Recovering Forces from Christoffel Symbols: A Derivation

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I don't think I've ever seen this discussed in a textbook, this is an attempt to throw some light on the connection between Christoffel symbols and forces.

In particular I want to derive the later as an approximation of the former, with some limitations on choices of coordinate systems (inertial frames).

I'll start off with assuming a cartesian coordinate system (t,x,y,z).
Next, we will add some "flatness" assumptions. Basically, to reduce the rank 3 Christoffel symbols to a rank 1 vector, we need the majority of them to be zero, or at least "small".

\Gamma^x{}_{tt}, \Gamma^y{}_{tt}, and \Gamma^z{}_{tt}[/tex] can be nonzero, the rest must be &quot;small&quot;.<br /> <br /> We will assume this to be the case, and check to see if it&#039;s sufficient for starters.<br /> <br /> Now that we have the right rank, we still need to worry about the fact that Christoffel symbols aren&#039;t tensors. So we take a close look at the transformation law:<br /> <br /> Wiki <a href="http://en.wikipedia.org/w/index.php?title=Christoffel_symbols&amp;oldid=551829920" target="_blank" class="link link--external" rel="nofollow ugc noopener">http://en.wikipedia.org/w/index.php?title=Christoffel_symbols&amp;oldid=551829920</a><br /> <br /> writes this as:<br /> <br /> \overline{\Gamma^k{}_{ij}} =&lt;br /&gt; \frac{\partial x^p}{\partial y^i}\,&lt;br /&gt; \frac{\partial x^q}{\partial y^j}\,&lt;br /&gt; \Gamma^r{}_{pq}\,&lt;br /&gt; \frac{\partial y^k}{\partial x^r}&lt;br /&gt; + &lt;br /&gt; \frac{\partial y^k}{\partial x^m}\, &lt;br /&gt; \frac{\partial^2 x^m}{\partial y^i \partial y^j}<br /> <br /> We can substitute in \partial y^i = \partial y^j = \partial t, since i and j must be equal to the index of t (usually zero) by our assumption.<br /> <br /> \overline{\Gamma^k{}_{ij}} =&lt;br /&gt; \frac{\partial x^p}{\partial t}\,&lt;br /&gt; \frac{\partial x^q}{\partial t}\,&lt;br /&gt; \Gamma^r{}_{pq}\,&lt;br /&gt; \frac{\partial y^k}{\partial x^r}&lt;br /&gt; + &lt;br /&gt; \frac{\partial y^k}{\partial x^m}\, &lt;br /&gt; \frac{\partial^2 x^m}{\partial t^2}<br /> <br /> We see that we need \frac{\partial^2 x^m}{\partial t^2} to be zero - this is true if we transform between inertial frames.<br /> <br /> This then leaves us with basically the basic tensor transformation law, except that forces transform quadratically with time.<br /> <br /> This is a bit troublesome, apparently we can&#039;t do arbitrary time transformations t-&gt;t&#039; if we wish to use forces. This makes sense, though, intiutively - it means as part of choosing an &quot;inertial frame&quot; we&#039;re stucking with choosing a regularly ticking clock if we wish to use forces. Then this factor becomes a simple scale factor.<br /> <br /> I think that&#039;s pretty much the basics, though due to the limitations on time transformation, some extra attention should be paid to Lorentz transforms to demonstrate why four-fources work relativistically.<br /> <br /> I feel there are some things lacking in this approach - it doesn&#039;t quite demonstrate that Lorentz transforms work, nor does it demonstrate, really, why you can use forces in polar coordinates. But I think it&#039;s a start.<br /> <br /> If anyone has seen anything similar to this in the literature, I&#039;d be interested in hearing about it.
 
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