build other real world devices that measure them (though maybe with less precision). But if we are not willing to even consider hypothetically what were to happen if we used those instead for all our current real world measurements, we become blind to how much the SI system and its standards for measurement determine. We won't be able to see that the invariance and shape of Maxwell as seen by experiments and theory is a logical consequence of SI standards and postulates underlying it. And because they determine both measurement and model in the same way, there is no possibility to create a violation. They cannot be proven wrong, as they are a convention. And we get stuck in using only a singular interpretation of spacetime which just isn't practical for a lot of problems, mainly in theoretical physics. Issues like failure to establish quantum gravity or the huge amounts of dark matter we need to postulate to make GR conform with observations may be related to this. Not sure if the latter two instances you do consider practical problems though.

Let me be clear, if we just discuss calculating satellite orbits around earth, all the considerations discussed here are entirely useless. Only when we look at problems of galaxy rotation anomalies, or the issues of describing the quantum interactions in a black hole, things start to look differently. It is not like we can put a Caesium clock inside a black hole to get 'real world measurements' from it (that's like talking about invisible pink unicorns). Generally I am not convicted the SI standards are well definierend for that scenario. Maybe you can only describe the situation in there by extrapolating a definition of a hypothetical clock device that corrects for all gravity effects.

4 hours ago, Markus Hanke said:

This depends on what exactly you mean by “geometry”. In the context of GR, this means the various tensor fields that describe the distribution of energy-momentum and the associated effects this has on the world lines of test particles. And these very much are invariant under diffeomorphisms, which is to say you can label the same physical events in different ways.

Ah, this is crucial information and now i unterstand your prior statements. But your are using the diffeomorphism very differently then i do. Now a diffeomorphism can pullback (https://en.wikipedia.org/wiki/Pullback_(differential_geometry)) all objects defined on manifold, be it points, vector fields, metrics and even a connection. If you do that, the target manifold becomes really the original one as the diffeomorphism is used almost like a map itself, so this is almost a mere change of coordinates.

But if we have two metrics, this doesn't help us at all, because such a diffeomorphism phi, the pullback metric g(phi) is not g' and same holds for the connection. Therefore in my case it makes no sense to apply the diffeomorphisms to any features linked to the geometry. We can only apply it to the points of the manifold but not even to the vector fields. And this is why:

Imagine you have clocks that correct SI atomic clocks for some effects and you use those in your measurements. One way to measure a force, it to let it affect test bodies and measure their acceleration, which is by its units is space / time^2 and therefore have a direct dependence of the spacetime geometry. Any corrections made to the time measurement standard will therefore change the outcome of the measurement, as the acceleration measured using the corrected clocks is different and hence the force itself. In SI we can see the dependency of all quantities on the spacetime geometry right from its units. So no, a change of metric does not keep even vector fields invariant as due to the change of perspective those are related yet actually different objects.

So we need to look a bit closer on the toolset of diff geo what it provides to handle this situation. So first let's have a look at the definition of a Riemann manifold (M,g):  https://en.wikipedia.org/wiki/Riemannian_manifold#Definition  - it is always a pair of two things and one is the defining metric. So if we have one smooth manifold M but two metrics on it, we have formally two Riemann manifolds (M,g) and (M,g'). When it comes to the connection, specifically Levi-Civita, we know that it is only defined on Riemann manifolds, as by definition it is explicitly defined via the metric: https://en.wikipedia.org/wiki/Levi-Civita_connection#Formal_definition. So two metrics, two Riemann manifolds, two LC-connections, only one smooth manifold M. When we use diffeomorphisms to transition between Riemann manifolds, we are no longer allowed pullback any objects that depend explicitly on the Riemann geometry. This situation is a part of diff geo that i haven't seen being used in physics, hence it won't be familiar for physicists to think in this way. 

4 hours ago, Markus Hanke said:

If that is what you want to do, you should take a look a teleparallel gravity. This model has no curvature (ie all geodesics remain parallel even globally), and all the information about gravity is contained in the form of torsion along world lines. Einstein himself investigated this in some detail.

