Abstract
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.
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Notes
- 1.
For simplicity, I will often just speak of the ‘energy–momentum tensor’ or even just of the ‘energy tensor’ of a material system, rather than of a mass–stress–energy–momentum density tensor. Note that T μν is not a tensor density in the mathematical sense: like the scalar field ρ in Newtonian theory, it is a tensor that represents a physical density, rather than a mathematical object that transforms as a tensor density.
- 2.
The custom in relativity theory is to count radiation like the electromagnetic field as ‘matter’. The left-hand side of the Einstein equations is often claimed to describe both the geometry of spacetime and the gravitational field. The main issues of the paper do not depend on whether one sees the metric field g μν as representing the geometry of physical spacetime, as ‘just another field, not intrinsically different from the electromagnetic field’, or as both at once. I will sometimes call g μν ‘the geometry of spacetime’, but people who do not like that and the ontological flavour of this choice of words should just substitute for it ‘the gravitational field’ or ‘the metric field’, without this altering the points made in this article. In Lehmkuhl (2008), I discuss the ways in which this alleged double role can be understood.
- 3.
Both equations also contain coupling constants, κ N = − 4πG and \({\kappa }_{E} = \frac{8\pi G} {{c}^{4}}\), where the latter is obtained by demanding that the Einstein equations should go over into the Poisson equation in the non-relativistic limit.
- 4.
See Hoefer (1994) for details.
- 5.
See Call No. 17447 of the Einstein Arcives at the University of Jerusalem, also found at the Einstein Papers project at the California Institute of Technology.
- 6.
The following section rests on discussions with Robert Geroch, Erik Curiel, Stephen Lyle, John Norton and David Malament, to whom I am very grateful for their help and patience. Needless to say, any remaining unclarities or misconceptions are surely mine.
- 7.
See Wald (1984, pp. 22–25).
- 8.
Cf. Hawking and Ellis (1973, p. 63).
- 9.
- 10.
For the above way of defining the energy–momentum tensor see Malamant (2007, p. 240).
References
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Acknowledgements
I am very grateful to Oliver Pooley, Harvey Brown, Jeremy Butterfield, Eleanor Knox, Robert Geroch, Stephen Lyle, Tilman Sauer, Edward Slowik and Stephen Tiley. Each of them read one or more than one version of this paper, and helped improve it significantly with their comments and suggestions.
I also thank Robert Geroch, Erik Curiel, Stephen Lyle and John Norton for very helpful discussions, in particular about the energy tensors of different fluid systems and their relation to spacetime structure, and David Malament and Eric Curiel for enlightening discussions about the more subtle dependencies of energy tensors on spacetime structure.
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Lehmkuhl, D. (2010). Matter(s) in Relativity Theory. In: Suárez, M., Dorato, M., Rédei, M. (eds) EPSA Philosophical Issues in the Sciences. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3252-2_16
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