I'm sure that most people on these forums are familiar with the similarities between compressible flow in aerodynamics and the relations between relativistic mass and rest mass in general relativity. A common example that's brought up frequently is the Prandtl-Glauert rule. After playing around a little bit with some non dimensional analysis of relativistic equations, I found a striking similarity that, at least in my mind, gives me hope that one day we'll be able to treat space-time like a fluid. I would not be surprised if this analysis (or something similar) has been done before. I am not sure, but I would welcome ideas and feedback.(adsbygoogle = window.adsbygoogle || []).push({});

The coefficient of pressure at a given point on an object traveling through a compressible medium (a gas) is defined:

[itex]C_{p,0}=[/itex][itex]\frac{p-p_{\infty}}{q_{\infty}}[/itex]=[itex]\frac{p-p_{\infty}}{\frac{1}{2}\rho_{\infty} v^{2}}[/itex]

The Mach number we all know and love is defined (where c is the speed of sound in the medium):

[itex]Ma=\frac{v}{c}[/itex]

Then, the Prandtl-Glauert rule is given:

[itex]C_{p}=\frac{C_{p,0}}{\sqrt{1-Ma^{2}}}[/itex]

Everyone should recognize the similarity between this and special relativity, but if not, I'll review below (where c is the speed of light in the medium, a vacuum in this case):

[itex]\beta=\frac{v}{c}[/itex]

[itex]m=\frac{m_{0}}{\sqrt{1-\beta^{2}}}[/itex]

However, it should be noted that Einstein came up with his at the turn of the century, late 1800's early 1900's (1907 is when it was published?) Prandtl and Glauert came up with theirs in 1930's or 1940's I think. (?) Anyway, I felt the need to try to carry the analogy a step further. Remember that the potential field created by an object of mass [itex]m_{0}[/itex] is:

[itex]V=-G\frac{m_{0}}{R}[/itex]

Well, then the analogy could extend to the field produced by a moving mass. Let's define a field coefficient [itex]C_{V,0}[/itex]:

[itex]C_{V,0}=\frac{V}{c^{2}}=\frac{-G m_{0}}{R c^{2}}[/itex]

Well, now that coefficient will change with [itex]\beta[/itex].

[itex]C_{V}=\frac{C_{V,0}}{\sqrt{1-\beta^{2}}}=\frac{-G m_{0}}{R c^{2}\sqrt{1-\beta^{2}}}[/itex]

So, assuming that this analogy is somehow correct (though it may not be), has it been done before? In my mind, what this tells us might be significant, it almost implies that the gravitational field created by an object is what slows it down. Just like pressure is what slows down an airfoil, the space-time equivalent (if it is), a gravitational field, is what creates "drag" on an object moving through space-time.

If this has been derived before (which I wouldn't at all be surprised), please don't slap me in the face too hard.

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# Aerodynamics/Special Relativity Analogy

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