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Minkowski's mechanics 
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#1
Mar2910, 12:15 AM

P: 611

I'm looking at Hermann Minkowski's 1908 lecture "Space and Time" and in section IV he introduces a law of motion in this way,
"The force vector of motion is equal to the motive force vector." He defines both of these terms in the previous two paragraphs, but his definition of "motive force vector" is rather convoluted. I was wondering if this law could be expressed more simply, or at least using different terminology like fouracceleration, fourmomentum, proper time, etc. Thanks. 


#2
Mar2910, 12:35 AM

Mentor
P: 11,869

How about "fourfource equals invariant mass times fouracceleration" or "fourfource equais the derivative of fourmomentum with respect to proper time"?
http://en.wikipedia.org/wiki/Fourforce 


#3
Mar2910, 03:40 AM

P: 611

but I thought those were (equivalent) definitions of the fourforce vector, rather than physical laws.



#4
Mar3110, 12:17 AM

P: 611

Minkowski's mechanics
Anyone else? I'm really just wondering how one arrives at
[tex] \vec {F} = \gamma ^3 m \vec {a} \parallel + \gamma m \vec {a} \perp [/tex] using fourvectors, or at least by starting with them. 


#5
Mar1512, 04:13 PM

P: 5

"I was wondering if this law could be expressed more simply, or at least using different terminology like fouracceleration, fourmomentum, proper time, etc."
No. It can't. I'm (quite) certain Minkowski's derivation does not define a single differentiable 4manifold in the Einsteinian sense. As he remarked to Sommerfeld the element of proper time is not a total differential. That is because the two frames of reference each have their 'proper space' as well as proper time, as can be seen by inspecting Minkowski's Fig. 1. ("oblique coordinates") 


#6
Mar1512, 05:14 PM

P: 261

[tex]f^\mu = qu_\nu F^{\mu \nu }[/tex] where F is the electromagnetic tensor. 


#7
Mar1912, 09:47 AM

P: 5

Another thing about Minkowski and 4forces. Is he not actually defining an 8dimensional space? There's (x,y,z,ct) and (x',y',z',ct').



#8
Mar1912, 01:23 PM

P: 261

The dimension of a space is the minimum number of coordinates you need to label the points in that space. (x,y,z,ct) and (x',y',z',ct') both describe the same point. 


#9
Mar2112, 07:46 AM

P: 5

You are right about one thing, that 8 are not always necessary and possibly not ever. as, at least in mechanical problems, 2 are degenerate. Furthermore "8" is only how the problem is defined, so any succesful solution will at least reduce this number to 7. Of course the goal is to reduce the number of dimensions, but this goal is to be balanced with the goal of unifying forces  so far we have been extraordinarily succesful in eliminated magnetism as a force independent of electricity  an accomplishment which is given equally by the Einstein/Lorentz approach and by Minkowski's approach. However, and what distinguishes Minkowski's relativity from Einstein's and Lorentz's, there is no way to reduce the number of dimensions below 5 (=61: x,y,z,ct,x',ct' in which one is dependent) for a system defined in 3D space when at rest. The reasoning is as follows: any mechanical motion defined on a 4D infinitesimal may be rotated (in 3D, not 4D 'Poincarefashion') so that the motion in the local frame can be considered to be along the xaxis only, making y'=y and z'=z. [Saha translation of 'Space & Time': "we can give the x, y, zaxes an arbitrary rotation about the nullpoint"] Electromagnetic problems, however, might preclude at least assuming y'=y, making the minimum number of dimensions 6, (71: x,y,z,ct,x',y'ct') because of Maxwell's crossproduct equations. [Saha translation: "the forcevector exerted by the first electron e (moving in any possible manner) upon the second electron," and ensuing equations] Note that these equations now involve motion along the yaxis, though the zcomponent is still zero. That the result is a "6vector" is confirmed in Sommerfeld's notes which accompany the Jeffery translation. (Dover edition) On the other hand, equation (2) of Einstein's General Relativity defines a linear relationship between the (plural) g, which aims at reducing the number of dimensions to 4. (i.e. 51 after t' is eliminated) This goal seems rather ambitious. The empirical results (the constancy of the speed of light) define a scalar constraint between the magnitudes of any two velocity vectors and this constraint produces a quadratic differential relationship. Where does the justification for linearity come from? How can we be sure that the 'g' become constants? (even complex constants would be just fine) 


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