In the spacetime formalism, the question of acceleration is answered in the absolute, even though it may be experimentally difficult. Simply, the inertial observer has no worldline with curvature. This does not that mean that there is an absolute frame of reference.
Note that, op. cit. Burgess *, absolute acceleration is used as a geometric concept, a mere manifestation of the deformation of the worldline of an accelerating particle, and does not imply some absolute space with respect to which the particle accelerates.
When Gauss defined the curvature of a surface, for example, he made explicit use of the way the surface sits in space. There is an extrinsic “curvature as bending” and an intrinsic “curvature as stretching”, that he also measured. The latter is used in the spacetime formalism.
Exactly in the same way, absolute inertial motion reflects the straightness of the worldline of an inertial particle and does not imply some absolute space with respect to which the particle moves with constant velocity.
* Gauss’ Theorema Egregium states that the Gaussian curvature can be determined purely in terms of lengths and angles measured within the surface — that is, in terms of the induced metric and its derivatives — and so is a property intrinsic to the surface itself (as opposed to an extrinsic property that depends on how the surface is embedded into an external R3, how it sits in space).
REFERENCES
H. Minkowski. Space and Time: Minkowski's Papers on Relativity. 2012. Online at: http://rgs.vniims.ru/books/spacetime.pdf
C. P. Burgess. General Relativity: the Notes. 2009. Online at: http://www.physics.mcmaster.ca/~cburgess/Notes/GRnotes.pdf
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Dear Ed,
If you are interested in the space-time relationship with the curvature of Gauss and the Egregium theorem, as well as the relationship of acceleration with gravity I recommend my paper, especially the part of general relativity and gravity
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