Determinism, Stokes' Theorem and Relativity

[Phrak]

I don't think these three: {Determinism, Stokes' Theorem, Relativity Theory}, are compatible.

The notion of determinism, as applied to spacetime physics, means that if we know everything on an R³ spacelike hypersurface at time t_a, we can predict what will be will be the state of things on an R³ spacelike hypersurface at some future time, t_c.

The Kelvin-Stokes theorem,

$$\int_{t_a}^{t_c}f(t')dt' = F(t_c)-F(t_a)$$​

seems to say something else:-

It tells me that what I see at time t_b, where t_a<t_b<t_c, is defined by the events at both t_a and t_c up to a constant of integration.

 

[Bill_K]

There are different kinds of boundary value problems. Typically you specify one value everywhere on the boundary, and the field inside is determined. For a second order field equation like the wave equation, the "value" can either be the field Φ itself or its normal derivative ∇Φ. With an appropriate choice of the surface of integration, the surface integral boils down to an integration over all space at t = t_a and t=t_b. As you point out this is unphysical, since it makes Φ appear to depend on its value at some future time. So you use instead Cauchy conditions, which specify both Φ and ∇Φ on just the initial surface t = t_a.

The situation is clearer for first order equations, which is why Dirac sought a set of first order equations to describe electrons. For a first-order matrix equation Φ_t = M Φ, you just need to specify Φ at t = t_a.

 

[Phrak]

As you point out this is unphysical, since it makes Φ appear to depend on its value at some future time. So you use instead Cauchy conditions, which specify both Φ and ∇Φ on just the initial surface t = t_a.

I didn't point out that it was nonphysical but take serious consideration. Consider ∇Φ to be the measurable quantity and Φ to be a gauge field, Φ <-- Φ' = Φ + dφ.

 

[pervect]

I don't think these three: {Determinism, Stokes' Theorem, Relativity Theory}, are compatible.

The notion of determinism, as applied to spacetime physics, means that if we know everything on an R³ spacelike hypersurface at time t_a, we can predict what will be will be the state of things on an R³ spacelike hypersurface at some future time, t_c.

Actually, this isn't quite the case. As the other posters mentioned, you need to know not only everything on the space-like hypersurface at some time, you also need the boundary conditions, even in classical mechanics.

For example, a photon or "light packet" could come in "from infinity" and upset things, unless one specifies the boundary conditions so this doesn't happen.

There is extensive discussion in Wald, "General Relativity", of the initial value formulation of GR, if you have the textbook. Chapter 10. So if you're trying to argue that GR doesn't have a well posed initial value formulation, you'll have to show where Wald went wrong...

 

[Phrak]

Actually, this isn't quite the case. As the other posters mentioned, you need to know not only everything on the space-like hypersurface at some time, you also need the boundary conditions, even in classical mechanics.

You can't get away with anything around here. I was attempting to state the issue in the simplest possible manner, but not too simple. I guess I didn't.

Of course, we need all the entire R^3 hypersurface encompassing a 4-volume, to be precise. This shouldn't distract from the question.

 

[Phrak]

In attempting to decern which fields are measurable and which are not I've asked, "Can charge density be measured without electric or magnetic fields?" Also, "Can electric and magnetic fields be measured without charge?" The answer seems to be no.

Given this (and please correct me if I am wrong) only the products of second and third derivatives of the 4-vector electromagnetic potential are at our disposal to determine the current, past, and future values of electromagnetic phenomena.

Bill_K, you've given much food for thought. In the above, I mean we can set up many textbook problems with bounary conditions in ∇Φ and Φ, where Φ is some tensor, but not all of these field values are really accessible by experimentation, and so these textbook execises are somewhat constrived and nonphysical.