Recent content by Yuras

You are missing the whole point. "Mass" and "mass of" are the same physical quantity.

Spectral theorem is complicated because it deals with infinite dimensional Hilbert space. If you are looking for intuition (i.e. not particularly interested in mathematical rigor), then I'd suggest to think about the finite dimensional case. If Hilbert space has dimension ##N##, then Hamiltonian...

I'm not particularly good in mathematical foundations of quantum mechanics, but I think the answer is the spectral theorem. Any solution has spectral decomposition, which is (I think) exactly what Griffith is talking about.

I never said they transform the same way. Of course they don't, and it's obvious enough to make it not worth discussing. I already explained what I meant by "they give the same number" and why transformation rules are irrelevant here. To be clear: I'm not denying your right to distinguish these...

We discussed this with @Sagittarius A-Star above, see #54. The transformation rules are irrelevant here. In one case you apply Lorentz transformations while in other case you change the observer's 4-velocity. The result is the same. At least that's my understanding of these concepts, and I'm...

Then I guess we are using different terminology again. Thanks! Fare enough. Though I don't think MTW actually distinguish between "energy" and "energy measured by". E.g. in the other quote (see #50) they say "No coordinates are needed to describe the fact that a specific emitter emitting a...

The energy as a timelike coordinate of the 4-momentum and energy as a scalar product of 4-momentum and the observer's 4-velocity. In #54 I literally applied both definitions relative to observer ##A## and got the same results. Then I applied both definitions relative to observer ##B## and again...

Great! So as far as I'm concerned we established that these two definitions of energy agree numerically, so for me they are just two different definitions of the same thing. Of course you are free to differentiate between them if you want.

Ah, I see what's going on here. You are answering a different question. I'm saying that timelike component of ##p## in the reference frame of ##A## is numerically the same as ##p\cdot u_a##. The same way for ##B## case. I'm asking: do you agree with this?

Oops, it probably sounds a bit passive-aggressive, sorry. @Dale It's possible that you already saw it, so instead of editing, I'll clarify it. I want clarity in this question because otherwise people will continue referring to this conversation claiming that I inverted the whole concept.

OK, I'm confused. I'll reread my message tomorrow and check for typos just in case. As I said above, I think the "as measured by" part is here for disambiguation. But if you insist on having this, then OK, I yield. At this point I'm mostly interested in defending against your point that I...

I though we agreed that these are two different definitions. What is your point here? Let's say we have a particle with 4-momentum ##p## and two observers ##A## and ##B## with 4-velocities ##u_a## and ##u_b##. To calculate the energy as measured by the observer ##A## you can transform ##p## to...

Fare enough. I interpret these adjectives as a way to disambiguate quantities instead of a completely new term, but there could be other interpretations. Though in general I have an impression that authors don't distinguish energy-as-the-timelike-component and energy-as-p-dot-u throughout the book.

Another quote from Misner etc (also section 2.8):

See for example Misner, Thorne, Wheeler, Gravitation, section 2.8: The geometric definition of energy is used multiple times throughout the section (and throughout the whole book IIRC.) I think you are missing the context here. The whole thread is about how the "standard" way to present SR is...