Death to mass or to force?

PeterDonis

Actually, you can equally well view quantum field theories like the Standard Model of particle physics as ways of making all interactions look like geometry the way gravity does, by adding "dimensions" to the gauge group on which the theory is defined. Basically it's the same idea as Kaluza-Klein theory: each "point" of spacetime is no longer viewed as a point but a multi-dimensional "internal" space with an internal geometry that is described by a gauge group; for the Standard Model the gauge group is SU(3) x SU(2) x U(1), modulo some technicalities, as described by John Baez here:

http://math.ucr.edu/home/baez/week253.html.

What we think of as "forces" are then described as connections on subgroups of the gauge group, the same way the "force of gravity" is described by a connection on spacetime; in other words, when you put all the forces together you are looking at geometry on some higher-dimensional space that includes what we normally call "spacetime" as a subspace, plus other subspaces for the other "forces" besides gravity. String theory works along the same lines.

PeterDonis

You may be implicitly switching between two views of "inertial forces" here. Strictly speaking, there is a key physical difference between "inertial forces" and "real forces": real forces are actually felt as acceleration; inertial forces are not. This is modeled in differential geometry as the covariant derivative of a worldline: it's zero for a body moving solely due to "inertial forces", but nonzero for a body subject to "real forces".

But often when we talk about "inertial forces", we forget that the actual "force" we feel is not due to the inertial force itself; it's due to the real force that is pushing us out of the geodesic path that the inertial force would have us follow. I feel a force sitting here on the surface of the Earth, and speaking loosely I might say this is the "force of gravity": but actually it's not, it's the force of the Earth pushing on me. A rock falling past me is moving due to the "force of gravity", but it feels no force. The principle of equivalence does not require me to say that I and the rock are equivalent; so IMO it doesn't require me to say that inertial forces and "real" forces are equivalent either.

atyy

OK, let me clarify my terminology. I'm essentially considering the geodesic equation a "force equation". In full GR, the geodesic equation is not fundamental. It acts on "test particles" which are again not fundamental. So Newton-Cartan as I understand it usually has a geodesic equation and test particles. Is there a Newton-Cartan or Newtonian gravity in which the geodesic equation is not fundamental, ie. something like GR where we write the full action as Einstein-Hilbert plus minimally coupled matter actions and the fundamental degrees of freedom (with respect to which we vary the action) are all fields?

PeterDonis

I think what pervect was referring to is that the "acceleration due to gravity", or "force of gravity", is modeled by the Christoffel symbols, which don't transform as tensors; you can make them all vanish locally at a given event by transforming to a local inertial frame at that event.

But the actual 4-acceleration of a worldline, i.e., the covariant derivative of its tangent vector, does transform as a 4-vector, i.e., a (1, 0) tensor. The 4-acceleration requires contracting the Christoffel symbols with the 4-velocity, which is how it can transform as a tensor even though the Christoffel symbols by themselves don't. (In a local inertial frame, all the Christoffel symbols vanish, but the 4-acceleration still has another term which is the actual partial derivative of the 4-velocity with respect to proper time, and in the local inertial frame that is nonzero for a non-geodesic worldline. In a non-inertial frame, for example a "static" frame using Schwarzschild coordinates around a massive body, the partial derivative vanishes but there the Christoffel symbols don't, and their contraction with the 4-velocity gives the 4-acceleration.)

PeterDonis

Ah, I see. I don't know if Newton-Cartan gravity has been given such a formulation. Obviously you can formulate Newtonian gravity using the Lagrangian formalism, but I don't know if anyone has ever checked to see if such a formulation will lead to the Newton-Cartan formalism by the same route as you can use to get GR from the Einstein-Hilbert action.

atyy

A quick google turns up Goenner, A variational principle for Newton-Cartan theory, whose abstract seems to indicate the answer is "not completely". Unfortunately, this paper is probably not freely available.

PeterDonis

It isn't. But the abstract does, as you say, imply that the Newtonian case works differently than the GR case.

atyy

As PAllen pointed out, the point particle concpet is not related to forces. One can always use the 3-force density in Newtonian physics, eg. the Navier-Stokes equations. So in the following "force" always means "force density".

OK, is the following the correct hierachy of concepts?