Only distantly related, as it discusses using an alternative geometry and maybe the flatness, but everything else is based on a different concepts, hence resulting in quite a different setup.

Edited Sunday at 01:02 PM by Killtech

[Markus Hanke]

16 minutes ago, Killtech said:

Arg, no, that is not what i was saying at all. You misunderstand me.

But you were explicitly saying that, I quote, we should “forget the old world entirely”, with old world referring to real-world measurements with clocks and rulers.

18 minutes ago, Killtech said:

I am interested in exploring the definition of SI second and meter and how they affect measurements made by experiments, the laws of physics those experiments perceive when using those standards and finally how that shapes the entire physical model that we use.

The choice of units has no impact at all on the physical outcome of experiments, or on the form of physical laws that are written in covariant notation. You can measure lengths and angles in meters and radians, or you can use fingers and degrees, but the apple will always fall when it is ripe, and it will do so radially downwards. No redefinition you do changes this physical process. So you might as well work with the simplest mathematical description of it, and safe yourself unnecessary complications.

31 minutes ago, Killtech said:

But there is more then one concept of time and of course we can build other real world devices that measure them

Of course you can - like a grandfather clock for example, which is influenced by local gravity conditions. Nothing wrong with that, but the question is how useful that is.

34 minutes ago, Killtech said:

We won't be able to see that the invariance and shape of Maxwell as seen by experiments and theory is a logical consequence of SI standards

No it isn’t. It’s a reflection of the fact that relative uniform motion has no influence on how electromagnetism works. Your laptop works in your living room just the same as it would in a rocket at 99% light speed wrt Earth. Of course you can always write down a model where this is not so, I just fail to see the point why you would do that, as you then aren’t describing what actually happens in the real world.

41 minutes ago, Killtech said:

One way to measure a force, it to let it affect test bodies and measure their acceleration, which is by its units is space / time^2 and therefore have a direct dependence of the spacetime geometry.

Are you talking about free fall here? There is no proper acceleration in the rest frame of a free falling particle.

44 minutes ago, Killtech said:

Any corrections made to the time measurement standard will therefore change the outcome of the measurement, as the acceleration measured using the corrected clocks is different and hence the force itself.

If you change the standard definition of time in a manner that makes it explicitly dependent on location, then \(a\left( t \right) \neq \ddot{s} \left( t \right)\), and thus Newton’s laws are no longer valid in their usual form F=ma. All laws of physics will take on a different form in this case. As stated previously, you can of course do this if you really want, I’m questioning only the point and usefulness in that.

54 minutes ago, Killtech said:

as by definition it is explicitly defined via the metric

For the connection to be of type Levi-Civita, it needs to preserve the metric and be torsion free, otherwise it can’t be said to be Levi-Civita (there are infinitely many possible types). But take careful note that you can have a connection (of any type) on a manifold without there being any metric, it’s just that you then can’t formally say that that connection is of type Levi-Civita. So it doesn’t make much sense to say the connection is explicitly defined via the metric, only that the connection is of that specific type in the presence of a metric.

1 hour ago, Killtech said:

So two metrics, two Riemann manifolds, two LC-connections, only one smooth manifold M.

If there’s only one smooth manifold, then there’s only one LC connection, though you can have as many metrics as you want. But you can’t have two “different” connections that are both Levi-Civita, that makes no sense. If your formal manifolds are both Riemann, an LC connection must preserve any metric on either one of these manifolds. If it doesn’t do that, then it’s either not an LC connection, or one of the manifolds is not Riemann, or it’s not a valid metric in the first place (not every notion of inner product is automatically a valid metric, there are conditions here too).

[swansont]

2 hours ago, Killtech said:

I am interested in exploring the definition of SI second and meter and how they affect measurements made by experiments, the laws of physics those experiments perceive when using those standards and finally how that shapes the entire physical model that we use.

And yet you bring up situations where these don’t matter, because the experiments don’t rely on them.

