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#19
Dec611, 11:33 AM

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PF Gold
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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. 


#20
Dec611, 11:40 AM

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#21
Dec611, 11:43 AM

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PF Gold
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But the actual 4acceleration of a worldline, i.e., the covariant derivative of its tangent vector, does transform as a 4vector, i.e., a (1, 0) tensor. The 4acceleration requires contracting the Christoffel symbols with the 4velocity, 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 4acceleration still has another term which is the actual partial derivative of the 4velocity with respect to proper time, and in the local inertial frame that is nonzero for a nongeodesic worldline. In a noninertial 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 4velocity gives the 4acceleration.) 


#22
Dec611, 11:47 AM

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PF Gold
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#23
Dec611, 12:36 PM

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#24
Dec611, 12:53 PM

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#25
Dec611, 01:10 PM

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As PAllen pointed out, the point particle concpet is not related to forces. One can always use the 3force density in Newtonian physics, eg. the NavierStokes 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, 3force, inertial mass Classical special relativity  3force and inertial mass no longer fundamental, instead we have 4force and invariant mass (which is not the same as inertial mass). The inertial mass is defined via the 3force and is now velocity dependent. Neither the 3force 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 3force nor 4force are fundamental, the dynamics is given by the Einstein field equations and equations of state such as Maxwell's equations. 4force 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 spin2 field. 


#26
Dec611, 01:13 PM

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PF Gold
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#27
Dec611, 01:36 PM

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#28
Dec611, 02:56 PM

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#29
Dec611, 02:58 PM

P: 476

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 selfinteraction associated to the field interaction. WheelerFeynman theory (an actionatadistance 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. 


#30
Dec611, 03:01 PM

P: 476

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. 


#31
Dec611, 04:06 PM

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PF Gold
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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). 


#32
Dec611, 04:14 PM

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PF Gold
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I'm a little confused by these two statements:
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. 


#33
Dec611, 04:55 PM

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


#34
Dec611, 05:05 PM

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PF Gold
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#35
Dec611, 05:22 PM

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BTW, now that we established that the E field is as real or as fake as the inertial mass, why is the latter considered so much more pedagogically harmful than the former? 


#36
Dec611, 05:41 PM

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PF Gold
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However, I get your main point. E could be obsoleted (and I note that Penrose, in all his discussions of EM in "Road to Reality" uses only F, not E or B). Yet most people (myself included) still find E and B useful. In sum, I'll grant you: touche. 


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