Just when the hubbub over the infamous hole argument seemed to have died down, and you thought it was safe go back into the placid pool of classical GTR (away from the white-water rapids of quantum gravity), John Earman has conjured up yet another monster to trouble poor old Ein- stein.1 This time, a sprinkling of the magic powder of the constrained Hamiltonian formalism has been employed to Thoroughly Muddled resurrect the decomposing flesh of McTaggart, who intones that the GTR (yes, the plain old vanilla GTR) implies that "no McTaggart genuine physical magnitude countenanced in GTR changes over time" (p. 6), i.e., that if the GTR is complete, we not only live in a block universe, but the block is "frozen", with no Or real physical quantities changing. I hope to drive a stake through the heart of the undead McTaggart and end this How to Abuse Gauge Freedom to new rampage before it has begun. Generate Metaphysical Monstrosities I choose the image of driving a stake through the heart with some care. There is much in this paper, especially in the latter sections, which will be of considerable use to phi- losophers of physics. Professor Earman has, as usual, mas- tered quite a lot of mathematical physics that would be be- Tim Maudlin yond the grasp of most of us, and has done us the great fa- vor of reviewing many technically demanding programs that are of especial interest to philosophers of physics. But With a Response by the motor of this project is supposed to be a very, very sur- John Earman prising feature of the "deep structure" (p. 6) of GTR: namely that according to the deep structure, nothing physically real changes. This, and only this, is the claim I seek to demolish. Even if I succeed, much of interest may be found in the dis- jecta membra of Professor Earman's paper. Philosophers' Imprint <www.philosophersimprint.org/002004/> 1 "Thoroughly Modern McTaggart: Or What McTaggart Would Have Said Volume 2, No. 4 If He Had Learned the General Theory of Relativity," Philosophers' Imprint August 2002 Vol. 2, No. 3 (August 2002): http://www.philosophersimprint.org/002003/. © 2002 Tim Maudlin and John Earman Tim Maudlin is Professor of Philosophy at Rutgers University. Tim Maudlin Thoroughly Muddled McTaggart Let's begin with what we agree on. Earman rejects the definition of an observable in GTR, which I dub the "Ob- original McTaggart's A-series/B-series argument. He agrees servables Argument". I will treat these two arguments sepa- that if one means by a "block universe" only a universe rately, since they rely on different principles. which can, in its entirety, be modeled by a single 4- dimensional manifold with Lorentz metric, then a block uni- The Hamiltonian Argument verse can contain real physical change. This is, I think, the The Hamiltonian Argument derives from the work of Dirac common sense view. The sort of models of the GTR we are on the "constrained Hamiltonian" formalism for presenting a most familiar with, the solutions of the Einstein Field Equa- physical theory. Earman presents this formalism and its in- tions, represent worlds in which things change: stars col- terpretation very compactly. I would like to explicate the lapse, perihelions precess, binary star systems radiate leading ideas in a somewhat different and, I think, more in- gravitational waves and increase their rate of spin. The rep- tuitive way. I will not touch on many of the technical details resentation, as a mathematical object, does not change, but which, I think, play no essential role in understanding the fi- that's just because no mathematical object changes. As Ear- nal result. Let us then approach the Hamiltonian argument man approvingly paraphrases Savitt: to have a picture of in a series of steps. animation, one doesn't have to provide an animated picture. First, we are to cast the GTR in Hamiltonian form. We But more than that is here conceded. Earman characterizes then notice that, so cast, the dynamics of the theory appears the common sense argument for physical change in the GTR to be indeterministic. Next we consider similar cases in as which apparent indeterminism arises because of a "gauge based on a naively realistic reading of the surface struc- ture of the theory-tensor, vector, and scalar fields on freedom" of the theory. We review a standard method for manifolds. But this naive reading must be radically removing this indeterminism by "quotienting out" the gauge modified if GTR is to count as a deterministic theory, freedom in the phase space of the theory. Applying this and the modification undercuts the common sense pic- standard method to the GTR does indeed restore the deter- ture of change by freezing the dynamics. [p. 7] minism of the theory—but at a price. The price is that the So whatever the new threat to change is supposed to be, it dynamics of the theory becomes "pure gauge"; that is, states does not appear in the "naive reading". The