2 hours ago, Killtech said:

Also Einstein uses in his gedankenexperiments a lot of clocks, yet this object is never actually defined or meaningfully explored. Implicitly all his considerations assume there is only one concept of time, which all real world clocks have to measure.

He uses ideal, perfect clocks, which don’t rely on the definition of any particular unit system, rather than real world clocks. And his concept of time is that of physics - it’s what’s measured by a clock.

2 hours ago, Killtech said:

Imagine you have clocks that correct SI atomic clocks for some effects and you use those in your measurements

I don’t know what this is supposed to mean. Can you cite an actual experiment, rather than a contrived one? What’s an “SI atomic clock”?

[Killtech]

2 hours ago, Markus Hanke said:

If you change the standard definition of time in a manner that makes it explicitly dependent on location, then a(t)≠s¨(t) , and thus Newton’s laws are no longer valid in their usual form F=ma. All laws of physics will take on a different form in this case. As stated previously, you can of course do this if you really want, I’m questioning only the point and usefulness in that.

Yes, this is exactly what i am doing. So far we agree what it means for the model. Now to the experiment:

2 hours ago, Markus Hanke said:

The choice of units has no impact at all on the physical outcome of experiments, or on the form of physical laws that are written in covariant notation. You can measure lengths and angles in meters and radians, or you can use fingers and degrees, but the apple will always fall when it is ripe, and it will do so radially downwards. No redefinition you do changes this physical process. So you might as well work with the simplest mathematical description of it, and safe yourself unnecessary complications.

1 hour ago, swansont said:

I don’t know what this is supposed to mean. Can you cite an actual experiment, rather than a contrived one? What’s an “SI atomic clock”?

As @Markus Hanke stated the change of standard definition of time makes the laws of physics take a different form, so it is more then a mere change of units. On the other hand it does not change the outcome of experiments directly. In that situation the original experiment cannot be directly compared with the model - at least not without expensive transformations (like you stated correctly before). So we look at how we could adjust our experimental setup to make its measurements conform with the new standard definition of time to make it comparable again. In practice this means exchanging all measuring devices to new ones that adopt the new standard, including mainly clocks.

Let's go through this process slowly, because it is fairly important to become aware of what it does. In an laboratory we can measure velocities by taking the time a test body needed from one point to another. A ruler gives us the distance between the points, a clock measures the time and the velocity is the quotient of both measurements. Now if change to the new standard where the clock in the laboratory have correction factor f(x) (which we assume is nearly constant in the area of the laboratory for this example), then this will result in obtaining a different numerical value for the velocity then in the original experiment, i.e. by a factor of 1/f(x). Similarly we can measure the acceleration as a change of velocity over time and with adopting the new standard we obtain a result differing by a factor of 1/f(x)^2. So if we were to measure the Coulomb force a charged plate capacitor applies on charged test bodies, we observe a slightly modified law Coulomb law, one that has an additional factor 1/f(x)^2 in it. Performing the same experiment in a different laboratory where the correction factor has another value f(y), we get observe the Coulomb force is affected by the physical field which we correct via f(x).

However, as you stated yourself due to the change of the standard definition of time, the laws of physics take on a different form, and so this experimental outcome will be consistent with that. If our experiments uses the very same standards as our model assumes, we will not be able to create any testable violation between reality and theory. This is what is important for me to highlight.

2 hours ago, Markus Hanke said:

If there’s only one smooth manifold, then there’s only one LC connection, though you can have as many metrics as you want.

I am sorry to correct you here but that statement is false. On a smooth manifold, you can have a connection but without the metric you cannot classify it as LC, only as torsion free. There are many torsion free connections on a smooth manifold and they differ by what they do to different metrics. If you whish to contest me on that, then please provide some literature to back up your statement.

I think you a mistake smooth manifold for a Riemann manifold, because then your statement would align with the fundamental theorem of Riemann geometry https://en.wikipedia.org/wiki/Levi-Civita_connection#Fundamental_theorem_of_(pseudo-)Riemannian_geometry. Two metrics which are not isometric towards each other produce different LC-connections.

