In that setup, distance is measured with a bunch of identical standard rulers and time with many identical standard clocks. This appeared to be your requirement for deciding what coordinate systems are eligible to be called a "reference frame".
In order to measure a velocity, one needs to be...
Wow, and yet a different distinction someone is making between a coordinate system and a frame of reference. It seems the answers are almost as numerous as the people. Fascinating.
Thank you for sharing your take on it.
If I have a 'grid' of rulers and clocks at the intersections to mark off...
I feel WannabeNewton explained his stance well.
D.H., you also are distinguishing reference frame and coordinate system, but I'm still having trouble understanding your stance. Maybe we can cut this short -- Are you trying to say the same thing as WannabeNewton?
...
The following only...
Let me try to summarize to see if I understand your use of the terms.
To you a frame is nothing more than the basis vectors for the tangent space at some point. Where-as coordinates are just the labels used for the points. It is possible, but not necessary, to construct a coordinate system...
I am interested to hear what distinction you feel there is.
From wikipedia, on "frame of reference":
"In physics, a frame of reference (or reference frame) may refer to a coordinate system used to represent and measure properties of objects, such as their position and orientation, at...
I apologize if I am being dense here, but I honestly can't tell if you missed my point.
The point was, Newton's first law does not define an inertial frame.
It is a necessary property, but not a sufficient property. I brought up that any linear transformations preserves Newton's first...
"mass" can mean a lot of things. The equation
E=mc^2
only holds for a system at rest, otherwise we need to include the momentum terms.
E^2 -(pc)^2 = (mc^2)^2
That "m" there is indeed a Lorentz scalar as you mentioned. It is also called the rest mass.
So for an atom, the rest mass _does_...
No. That is a necessary, but not sufficient condition. Otherwise any linear transformation from an inertial coordinate system would give another inertial coordinate system.
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It is incredibly hard to define an inertial frame precisely. Everyone 'knows' what we mean, so most books just...
So in what sense does that vanishing have anything to do with the Bianchi identity?
Maybe it is a bit of a misnomer that those parts of Maxwell's equations come from the Bianchi identity, because now it feels like it is independent of the Bianchi identity.
Wait, why is that first term equal to...
Thanks for the info on invariants. This is quite interesting.
I'm really curious about this now. Is this discussed in Jackson? Maybe its time I got a copy of Landau-Lifshitz.
The leap from quadratic invariants to all invariants seems almost mystical.
Is this true for any antisymmetric...
I don't see how the Kulkarni–Nomizu product will help us here. Maybe I'm missing something.
But your notes on the Bianchi identity gives me some ideas.
##F_{ab} = \nabla_a A_b - \nabla_b A_a = 2 \nabla_{[a} A_{b]}##
So this gives at a point for some ##\alpha##:
##F^{ab}F^{cd} = 4 (\nabla^{[a}...
Can you expand on this?
How do you know there are only two algebraically independent invariants?
I might agree that there are only two algebraically independent quadratic invariants. But I'm not sure how to actually prove that, let alone leap to there not being any higher order invariants.
In...
The Reimann curvature tensor has the following symmetry resulting from a Bianchi identityR_{abcd}+R_{acdb}+R_{adbc}=0
The derivative of the electromagnetic field tensor also yields some of Maxwell's equations from a Bianchi identity\partial_\gamma F_{ \alpha \beta } + \partial_\alpha F_{ \beta...
Your questions seem to presuppose he is aware his ideas are crackpot, which I don't believe is the case. Or at least I'd prefer to believe that he means well, even if at times he makes it hard to believe so.
What makes me really upset is that he was given a grant from some physics foundation...
Check out vol. 6 and I think you'll immediately see what we're talking about. That's where he finally reveals his crackpot theory in its full "glory".
An example of serious problems before he reveals his big pet theory of everything is his gravitation section where he "derives" everything from...