On the Definition of an Inertial Frame of Reference

A transformation of spatial coordinates is a transformation of space-time coordinates, just a special case. A rotation is a special kind of Lorentz transform, even though it doesn't act on time coordinates.

It depends what you mean by 'the wave equation'. The wave equation typically used by anyone doing basic mechanics is the flat space form, ie \(\partial^{2}_{t}\phi = \Delta \phi\). Would someone doing GR call this 'the wave equation'? No, they'd use the general form which Guest has given, the one where partials become covariant, ie \(\nabla_{a}\nabla^{a}\phi\) where a goes from 0 to N, where N is the number of spatial directions. That is covariant and it reduces to the standard one when you have flat Cartesian systems.

I can see why you're saying what you're saying but I seriously think you're not understanding the points Guest is trying to make. If someone knows the standard special relativity formulation of something then to upgrade to full GR typically you change all partial derivatives to covariant ones, \(\partial \to \nabla\). This means that if you view the special relativity formulation as the general relativity one for a specific coordinate system in a specific space-time then it is covariant but only because you know you've got connection terms which just happen to be zero. If you don't know you should include connection terms then the special relativity formulation isn't covariant. It's a subtle different but an important one and I think you're labouring under the misconception that Guest doesn't know the difference.

Your comments about him and Eugene being made for one another (or whatever it was you specifically said) are just ad homs since at no point has Guest been saying Eugene's paranoid nonsense about physicist is right but instead he's been making a point about the general concept of coordinate transformations and the mathematical formalism behind them. Implicit definition of coordinate transformations doesn't make them invalid, as the consistency of a (supposed) coordinate transformation is entirely independent of how they are defined, it rests in the interdependency between old and new parameters throughout the manifold.

I'm sure Guest would have no problem rolling his sleeves up and going to town on the subject of tangent bundles, gauge connections, coordinate atlases etc but I wonder whether you'd be up to it. Every time Guest or I have pushed any kind of detail you've not replied in kind, you just reply with "You don't know what you're on about". Since clearly there's wires being crossed surely it would be to everyones benefit (ie those of us who claim to understand the details) if the discussion became more formal and we were all very precise with our definitions and notion and provided algebra where needed? Even Eugene, for all his delusions, at least realises attempts at mathematical formalism are needed to have any hope of getting his claims listened to. Out of the people in this thread who've pitched in I've seen the least mathematics from you[. Hopefully its just because I'm not entirely paying attention and you've not shown your best side so if we all kick up the details and formalism a few gears maybe that'd go a long way to sorting out whatever the problems seem to be?

 

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No, it specifically identifies certain coordinate systems. That's the point of it. Inertial coordinate systems are those in which \(g_{\mu\nu} \,=\, \eta_{\mu\nu}\). In most coordinate systems, \(g_{\mu\nu} \,\neq\, \eta_{\mu\nu}\). Hence, most coordinate systems are non-inertial. By definition. Just like most colours aren't blue.

If the definition of "inertial coordinate system" was generally covariant, every coordinate system would be "inertial". In that case, the qualifier "inertial" would become redundant, and we may as well drop it and just call them "coordinate systems".

 

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So you're saying that there are laws of physics that are not generally covariant? Do mathematicians agree that there is no way to generalize Newton's first law of motion to make that law generally covariant?

 

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No, I never said that.

 

In Newtonian physics the 1st law says that an object remains at rest or at a constant velocity if no external force acts on it, ie it moves along straight lines in the Newtonian space-time \(\mathbb{R}^{3} \times \mathbb{R}\). Straight lines in that space (by which I mean the manifold equipped with the relevant metric, not specifically the spatial parts) are geodesics and so the logical choice for an upgrade to some kind of covariant formulation would be to consider geodesics in whatever space you're interested in and geodesics can be worked out without too much bother in plenty of GR systems.

This is confirmed when you consider Newton's 2nd law, from which the 1st law can be seen to be an implementation of, upgraded to a relativistic formulation. Its (literally) a homework exercise to derive the geodesic equation from standard GR considerations and it take the form

\(\ddot{x}^{c} + \Gamma_{ab}^{c}\dot{x}^{a}\dot{x}^{b} = 0\).

This is what you get when you do the Euler-Lagrange equations for a test particle in a space-time with the metric which defines \(\Gamma\), just as you get Newton's 2nd law by considering the Lagrangian \(\frac{1}{2}m\dot{x}\cdot \dot{x}\) in a system with constant potential (as forces are obtained by considering the derivatives of potentials). This is how you can work out the GR prediction for the orbital precession of Mercury, which is something GR does better than Newton's work.

Haven't you ever seen the geodesic equation before?

 

So, you can't do it. As expected.

 

GR is not the subject of the debate. Covariant derivatives don't even enter the statement of the problem. Guest simply threw up a lot of red herrings that show that he didn't even understand the issue. The issue is very simple : Shubert's crackpot transforms do not:

-allow for the derivation of simple things like speed composition, Doppler effect

-allow for the covariant transformation of Maxwell's equations

There are many other defects but this is basically what the discussion is all about.

