4 This is all the more striking as Einstein had vigorously opposed singularities in debates with de Sitter and Weyl, 5 and had pioneered a restricted version of the T approach with Marcel Grossmann in 1913. Instead, they assume the vacuum field equations, R μ ν = 0, and allow for singularities to appear in the spacetime metric in regions supposedly filled with matter. They explicitly consider the T approach and dismiss it in favour of not introducing any energy-momentum tensor whatsoever. However, Einstein and Grommer do not endorse the same approach of deriving geodesic motion. Like Brown in the quote above, Einstein and Grommer emphasise how different this would make GR as compared to previous field theories. 3Įinstein wrote his first paper on the possible derivability of geodesic motion from the field equations in 1927, together with Jakob Grommer. As Malament (2012) and Weatherall (2012) have pointed out, the T approach necessarily assumes that T μ ν obeys certain energy conditions. I have called this general approach of deriving the geodesic motion of matter ‘the T approach’ in Lehmkuhl (2017). Identifying geodesic motion with inertial motion, Brown describes how the Einstein equations imply that the Bianchi identities ∇ μ G μ ν ≡ 0 imply that ∇ μ T μ ν = 0, which in turn implies that test particles move on the geodesics of the affine connection compatible with the metric g μ ν governed by the Einstein equations. For the first time since Aristotle introduced the fundamental distinction between natural and forced motions, inertial motion is part of the dynamics. GR is the first in the long line of dynamical theories, based on the profound Aristotelian distinction between natural and forced motions of bodies, that explains inertial motion. It is this result that Brown had in mind when he wrote that 2 It is about the fact that the Einstein equations imply that (test) particles move on geodesics of the spacetime metric that is subject to the Einstein equations. G μ ν = κ T μ νīut if T μ ν is the mass-energy of matter, how do we read ‘Spacetime tells matter how to move’ into this? The answer is that the first part of Wheeler's sentence is best not interpreted as being directly about the Einstein field equations. Yes, the Einstein equations can be interpreted as telling us that the presence of the mass-energy of matter (represented by T μ ν) in a given region of spacetime increases the curvature in that region (represented by G μ ν : = R μ ν − 1 2 g μ ν R) i.e. Once we start pondering these questions, we quickly see that only the second part of the sentence is straightforwardly about the Einstein field equations. What does ‘spacetime’ refer to in this slogan? How does it tell matter how to move? Or does it? One of the most enduring series of questions of his 2005 book ‘Physical Relativity’ is due to a raised eyebrow brought about by the fact that everybody else was so mesmerised by Wheeler's beautiful slogan. Harvey Brown's brilliance, first and foremost, lies in putting his finger on a puzzle, and about not being seduced by a pretty picture of the physics. Spacetime tells matter how to move matter tells spacetime how to curve. Generations of physicists and philosophers of physics have been struck by the simplicity and imagination-inducing power of Wheeler summarising the meaning of Einstein's field equations thus 1: John Wheeler was a brilliant physicist, but his most outstanding talent lay in capturing the essence of physics in a picture, in a slogan. The reason he preferred singularities was that he hoped that their mathematical treatment would give a hint as to the sought after theory of matter, a theory that would do justice to quantum features of matter. As a result, Einstein saw energy-momentum tensors and singularities in GR as placeholders for a theory of matter not yet delivered. I show that Einstein saw GR as a hybrid theory from very early on: fundamental and correct as far as gravity was concerned but phenomenological and effective in how it accounted for matter. Drawing on hitherto unknown correspondence between Einstein and George Yuri Rainich, I then show step by step how his work on the vacuum approach came about, and how his quest for a unified field theory informed his interpretation of GR. The paper first investigates why Einstein was so skeptical of the energy-momentum tensor and its role in GR. In addressing this question, Einstein himself always preferred the vacuum approach to the problem: the attempt to derive geodesic motion of matter from the vacuum Einstein equations. In this paper I describe the genesis of Einstein's early work on the problem of motion in general relativity (GR): the question of whether the motion of matter subject to gravity can be derived directly from the Einstein field equations.
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