new problem is of the mathematical model which we had originally taken to not to be founded, for example, on the absence of a preferred represent physically different conditions occurring at differ- foliation of space-time into instantaneous spaces, since no ent times are now deemed equivalent since they are related such foliation exists in the "naive reading". Rather, the by a "gauge transformation". We find that what we took to problem only appears when the GTR is recast in a way be an "earlier" state of the universe is "gauge equivalent" to somewhat different from its usual presentation. what we took to be a "later" state. If gauge equivalent states There are two wholly distinct arguments for this new are taken to be physically equivalent, it follows that there is "problem of change" for GTR. One argument involves re- no physical difference between the "earlier" and the "later" states: writing the GTR Hamiltonian form, which I will call the there is no real physical change. "Hamiltonian Argument". The other turns on the proper The key step to this argument lies in the technique for 2 Tim Maudlin Thoroughly Muddled McTaggart removing indeterminism by quotienting the phase space. In satisfies the equations of motion. order to understand the nature and prima facie justification of Now, it is well known that if a classical electromagnetic this apparatus, it is best to begin on familiar territory: gauge field satisfies Maxwell's equations, then the field can be rep- freedom in classical electromagnetic theory. resented by vector and scalar electromagnetic potentials, A Φ If we take the ontology of Maxwellian electrodynamics at and , such that face value, we get a picture like this. The complete electro- magnetic state of the universe at some moment is specified E = -grad Φ - ∂A/c ∂t by the values of the electric and magnetic fields at every point of space at that moment, together with their first de- and rivatives. (This gives the free field. We may want to add the distribution of electric charge and its flux to this state, in B = curl A. which case there will be constraints between the field values and the charge densities, but these complications are not of It is equally well known that the relation between the poten- the moment here.) Maxwell's equations then provide the tials and the fields is many-one: different scalar and vector dynamics of this system: they specify how the electric and potentials yield the very same electric and magnetic fields. magnetic fields evolve through time. Generically, the state A mathematical operation changing one pair of potentials of the universe changes deterministically under this dy- into another that yields the same fields is a gauge transforma- namics. tion, and the potentials themselves are said to be gauge We can reformulate this theory in Hamiltonian form as equivalent. The justification for this terminology is clear: follows. We begin with a Hamiltonian function that, since the fields are taken to be the fundamental ontology, roughly, specifies the total energy of the system. We then potentials that are gauge equivalent are taken to represent construct a phase space of the system. Each point in the the very same physical state of affairs. The freedom to choose phase space specifies the complete global distribution of the among gauge-equivalent potentials is not a physical degree electric and magnetic fields, as well as their conjugate mo- of freedom: it rather results from the fact that we have many menta. The time evolution of the system is specified by distinct mathematical objects all of which represent the same Hamilton's equations: the rate of change of any of the ca- physical state. nonical variables is given by a partial differential of the Now suppose we wish to formulate the dynamics of the Hamiltonian. The net result of all this is that the history of theory in terms of the potentials rather than the fields. the entire system is now represented by a trajectory through Without any further ado, we should automatically expect phase space, parameterized by time, which solves Hamil- that (unless something is done), the dynamics in terms of the ton's equations. And since the development of the electro- potentials ought to be indeterministic. For if the original magnetic field in the original (Maxwell) theory was determi- dynamics implies that a state of the electromagnetic field E0- nistic, we expect the dynamics here to be deterministic: each B0 will evolve, after a period of time, into E1-B1, then we point in phase space will belong to a unique trajectory which should expect the new dynamics only to demand that a pair 3 Tim Maudlin Thoroughly Muddled McTaggart of potentials which yields (by the equations given above) E0- namics of electromagnetic theory in terms of potentials B0 ought to evolve into some pair of potentials which yields rather than fields will yield an indeterministic dynamics, E1-B1.