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.

Topics
Einstein field equations, General relativity, Physicists
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
);
Google Scholar
 
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
);
Google Scholar
Crossref
Search ADS
 
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
);
Google Scholar
Crossref
Search ADS
 
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
).
Google Scholar
Crossref
Search ADS
 
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→f−1(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′|Ē, it is helpful to use a second radial coordinate r̄ which coincides with r. Let E have the coordinate r=r0 in G(r) and Ē have the coordinate r̄=r̄0=f−1(r0) in G′(r̄). Then the transformation r̄=f−1(r) carries G(r)|r0 to G′(r̄)|r̄0. For example, the radial term transforms properly: (1−2m/r0)−1dr2=(1−2m/f(r̄0))−1(f′(r̄0)dr̄)2.
6.
C.
Misner
, “
Precis of General Relativity
,” gr-qc/9508043.
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.  
This content is only available via PDF.
© 2001 American Association of Physics Teachers.
2001
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.