# Minkowski's mechanics

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 four-acceleration, four-momentum, proper time, etc.

Thanks.

jtbell
Mentor
How about "four-fource equals invariant mass times four-acceleration" or "four-fource equais the derivative of four-momentum with respect to proper time"?

http://en.wikipedia.org/wiki/Four-force

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

Anyone else? I'm really just wondering how one arrives at

$$\vec {F} = \gamma ^3 m \vec {a} \parallel + \gamma m \vec {a} \perp$$

using four-vectors, or at least by starting with them.

"I was wondering if this law could be expressed more simply, or at least using different terminology like four-acceleration, four-momentum, proper time, etc."

No. It can't. I'm (quite) certain Minkowski's derivation does not define a single differentiable 4-manifold 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")

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but I thought those were (equivalent) definitions of the four-force vector, rather than physical laws.
In SR, $f^\mu =\frac{dp^\mu}{d\tau}$ is no more a definition of four-force than $\mathbf{F}=\frac{d\mathbf{p}}{dt}$ is a definition of Newtonian force. What F=dp/dt does is tell you how a particular force is related to the motion of a body. Newton, when he gave his law F=dp/dt, also gave an example of a force: gravitation. Gravitational forces are given by F=GMm/r2. In SR an example of a force is the Lorentz four-force which is given by:

$$f^\mu = qu_\nu F^{\mu \nu }$$

where F is the electromagnetic tensor.

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

Another thing about Minkowski and 4-forces. Is he not actually defining an 8-dimensional space? There's (x,y,z,ct) and (x',y',z',ct').
What do you mean? If this were true then you could just as easily say that Minkowski space is 12-dimensional because there's also (x'',y'',z'',ct''). Actually, there are in infinite number of frames that can describe the points in Minkowski space, so you would end up having to say that it is infinite-dimensional.

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.

If this were true then you could just as easily say that Minkowski space is 12-dimensional...

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.
(x,y,z,ct) and (x',y',z',ct') do describe the same point but the question is, can you get to another point which is a '4-distance' ds away through a linear transformation? As an example of a case in which this does not hold, consider the differential elements in spherical coordinates. They are related in a nonlinear fashion.

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 (=6-1: 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 'Poincare-fashion') so that the motion in the local frame can be considered to be along the x-axis only, making y'=y and z'=z.
[Saha translation of 'Space & Time': "we can give the x, y, z-axes an arbitrary rotation about the null-point"]

Electro-magnetic problems, however, might preclude at least assuming y'=y, making the minimum number of dimensions 6, (7-1: x,y,z,ct,x',y'ct') because of Maxwell's cross-product equations.
[Saha translation: "the force-vector 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 y-axis, though the z-component is still zero.
That the result is a "6-vector" 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. 5-1 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)