Edited Sunday at 04:50 PM by Killtech

[Killtech]

3 hours ago, Markus Hanke said:

If you change the standard definition of time in a manner that makes it explicitly dependent on location, then a(t)≠s¨(t)

One addendum though: in your example a(t) is the acceleration as defined in the previous standard.

But if we think of proper acceleration and proper velocity, then those concepts are defined in terms of the proper time. Now as the standard definition of time changes, it reflects directly on proper time tau which is exchanged for tau' in the new standard. This in turn defines a new proper velocity in the new standard. same goes for the proper acceleration.

So if the new standards dictates a change from tau to tau', it also requires switching from w to w' for the proper velocity and a to a' for the proper acceleration. And indeed a(t) != a'(t). Only s(t) = s'(t) is unaffected. Note that due to the change in geometry we attribute the same physical information differently between physical quantities and the geometry.

So we have a'(tau') = d^2/d tau'^2 s'(tau') != a(tau')

Now, where can i find how to use latext here?!?

Edited Sunday at 06:17 PM by Killtech

[swansont]

1 hour ago, Killtech said:

As @Markus Hanke stated the change of standard definition of time makes the laws of physics take a different form, so it is more then a mere change of units.

That’s much more than you were claiming, though. A change in the definition of time is not just changing the definition of the unit. You were arguing the opposite - that the unit definition drives the laws. “the laws of physics those experiments perceive when using those standards” Our unit definitions are based on our known laws of physics, which are interdependent. 

If you redefine what acceleration means, a whole bunch of stuff could chamge. That a whole other argument.

1 hour ago, Killtech said:

However, as you stated yourself due to the change of the standard definition of time, the laws of physics take on a different form, and so this experimental outcome will be consistent with that. If our experiments uses the very same standards as our model assumes, we will not be able to create any testable violation between reality and theory. This is what is important for me to highlight.

No, as Markus stated, in a completely different argument about laws, rather than unit definitions.

[Killtech]

4 minutes ago, swansont said:

Our unit definitions are based on our known laws of physics, which are interdependent. 

If you redefine what acceleration means, a whole bunch of stuff could chamge. That a whole other argument.

Sorry for the miscommunication and the wrong wording. Didn't know how else to describe it. 

Highlighting what this interdependence means is my main focus here and what it may lead to. The logical consequences of interdependencies are rarely trivial. Specifically what it means for the isotropy of light and the constancy of c.

[studiot]

Well I just don't get this vendetta against the SI system.

Of the seven base units only one is a pure number that is completely unambiguous.

The rest require a real physical standard to measure against, somewhere along the line.

Of course alternative sets of base units and quantities can be used, for example in fluid and Continuum Mechanics more generally Force is sometimes used instead of Mass.

 

Just now, Killtech said:

Specifically what it means for the isotropy of light

Isotropy of light ?

 

Just now, Killtech said:

and the constancy of c.

Do you fully appreciate the difference between invariance and constancy ?

A quantity may be one or both or neither.

 

[swansont]

1 hour ago, Killtech said:

Sorry for the miscommunication and the wrong wording. Didn't know how else to describe it. 

Highlighting what this interdependence means is my main focus here and what it may lead to. The logical consequences of interdependencies are rarely trivial. Specifically what it means for the isotropy of light and the constancy of c.

If you change what the laws of physics are, you throw GR and electrodynamics out the window. So there is no guarantee of an invariant c. Come up with laws that hold in an accelerating frame, for instance.

[Markus Hanke]

22 hours ago, Killtech said:

On a smooth manifold, you can have a connection but without the metric you cannot classify it as LC, only as torsion free.

Yes, this is what I said in my post (I’m quoting my self):

On 3/30/2025 at 4:18 PM, Markus Hanke said:

But take careful note that you can have a connection (of any type) on a manifold without there being any metric, it’s just that you then can’t formally say that that connection is of type Levi-Civita.

