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Let observer $\mathscr{O}_1$ be in a freely falling frame and following the trajectory $\gamma_1(\tau)$. Consider another observer $\mathscr{O}_2$ who is not in a freely falling frame and has trajectory $\gamma_2(\tau)$. Let also the trajectories $\gamma_1(\tau)$ and $\gamma_2(\tau)$ meet at the space time event $A$. We also assume that we have two particles particle $P_1$ has a world line that also passes through event $A$ and particle $P_2$ has a world line that does not.

My question is:

To what extent can the observers $\mathscr{O}_1$ and $\mathscr{O}_2$ describe the properties of the particles $P_1$ and $P_2$ using the formula of Special relativity?

My Guess

(The following is provided as an idea of what I am looking for, this probably isn't correct)

I have been reading around and although am still fairly confused about this topic, think I have some idea.This answer seems to indicate the following: Observer $\mathscr{O}_2$ can explain the properties of the particle $P_2$ using special relativity, as long as no derivatives are used. They can do this using a special set of coordinates called a 'local observers frame' (see Moore 2013, p. 145) in which the metric is Minkowski. This means that this measures for particle $P_1$ an energy: $$E(P_1)=-p^\mu(P_1) u_{\mu}(\mathscr{O}_2) $$ but for particle $P_2$ the same relation does not hold: $$E(P_2)\ne-p^\mu(P_2) u_{\mu}(\mathscr{O}_2) $$ Observer $\mathscr{O}_1$ can also do one step better then $\mathscr{O}_2$ in that they can explain the properties of $P_1$ using special relativity with formulas that have up to one derivative but it to can't use special relativistic formulas to explain the properties of $P_2$.

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  • $\begingroup$ Two bowling balls are rolling along the ground. To what extent will the assumption the ground is flat be an accurate approximation? $\endgroup$ – AGML Mar 27 '17 at 18:34
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    $\begingroup$ @AGML To a very good one. But what about this: Two bowling balls are rolling along the ground you are accelerating past them at $10^{100}ms^{-2}$. To what extent will the assumption the ground/space-time is flat be an accurate approximation? :) $\endgroup$ – Quantum spaghettification Mar 27 '17 at 18:39
  • $\begingroup$ It seems to me that the reason why your question has not been answered yet is that it is not complete and unambiguous, certain choices have not been taken. In particular, you mention only one event. That is not sufficient for the measurement of velocity u. If you define all events in your question, your question might answer by itself. $\endgroup$ – Moonraker Mar 28 '17 at 12:33
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Here is my answer after some further research. I believe it is correct but would like confirmation. If it is indeed so I will award the bounty to an expansion on this answer - maybe including a explanation of the case without diagonal metrics.

The Hypothesis of Locality

The idea set out in this question is based on the 'Hypothesis of locality' which as stated in [1] (which is the source unless otherwise stated) is given by:

Hypothesis of locality: Locally (i.e., over infinitesimal regions of space and time), neither gravity nor acceleration changes the length of a standard rod or the rate of a standard clock relative to a nearby freely falling (i.e., inertial) standard rod, or standard clock instantaneously co-moving with it.

This hypothesis is, however, not generally true - but is true in the case where the metric in the non-inertial frame is diagonal or can be written in a diagonal form.

This is, however, nearly always the case.

The momentum-energy example

The argument that: $$E(P_1)=-p^\mu(P_1) u_{\mu}(\mathscr{O}_2)$$ then proceeds as follows. Let $\mathscr{O}_3$ be a freely-falling observer co-moving with $\mathscr{O}_2$ at event $A$. Then for $\mathscr{O}_3$ the space-time is locally Minkowskian and he/she therefore measures an energy given by the special relativistic formula: $$\tilde E(P_1)=-p^\mu(P_1) u_{\mu}(\mathscr{O}_3)$$ On the assumption of the validity of the Hypothesis of locality we must therefore have $$E(P_1)=\tilde E(P_1)$$ where $E(P_1)$ is the energy measured by observer $\mathscr{O}_2$ and $\tilde E(P_1)$ is that measured by observer $\mathscr{O}_3$. Since $u_\mu(\mathscr{O}_3)=u_\mu(\mathscr{O}_2)$ by definition we therefore have: $$E(P_1)=-p^\mu(P_1) u_{\mu}(\mathscr{O}_2)$$ as stated in the question.

References

R.D.Klauber, ”Toward a Consistent Theory of Relativistic Rotation”, in ”Relativity in Rotating Frames”, ed. G.Rizzo and M.L.Ruggiero, ed. Kluwer, 2004.

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    $\begingroup$ You have your answer there, it is correct, and this is also how this works in general relativity. (It would be impossible to even start constructing GR without the hypothesis of locality.) I also recommend the first chapter "The Hypothesis of Locality and its Limitations" by Bahram Mashoon from Relativity in Rotating Frames. $\endgroup$ – Void Mar 28 '17 at 9:26

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