In general relativity, space and time are inseparable from a gravitational field: no field, no spacetime. This is a lesson of Einstein’s *hole argument*. We use a simple transformation in a Schwartzschild spacetime to illustrate this. On the basis of the general theory of relativity … space as opposed to “what fills space” … has no separate existence. … There is no such thing as an empty space, i.e., a space without [a gravitational] field. … Spacetime does not claim existence on its own, but only as a structural quality of the field. Albert Einstein, 1952.

## REFERENCES

1.

A. Einstein,

*Relativity: The Special and the General Theory*(Methuen, London, 1952), p. 155.2.

For an account of the history of general relativity, see R. Torretti,

*Relativity and Geometry*(Pergamon, Oxford, 1983).3.

E. Taylor and J. Wheeler,

*Spacetime Physics*(Freeman, San Francisco, 1966).4.

R. Torretti, Ref. 2;

J.

Norton

, “How Einstein found his field equations: 1912-1915

,” Hist. St. Phil. Sci.

14

, 253

–316

(1984

);J. Stachel, “What a physicist can learn from the History of Einstein’s Discovery of General Relativity,” in

*Proceedings of the Fourth Marcel Grossman Meeting on General Relativity*, edited by R. Ruffini (Elsevier, Amsterdam, 1986), pp. 1857–1862;J. Norton, “Einstein, the hole argument, and the reality of space,” in

*Measurement, Realism and Objectivity*, edited by J. Forge (Dordrecht, Boston, 1987), pp. 153–188;J.

Earman

and J.

Norton

, “What Price Substantivalism? The Hole Story

,” Br. J. Philos. Sci.

38

, 515

–525

(1987

);J. Stachel, “Einstein’s search for General Covariance, 1912–1915,” in

*Einstein and the History of General Relativity*, edited by D. Howard and J. Stachel (Birkhäuser, Boston, 1989);J.

Butterfield

, “The Hole Truth

,” Br. J. Philos. Sci.

40

, 1

–28

(1989

);J. Earman,

*World Enough and Spacetime*(MIT, Cambridge, 1989);J. Stachel, “The Meaning of General Covariance,” in

*Philosophical Problems of the Internal and External Worlds*, edited by J. Earman (University of Pittsburgh Press, Pittsburgh, 1993);R.

Rynasiewicz

, “The Lessons of the Hole Argument

,” Br. J. Philos. Sci.

45

, 407

–436

(1994

).John Stachel revived interest in the hole argument by pointing out its subtleties. [Stachel, 1989. The paper was written in 1980 (Stachel, 1993).] Earlier commentators did not fully realize the argument’s significance, because they concluded that Einstein made a simple mistake in it.

5.

Since $r\u2192f\u22121(r)$ is an active point transformation, only one radial coordinate is involved. Nevertheless, in order to see that the transformation carries $G|E$ to $G\u2032|\u0112,$ it is helpful to use a second radial coordinate $r\u0304$ which coincides with

*r*. Let*E*have the coordinate $r=r0$ in $G(r)$ and $\u0112$ have the coordinate $r\u0304=r\u03040=f\u22121(r0)$ in $G\u2032(r\u0304).$ Then the transformation $r\u0304=f\u22121(r)$ carries $G(r)|r0$ to $G\u2032(r\u0304)|r\u03040.$ For example, the radial term transforms properly: $(1\u22122m/r0)\u22121dr2=(1\u22122m/f(r\u03040))\u22121(f\u2032(r\u03040)dr\u0304)2.$6.

7.

S. Adler, M. Bazin, and M. Schiffer,

*Introduction to General Relativity*(McGraw–Hill, New York, 1965), Chap. 9.8.

R. Hawking and R. Ellis,

*The Large Scale Structure of Spacetime*(Cambridge U.P., Cambridge, 1973), Chap. 7.9.

C. Misner, K. Thorne, and J. Wheeler,

*Gravitation*(Freeman, San Francisco, 1973), p. 837.
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