Newtonian mechanics and gravity - Galilean inertial frames, 3-force, inertial mass

Classical special relativity - 3-force and inertial mass no longer fundamental, instead we have 4-force and invariant mass (which is not the same as inertial mass). The inertial mass is defined via the 3-force and is now velocity dependent. Neither the 3-force nor inertial mass are fundamental, just as E and B fields are not fundamental (I would especially like commentary on this in the light of Bell's "Lorentzian pedagogy" where we say everything can be done in one Lorentz inertial frame).

Quantum special relativity - no concept of force at all. The dynamics are entirely given by Hamiltonian or Lagrangian formalisms.

General relativity - neither 3-force nor 4-force are fundamental, the dynamics is given by the Einstein field equations and equations of state such as Maxwell's equations. 4-force and the geodesic equation are derived concepts.

Quantum general relativity - no concept of force at all, same formalism as quantum special relativity, with gravity being a spin-2 field.

PAllen

I'd go along with this. It is certainly a very reasonable point of view, IMO.

PeterDonis

These all look OK to me.

You can do everything in one inertial frame, but which one you pick will depend on your own state of motion. And we don't fully control our state of motion. For example, the Earth orbits the Sun, meaning that, from the point of view of, say, an inertial frame in which the Sun is at rest, we are constantly changing our state of motion. (Also, of course, we sitting on the surface of the Earth are not really in an inertial frame to begin with, since we feel acceleration.) So from a fundamental physics point of view, we can't privilege any specific inertial frame.

atyy

Ok, but is there anything I can't do if I use 3-force, inertial mass and E and B fields in any particular inertial frame? In other words, is the only thing I lose manifest Lorentz covariance, or do I actually lose any physics (including the ability to predict what the physics is like in any other frame, inertial or not)?

juanrga

The outdated concept of relativistic mass was abandoned because was a source of many confusion. The mass of an electron is m_e, which is independent of its velocity.

The problems with point particles in Maxwell theory are not solved by introducing extended distributions of charge (Poincaré stresses). The problem is not in the existence of a force but in the self-interaction associated to the field interaction. Wheeler-Feynman theory (an action-at-a-distance theory) uses forces but solves the difficulties because abandon the interaction through fields.

What you say about field theory is just the inverse. Field theory is the old theory and the modern theories under development agree on that the concept of field is approximated only. Sometimes we use the term «effective theory» to emphasize that field theory is not fundamental.

juanrga

This is already the present in research.

All modern and satisfactory theories of gravity consider gravity a real force and consider that the geometrical description given by GR is only valid as a first approximation.

PeterDonis

I don't think so, but you were talking about what different theories consider "fundamental". Standard SR does not consider anything that is frame-dependent to be "fundamental"; only frame-invariant objects are "fundamental". So, for example, the electromagnetic field tensor [itex]F_{ab}[/itex] would be fundamental, but any particular decomposition into E and B fields would not be, since that is frame-dependent.

Perhaps it's also worth noting that, just as you can calculate everything using a single inertial frame, you can calculate things without using a frame at all. Any number that you can actually measure in an experiment can be expressed as a scalar (PAllen made this point in a recent thread on a similar subject), meaning you can express it without ever having to commit yourself to any specific frame, just write the expression in terms of contractions of vectors and tensors (which you can do in abstract index notation, without ever specifying a particular set of components).

PeterDonis

I'm a little confused by these two statements:

But the theory of gravity as a "real force" is a field theory; it's the theory of a massless spin-2 field, or the fancier versions given in string theory (is string theory one of the "modern and satisfactory theories of gravity"?), which are also field theories, just not field theories based on point particles. If the original massless spin-2 field theory is only an "effective" theory, are the string theory versions "effective" too?

Other "modern" theories of gravity (I don't know whether you consider them "satisfactory"), such as loop quantum gravity, don't look to me like theories of gravity as a "real force"; they look more like theories of geometry.

atyy

Yes. I realise you were addressing the poetic aspects of my question. I just wanted to check that there wasn't an underlying technical issue I was missing.

Now I just need to find a way to make the E field a scalar:)

PAllen

Ah, but if you try to measure the E field there are two things to consider: the EM field and an instrument. The world line of the instrument (e.g. its motion in your chosen frame) interacts with the field, producing a measurement consisting of one or more scalars. The instrument may well measure magnetic field strength due to its motion in your coordinates (in which, say, the E/M field is pure coulomb).