But my main point was rather that once a metric is established to be of type Levi-Civita, then all its characteristics are uniquely determined, so you can’t have two “different” connections that are both LC. What changes according to the metric are only the connection coefficients, not the connection itself. That’s an important difference.

22 hours ago, Killtech said:

Yes, this is exactly what i am doing. So far we agree what it means for the model.

Yes, I get what you are trying to do. Unfortunately I don’t think you have grasped the concerns I have tried to level at this idea, perhaps because they got buried in technical arguments. So let me try a more practical approach.

In the first instance, consider this simple scenario - let’s say you have a box that contains a quantity of muons (a bit contrived, I know, but bear with me). The box is locally in an inertial frame, and otherwise isolated from any external influences. There’s no spatial motion in the frame of the observer, the box just sits there and ages in time.

I’d like to use your own earlier example of a metric here, where the 00-component is unity, and the notion of time is your own adapted “new time”, not SI seconds; I will be using the letter u for this, to distinguish it from ordinary time t. In this spacetime, the geometric length of the muons’ world lines between two events A and B then is

\[s\prime =\int\limits_{B}^{A} ds=\int_{B}^{A} \sqrt{g_{\mu \nu}dx^{\mu}dx^{\nu}}=\int\limits_{B}^{A} \sqrt{g_{00}} du=\int\limits_{B}^{A} du=\bigtriangleup u+C\]

so it is just simply the difference in u’s (we can choose C=0 for simplicity). Let’s say the two events are 1 second u-time (not t-time!) apart, and at u=A the box contains X muons. My question is: how many muons are left at u=B, ie after 1 second u-time? All I’m after is a percentage of the original number of particles X, so nothing to do with any units.

I’m interested to see how you go about solving this - which, in ordinary physics, would be an almost trivially simple problem.

21 hours ago, Killtech said:

Now, where can i find how to use latext here?!?

Like so:

\ [ Latex code \ ]

just without the space between backslash and angle bracket.

[Killtech]

1 hour ago, Markus Hanke said:

I’m interested to see how you go about solving this - which, in ordinary physics, would be an almost trivially simple problem.

See, this is unfair. I demand access to write latex formulas... no seriously, how do you do that around here?

But i get what you are highlighting. in u-standards muon decay has a dependence on new physical laws and fields that appear in the new standard, hence the naive approach (utilizing classic/relativistic physics only) would indeed require an integration of the now location dependent decay rate along the worldline to get an answer.

On the other hand, if u is not some arbitrary time standard but indeed obtained from pure gravity correction, then this question becomes quite a bit more interesting. As far as i am aware we have not much experimental data on if or how particle decay channels may change in strong gravitational environments - nor do we have a theory to make such predictions. But we do know particle decays change in say strong electric fields (think photon e+e- pair production). there is no obvious reason (for me) to exclude such a possibility for gravity.

So, in u time standard, the corrections from gravity fields should result in producing some kind of physical 2-tensor field (at least) representing gravity and it should have interactions with both the EM-field and leptonic fields, hence after quantization (which should be straight forward in the flat spacetime u produces), we will get a very different Lagrangian. Around weak gravity fields this new u-standard-G+QED will then have to explain why (u-)gravitons slow down muonic decays. but around strong gravitational fields, i won't dare make any predictions.

Maybe this makes my approach appear a little bit more reasonable?

1 hour ago, Markus Hanke said:

But my main point was rather that once a metric is established to be of type Levi-Civita, then all its characteristics are uniquely determined, so you can’t have two “different” connections that are both LC. What changes according to the metric are only the connection coefficients, not the connection itself. That’s an important difference

Well, in one sense there can be only one LC-connection at a time, as it is unique. But as it is bound to the metric, and we have two distinct ones, the meaning of LC becomes ambiguous. So in oder to keep it clear, we need to say either a g-LC or g'-LC connection. Note that for g-LC connection \( \nabla \) we have \( \nabla g = 0 \neq \nabla g'\). Yet \( \nabla' g' = 0\) using the g'-LC connection \( \nabla' \). 

Edited 8 hours ago by Killtech