Determining Metric of Space-Time (H.P. Robertson 1949)

[ConformalGrpOp]

In a paper published in Reviews of Modern Physics in 1949, http://journals.aps.org/rmp/pdf/10.1103/RevModPhys.21.378 , H.P. Robertson provided an analysis of the physical implications of the Michelson/Morley, Kennedy and Thorndike, and Ives and Stilwell experiments which seems definitive with respect to the points he sought to address. But, in his conclusion, he wrote something that, to some degree, is puzzling, or in some sense, seems unexpectedly incomplete. He writes:

[T]he three second-order optical experiments of Michelson and Morley, of Kennedy and Thorndike, and of Ives and Stilwell, furnished empirical evidence which, within the limits of the inductive method, enables us to conclude that the three parameters (g₀, g₁, g₂), may be taken as independent of the motion of the observer. The kinematics im kleinen of physical space-time is thus found to be governed by the Minkowski metric, whose motions are the Lorentz transformations, the background upon which the special theory of relativity and its later extension to the general theory are based.

My first question is, what exactly did he intend to mean when he used the term "im kleinen" (trans: "in the small", or "in small scale"), and why did he stop there and not demonstrate that the Minkowski metric governs the kinematics, "im grossen" of space-time?

Second, setting aside general relativistic and other "field" effects, does it mean that Robertson conceived that it was possible that space-time on a large scale might not be governed by the Minkowski metric? Why would/wouldnt he have been able to resolve the question at that time? To what extent has this question been addressed or resolved experimentally since Robertson's paper?

 

[Nugatory]

Read "im kleinen" as "locally" and he's saying that spacetime is locally Minkowskian everywhere. It's not globally Minkowskian because of general relativistic effects.

 

[Dale]

Why would/wouldnt he have been able to resolve the question at that time

The experiments he was using were accurate to second order only. So they couldn't resolve the question on scales large enough for third or higher order effects to matter.

 

[ConformalGrpOp]

Thank you Nugatory and DaleSpam.

I have since done some further reading on tests of SR, e.g.:

http://relativity.livingreviews.org/Articles/lrr-2005-5/download/lrr-2005-5Color.pdf
http://arxiv.org/pdf/hep-ph/9703240v3.pdf
http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html#Domain_of_Applicability

which, together with your comments, answer the points of inquiry I had with respect to Robertson's remarks.

I came across a paper about, and have been researching the subject of spherical wave transformations of Maxwell's equations which result in velocity independent redshifts, i.e, light propagating with wave numbers that evolve as a function of distance/time. The transformations are not Poincare invariant, but preserve causality. Locally the waves appear to propagate in a space-time governed by the Minkowski metric. In fact, since the transformations are conformal, a local observer is not able to distinguish the fact that the radiation is propagating in a space-time that is non-Minkowskian (except by a rather "large" scale experiment). But, I think it is clear from the literature that Nugatory's reply to my inquiry is correct, that Robertson's remark relates to the fact that SR has been understood as the "local" limit of GR, and not to the fact that the second order optical experiments discussed in his paper were "local" lab experiments, per se.

 

[sardar]

I would like to provide the following response:

Firstly, the term "im kleinen" in this context refers to small scale or local effects. Robertson was specifically referring to the kinematics of physical space-time, which deals with the behavior of objects and particles in motion. He was not discussing the overall structure or curvature of space-time, which is a larger scale phenomenon. Therefore, he used the term "im kleinen" to specify that his analysis and conclusions were limited to the kinematics of space-time at a small scale.

Secondly, Robertson's paper primarily focused on the special theory of relativity and its implications for the Minkowski metric. He did not delve into the general theory of relativity, which deals with the curvature of space-time on a larger scale. It is possible that he did not have enough evidence or data at the time to make conclusions about the Minkowski metric at a larger scale. Additionally, the general theory of relativity was still a relatively new concept at the time of his paper and was not fully understood or accepted by the scientific community.

Since Robertson's paper, there have been numerous experiments and observations that support the use of the Minkowski metric as the governing metric of space-time at both small and large scales. For example, the behavior of particles in high-energy collisions at the Large Hadron Collider (LHC) is consistent with the predictions of the Minkowski metric. Additionally, the observations of gravitational lensing and the behavior of objects in the presence of strong gravitational fields also support the use of the Minkowski metric on a larger scale.

In conclusion, while Robertson's paper may seem incomplete in some aspects, it was a significant contribution to our understanding of the Minkowski metric and its role in the kinematics of space-time. Since then, numerous experiments and observations have further supported the use of the Minkowski metric at both small and large scales, but it is still an active area of research and there may be new discoveries that could challenge our current understanding.