Dennis Lehmkuhl

Mass–Energy–Momentum: Only there Because of Spacetime?

I describe how relativistic field theory generalizes the paradigm property of material systems, the possession of mass, to the requirement that they have a mass–energy–momentum density tensor Tμν associated with them. I argue that Tμν does not represent an intrinsic property of matter. For it will become evident that the definition of Tμν depends on the metric field gμν in a variety of ways. Accordingly, since gμν represents the geometry of spacetime itself, the properties of mass, stress, energy, and momentum should not be seen as intrinsic properties of matter, but as relational properties that material systems have only in virtue of their relation to spacetime structure. 1 Introduction 2 The Concept of Matter in Relativistic Field Theory   2.1 A short history of energy–momentum tensors   2.2 Metaphysical matters 3 Explicit Dependence of Energy Tensors on the Metric Field 4 Metric Dependence in Lagrangian Theories   4.1 The basic ideas of Lagrangian field theory   4.2 Enter the energy tensor   4.3 Different kinds of coupling 5 Metric Dependence in General   5.1 Definitional dependence at the level of the matter fields   5.2 Definitional dependence at the level of conditions on the energy tensor   5.3 Abstract definitional dependence   5.4 Interpretational dependence 6 What Kind of Property is Mass–Energy–Momentum Density?   6.1 A relational property?   6.2 Relational and essential? 7 Conclusion 1 Introduction 2 The Concept of Matter in Relativistic Field Theory   2.1 A short history of energy–momentum tensors   2.2 Metaphysical matters   2.1 A short history of energy–momentum tensors   2.2 Metaphysical matters 3 Explicit Dependence of Energy Tensors on the Metric Field 4 Metric Dependence in Lagrangian Theories   4.1 The basic ideas of Lagrangian field theory   4.2 Enter the energy tensor   4.3 Different kinds of coupling   4.1 The basic ideas of Lagrangian field theory   4.2 Enter the energy tensor   4.3 Different kinds of coupling 5 Metric Dependence in General   5.1 Definitional dependence at the level of the matter fields   5.2 Definitional dependence at the level of conditions on the energy tensor   5.3 Abstract definitional dependence   5.4 Interpretational dependence   5.1 Definitional dependence at the level of the matter fields   5.2 Definitional dependence at the level of conditions on the energy tensor   5.3 Abstract definitional dependence   5.4 Interpretational dependence 6 What Kind of Property is Mass–Energy–Momentum Density?   6.1 A relational property?   6.2 Relational and essential?   6.1 A relational property?   6.2 Relational and essential? 7 Conclusion

Einstein, the reality of space and the action–reaction principle

Einstein regarded as one of the triumphs of his 1915 theory of gravity - the general theory of relativity - that it vindicated the action-reaction principle, while Newtonian mechanics as well as his 1905 special theory of relativity supposedly violated it. In this paper we examine why Einstein came to emphasise this position several years after the development of general relativity. Several key considerations are relevant to the story: the connection Einstein originally saw between Machs analysis of inertia and both the equivalence principle and the principle of general covariance, the waning of Machs influence owing to de Sitters 1917 results, and Einsteins detailed correspondence with Moritz Schlick in 1920.

Towards a Theory of Spacetime Theories

I begin by reviewing some recent work on the status of the geodesic principle in general relativity and the geometrized formulation of Newtonian gravitation. I then turn to the question of whether either of these theories might be said to “explain” inertial motion. I argue that there is a sense in which both theories may be understood to explain inertial motion, but that the sense of “explain” is rather different from what one might have expected. This sense of explanation is connected with a view of theories—I call it the “puzzleball view”—on which the foundations of a physical theory are best understood as a network of mutually interdependent principles and assumptions.

General relativity as a hybrid theory: The genesis of Einstein's work on the problem of motion

In this paper I describe the genesis of Einsteins 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. 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. The paper first investigates why Einstein was so skeptical of the energy-momentum tensor and its role in GR. 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. 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. As a result, Einstein saw energy-momentum tensors and singularities in GR as placeholders for a theory of matter not yet delivered. 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.

Chapter 5 Is Spacetime a Gravitational Field

Abstract I point out that the often voiced claim that in the general theory of relativity (GR) geometry and gravity are ‘associated’ with each other can be understood in three different ways. The geometric interpretation asserts that gravity can be reduced to spacetime geometry, the field interpretation claims that the geometry of spacetime can be reduced to the behaviour of gravitational fields, and the egalitarian interpretation affirms that gravity and spacetime geometry are conceptually identified. I investigate different versions of each interpretation and argue that an egalitarian interpretation is the one most faithful to the formalism of GR. I then briefly review two rival theories of GR, Brans–Dicke theory and Rosens first bimetric theory, thereby showing that this is not the case for every modern theory of gravity, and that hence the one-to-one correspondence between geometry and gravity is a peculiar feature of GR.

Matter(s) in Relativity Theory

I describe how relativistic field theory generalises the paradigm property of material systems, the possession of mass, to the requirement that they have a mass–energy–momentum density tensor T μν (energy tensor for short) associated with them. I argue that T μν is not an intrinsic property of matter. For it will become evident that the matter fields Φ alone are not sufficient to define T μν; its definition depends on the metric field g μν in a variety of ways. Accordingly, since g μν represents the geometry of spacetime itself, the properties of mass, stress, energy and momentum should not be seen as intrinsic properties of matter, but as relational properties that material systems have only in virtue of their relation to spacetime structure.

Introduction: Towards A Theory of Spacetime Theories

The title of this book—Towards a Theory of Spacetime Theories—is an attempt at false modesty. Or, rather, maybe: an act of an unreasonable raising of the chin in the face of a task supposedly impossible to master. After all, we do not even have a comprehensive map of the solution space of general relativity; by far the most established and most investigated spacetime theory; how then are we supposed to draw a map of the space of spacetime theories in which general relativity itself is but one little point? It seems a daunting and impossible task. And still, we cannot afford not to take it on.