Trying to understand one-forms and vectors via Schutz's A First Course In General Relativity.

contraction of one-form and vector)

His example uses a spacetime diagram, a scalar field phi, a curve (worldline) parametrized using proper time and the tangent vector (4-velocity). I think I understand, but I can't see the physical significance of the scalar field phi. In plain English, in the spacetime diagram, what does or might phi represent?

  • $\begingroup$ This doesn't answer your question, but: There is something mysterious and tantalizing about fundamental scalar fields. Why don't we have any? (Moot if the Higgs actually exists.) Scalar fields are invoked to create inflation. They can easily lead to classical violations of the energy conditions of GR. If we were going to design a universe to run on laws more Machian than GR, the best known way to do it would be Brans-Dicke gravity, which involves a scalar field. I'm tempted to make a new law of physics: no fundamental scalar fields. It would be a very powerful law. $\endgroup$ – Ben Crowell Jul 29 '11 at 1:42
  • $\begingroup$ @Ben: well, from the point of view of almost all physics, all you need is fermions (to get Pauli's exclusion to obtain variety of atoms) and gauge bosons. One only needs scalar fields for advanced stuff like electroweak SSB or the inflation you mention. Neither of which is relevant for the present universe barring the LHC and black holes (and possibly few other extreme objects) and one certainly imagine an universe without these objects (certainly the LHC). Similar question is why we need three generations of matter. Well, we don't, except for very subtle effects in high-energy regimes. $\endgroup$ – Marek Jul 29 '11 at 9:12

Scalar field is a field of scalars (don't blame me, you invited Captain Obvious yourself). In other words, it's just a function on the manifold. If the manifold were the surface of the Earth (I hope you don't mind I move to 2+0 dimensions for a bit), it could e.g. be the ground temperature in certain moment.

Moving up to 2+1 dimensions, it can again be a field of temperatures but now with time record included. So, your curve $\gamma$ records your wandering around the globe and ${\rm d} \phi \over {\rm d} \tau$ just tells you how the temperature you experience varies both in space and time.

Moving up one further dimension still, a 3+1-dimensional field could be e.g. temperature anywhere in the atmosphere (and again also in time).

Now, in particle physics, scalar fields play a whole different role. They are not connected to familiar things like the temperature above and one instead has to think about them purely abstractly, relating them to things such as EM waves and photons. EM wave, after all, is just a vector field (meaning having spin 1). Similarly, an electron is a particle associated do a Dirac field (which is a fermionic field with spin 1/2). Finally a scalar field has spin 0. I hear, one of the recently-gone-famous fields in this class goes by the name of Higgs :)

  • $\begingroup$ Can't decide whether I like the clarity or the humour more in this answer: +1 ;) $\endgroup$ – qftme Jul 29 '11 at 8:58
  • $\begingroup$ Marek - thanks. I guess I was looking for what phi might actually mean in the context of GR, of contracting a vector with a gradient one-form. If everything else in the equation has a physical meaning, surely phi must have as well. Otherwise it all seems a bit too abstract. Later on Schutz talks about the flux across a surface and how a one-form defines a surface, and says, "this is one of the first concrete physical examples of our definition of a vector as a function of one-forms into real numbers". That sounds interesting. $\endgroup$ – Peter4075 Jul 29 '11 at 10:01

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