Hole argument

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In general relativity, the hole argument is a "paradox" which much troubled Albert Einstein on the road to his famous field equation. It is an argument against manifold substantialism, a doctrine that views the manifold of events in spacetime as a "substance" which exists independently of the matter within it.

The below is work in progress - not all references included yet

1 Einstein's hole argument
2 The resolution
3 Spacetime substantivalism
4 Philosopher's take on the Hole argument
5 References

[edit] Einstein's hole argument

General covariance is the idea that the laws of nature must be the same in all reference frames, and hence all coordinate systems. This is a principle Einstein had elevated as a criterion for selecting the field equations of a theory of gravitation. In 1912, in the process of formulating general relativity, Einstein realised something he found rather alarming which was a direct consequence of general covariance. He demonstrates these troubles with his so-called "hole argument" to which we now turn.

It begins with an utterly straightforward mathematical observation. Here is written the simple harmonic oscillator differential equation twice

${d^2 f(x) \over dx^2} + f(x) = 0$ (1)

${d^2 g(y) \over dy^2} + g(y) = 0$ (2)

except in (1) the independent variable is $x$ and in (2) the independent variable is $y$ . Once we find out that a solution to (1) is $f (x) = cos(x)$ , we immediately know that $g (y) = cos(y)$ solves (2). This observation combined with general covariance has profound implications for GR.

Assume pure gravity. Say we have two coordinate systems, $x$ -coordinates and $y$ -coordinates. General covariance demands the equations of motion have the same form in both coordinate systems, that is, we have exactly the same differential equation to solve in both coordinate systems except in one the independent variable is $x$ and in the other the independent variable is $y$ . Once we find a metric function $g a b (x)$ that solves the EQM in the $x$ -coordinates we immediately know (by exactly the same reasoning as above) that the same function written as a function of $y$ solves the EOM in the $y$ -coordinates. As both metric functions have the same functional form but belong to different coordinate systems, they impose different spacetime geometries. Thus we see that we have generated a second distinct solution. Given any solution to Einstein's equations there exist other distinct solutions, one for each permissible coordinate system.

Now, consider the situation depicted by Figure 2. Initially the universe is filled by matter but a hole forms which later goes away. Consider two distinct metrics $g a b (x)$ and $\tilde{g}_{ab} (x)$ both solutions of Einstein's equations which are equal everywhere in $M$ except for the hole. We introduce a spacelike (initial data) surface such that the hole is entirely in the future of it. Since the metrics are equal everywhere outside, they do have the same set of initial data on the surface.

Figure 2 (Reproduced from Rovelli's paper gr-qc/9910079)

Now comes the problem: the distance between two distinct points $P$ and $Q$ both inside the hole, is different depending on which of the two metrics you choose, although both metrics have the same initial conditoins. We conclude that general relativity does not determine the distance between these two space-time points.

Einstein found this unacceptable and spent the next three years frantically, (frantically because Hilbert — maybe the best mathematician in the world at that time — had also thrown himself into the problem), looking for non-generally covariant field equations. However, by 1915 Einstein had returned to general covariance. He realized how determinacy and the hole argument could peacefully coexist.

[edit] The resolution

The hole argument was resolved in 1915. Einstein had realized that there was a mistaken assumption about the nature of spacetime and in dropping this assumption there would no longer be any incompatibility between general covariance and determinism. To understand this let us see how the hole argument was resolved.

The idea was to define locations using physical objects, for example particles. (The lesson to be learned from the hole argument doesn't depend on whether or not the physical objects affect the gravitation field or not. The important point is that physical objects move along geodesics). For simpicity we consider only test particles. Consider the arrangement in figure 3. We have four particles labeled by $A$ , $B$ , $C$ and $D$ . The particles $A$ and $B$ intersect in $i$ and similarly the particles $C$ and $D$ intersect in $j$ . These particles start at the initial surface and their geodesics are found by solving the equations of motion. After we have performed the active diffeomorphism we need to solve to find the geodesics for the new metric $\tilde{g}_{ab}$ . The distances between such defined locations is deterministic. This is because the tragectories are dragged across together with the metric by the active diff transformation. This is because we solve to find the geodesics for the transformed metric. A deterministic quantity is the distance between the two particles in figure 3. So physical geometry is invariably define with respect to matter degrees of freedom (or in principle using degrees of freedom of the gravitational field itself).

Figure 3 (Reproduced from Rovelli's paper gr-qc/9910079)

What Einstein construed from the solution of the hole argument is that it is meaningful to refer to a location as a place where two freely falling particles intersect; however it is not meanigful to refer to a location as a point in spacetime (a spacetime event) because the distance from one such point to another is in undetermined in GR. That is spacetime points have, in themselves, no physical significance. In Einstein's own words:

"All our spacetime verifications invariably amount to a determination of spacetime coincidences. If, for example, events consisted merely in the motion of material points, then ultimately nothing would be observable but the meeting of two or more of these points." (Einstein, 1916, p.117).

[edit] Spacetime substantivalism

[edit] Philosopher's take on the Hole argument

[edit] References

Albert Einstein, H. A. Lorentz, H. Weyl, and H. Minkowski, The Principle of relativity (1916).
Carlo Rovelli, Quantum Gravity, Published by Cambridge University Press Year=2004 ID=ISBN 0-521-83733-2
Norton, John, The Hole Argument, The Stanford Encyclopedia of Philosophy (Spring 2004 Edition), Edward N. Zalta (ed.)
Iftime, Mihaela and Stachel, John, "The Hole Argument for Covariant Theories", in GRG Springer (2006), Vol.38, No 8, 1241-1252; e-print available as gr-qc/0512021
d'Inverno, Ray (1992). Introducing Einstein's Relativity. Oxford: Oxford University Press. ISBN 0-19-859686-3. See section 13.6.
``Physics Meets Philosophy at the Planck Scale (Cambridge University Press).
Joy Christian, Why the Quantum Must Yield to Gravity, e-print available as gr-qc/9810078 . Appears in ``Physics Meets Philosophy at the Planck Scale (Cambridge University Press).
Carlo Rovelli and Marcus Gaul, Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance, e-print available as gr-qc/9910079.