 

Your logical choice is incomplete. What other equations are you going to add to the geodesic equation?

Yes. It is a standard exercise. I had to do it in a graduate course in general relativity at UCSD. I also derived it using the Euler-Lagrange equations. The derivation is found in many books that I had already seen. What does your song and dance have to do with you not being able to answer my question?

In other words, Guest254 must enjoy banging his head on objects of near infinite density and you don't recognize elementary algebra.

 

So do you believe that there are laws of physics that are not generally covariant?

 

Depends what the Lagrangian for the system I'm interested in is. As I explained, that's the expression for Newton's 2nd law in the absence of an external force on the object. Adding in forces or other particles or fields is straightforward, its the basic method for pretty much all of relativity and quantum field theory. My point was that the formalism to extend basic mechanics into systems which are non-Euclidean or which make use of non-Cartesian coordinates is well known, even to undergraduates.

If you're familiar with it why did you ask your question? Do you not see how it applies to your question?

Where did Guest say that? His point seemed to be that assuming the transformation Eugene said meets the criteria associated with valid coordinate changes (explicit definitions not being a criteria, contrary to things you've said) there's nothing in principle wrong with what Eugene is trying to do. Whether or not those specific definitions Eugene has given actually meet the relevant criteria seems completely beside his point. You seem to think Guest is defending Eugene's work specifically, which doesn't seem to be the case.

Why are you struggling to get this? I imagine Guest would have said those things in any thread where you'd said as you have about the notions of covariance and ways to define transformations, Eugene's equations are irrelevant.

Do you have trouble understanding what he said or did you ask the same question twice accidentally? Why should Nature care what method of parametrisation someone uses to describe a system? Does Nature care we're talking about physics in English and not French?

 

How long will you go on beating your chest and making LOUD NOISES until you realise people can see straight through you? Even after several explanations (good ones at that, if I do say so myself) you're still utterly confused when it comes to the notion of general covariance. You've been corrected on things as basic as Christoffel symbols and basic calculus of curvilinear coordinates, and yet you continue to beat your chest and name-call. Hell, you seemed to be confused in how to apply the chain rule (standard high school mathematics)!

GR has nothing to do with any of the discussions here. This is simple tensor calculus and curvilinear coordinates. These are things taught to 1st and 2nd year undergraduates.

What you've done is to assume you know better than others on here. You've then attempted to enforce this position with an aggressive posting style. Inadvertently, you've made an utter fool of yourself in demonstrating you're unfamiliar with basic tensor calculus and more importantly, Einstein's notion of general covariance. In an attempt to uphold your position of authority, you've continued with the aggressive posting style and instead of admitting fault (or complete ignorance), you've called people names or USED CAPITAL LETTERS. You're now sitting in a big hole, and those who know what they're talking about are looking down on you.

How does it feel down there?

 

The question is if there is a straight answer for the simplest system imaginable. The Riemann curvature tensor is zero and particles don't interact.

 

As long as you keep showing that you are an idiot and a pretender.Are you Shubert's sockpuppet? You two are like identical twins.

 

Nah, just pointing the holes in your crackpot theory.

 

You say this, and yet everyone has to keep on correcting you! Come on, we're all friends here - you're feeling a little sheepish, aren't you?

 

Listen,

What you have been asked to do (and evaded repeatedly) is to show whether the equation:

\(\Bigg(\nabla^2 - { \mu\epsilon } {\partial^2 \over \partial t^2} \bigg) \mathbf{E}\ \ = \ \ 0\)

is invariant wrt the Shubert crackpot transform. Stop the prancing, roll up your sleeves and start calculating. This should call your bluff(s).

You already stumbled at the baby step of calculating \({\partial \over \partial t} \). I need to break down the problem in small steps for you such that you don't weasel out like your alter ego, Shubert. If you can't do it you might ask him to help you out.

 

That's not obvious to me. Do you have a mathematical proof?

 

A course in basic mechanics wasn't recommended for my math degree. I never studied mechanics seriously. Please post a link to a generally covariant equation or set of such equations that generalizes inertial frames of reference and/or Newton's first law of motion to the least extent possible. Please assume that the Riemann curvature tensor is zero and particles don't interact. As I said before, I'm looking for a straight answer to the simplest system imaginable.

 

\(\frac{dp}{d \tau}=F_M\)

where \(p=\frac{m_0v}{\sqrt{1-(v/c)^2}}\),\(F_M=\frac{F}{\sqrt{1-(v/c)^2}}\)

Now, how about you solved the simple kinematics problem you've been challenged? No more weaseling. Shubert.

 

So you are admitting that the orthodox understanding of an "inertial coordinate system" defined by the equation \(g_{\mu \nu} \,=\, \eta_{\mu\nu}\) is not generally covariant. But I assert that Newton's first law of motion is a law of physics and that my definition of an inertial frame of reference is ultimately generally covariant. So the sacred oracles of the ancient opinions may be rejected and I have the moral right to exalt the pure elegance in The Quintessence of Axiomatized Special Relativity Theory and to campaign against definitions that are not generally covariant. Clearly, mathematicians have the right to modify poor definitions. Beauty and consistency trump tradition.