Aerodynamics/Special Relativity Analogy

[jlefevre76]

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.

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

$C_{p,0}=$$\frac{p-p_{\infty}}{q_{\infty}}$=$\frac{p-p_{\infty}}{\frac{1}{2}\rho_{\infty} v^{2}}$

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

$Ma=\frac{v}{c}$

Then, the Prandtl-Glauert rule is given:

$C_{p}=\frac{C_{p,0}}{\sqrt{1-Ma^{2}}}$

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):

$\beta=\frac{v}{c}$

$m=\frac{m_{0}}{\sqrt{1-\beta^{2}}}$

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 $m_{0}$ is:

$V=-G\frac{m_{0}}{R}$

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

$C_{V,0}=\frac{V}{c^{2}}=\frac{-G m_{0}}{R c^{2}}$

Well, now that coefficient will change with $\beta$.

$C_{V}=\frac{C_{V,0}}{\sqrt{1-\beta^{2}}}=\frac{-G m_{0}}{R c^{2}\sqrt{1-\beta^{2}}}$

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.

 

[jlefevre76]

I'm overdue for bumping this, so here goes: BUMP!

Also, to correct a few things, since I'm going for an analogy, it's not perfect. When I say the gravitational field is what "slows it down," I don't mean that literally. I just mean that it keeps something from going faster (specifically, keeps things from breaking the light barrier, so far as we have ever observed). So, when I use terms like "slows down" and "drag," I don't mean them literally. For instance, say that instead of aerodynamic forces acting on an aircraft kept it from going faster than the speed of sound (which was the case until after WW2), say that the mass of the aircraft increased (ridiculous, I know). But the effect would be essentially the same, we would not be able to supersede the sound barrier. In relativity, instead of an opposing force limiting the acceleration in $\sum F=ma$, the increasing mass is what drives the acceleration to zero.

Any relevant sources anyone has that explores this thought approach would be interesting to read, if anybody would like to share a link or comment at all.

 

[Nugatory]

The analogy you are making is unconvincing because it implies that an object's velocity relative to this fluid-like spacetime is measurable, which leads to a preferred reference frame. This in turn leads to all of the problems of pre-relativistic ether theories.

 

[jlefevre76]

The analogy you are making is unconvincing because it implies that an object's velocity relative to this fluid-like spacetime is measurable, which leads to a preferred reference frame. This in turn leads to all of the problems of pre-relativistic ether theories.

Ah, this is good. Then, I guess my question is, in your opinion, will there ever be a theory that can draw an analogy between relativity and compressible fluid flow, or does the requirement for having reference frames make it impossible for there to ever exist such an analogy? As I remember from undergrad physics, relativity reference frames were generally non-intuitive when coming from a Newtonian sort of view of things. I always had trouble with them (which is why mechanical engineering was the best choice of major for me as everything jives with Newtonian physics a little better).

 

[WannabeNewton]

Space-time is not a fluid and you cannot model it as such. That being said, I have two comments. The first being that the general formalism of fluid flow and hydrodynamics in particular is an extremely useful, essentially necessary, and elegant tool in studying the kinematics and dynamics of bodies in space-times. The second is that in very limited scenarios it is conceptually useful to think about space-time in terms of fluid flow. I'll give one example and link a paper to another

Consider a stationary, axisymmetric asymptotically flat space-time. Intuitively this represents the space-time outside of a rotating star. Because the star is rotating, the space-time itself is rotating i.e. the space-time has an angular momentum. But how does one think of this conceptually? Well since the space-time is asymptotically flat, there exists at spatial infinity a Minkowskian inertial frame. The family of observers at rest with respect to this asymptotic Minkowski frame define a fluid in space-time with time-like tangent field $\xi^{\mu}$. It can be shown that $\xi^{\mu}$ has non-vanishing vorticity $\omega^{\mu} = \epsilon^{\mu\nu\gamma\delta}\xi_{\nu}\nabla_{\gamma}\xi_{\delta}$ if and only if the angular momentum of the space-time vanishes. In other words if I place a paddle wheel in the space-time at rest with respect to the asymptotic Minkowski frame then there will be a torque exerted on the paddle wheel, causing it to precess locally, due solely to the vorticity of $\xi^{\mu}$ which is itself due solely to the rotation of the space-time. So you can think of the rotation of the space-time in terms of the vorticity of this fluid. That being said, this analogy has its limitations and if taken too far can lead to conclusions that are at odds with calculations of gyroscopic precession in rotating space-times.

The other example can be found here: http://arxiv.org/pdf/gr-qc/9712010v2.pdf

 

[Nugatory]

Ah, this is good. Then, I guess my question is, in your opinion, will there ever be a theory that can draw an analogy between relativity and compressible fluid flow, or does the requirement for having reference frames make it impossible for there to ever exist such an analogy?

We've piled up enough experimental evidence to the contrary that I don't expect such an analogy to ever be successful.

For example: Imagine measuring the resistance to acceleration of an object under a constant force along two perpendicular axes of a device at rest relative to an observer. The fluid pressure analogy will predict different results on the two axes unless we just happen to be at rest relative to the hypothetical fluid, and even then all we need to do is wait a few hours so that the rotation of the Earth has changed the direction we're moving in, and then repeat the experiment.

No such anisotropy has ever been observed, and if it existed all sorts of things (GPS, satellite orbits) wouldn't work the way they do.

 

[jlefevre76]

Hmmm, for now, I think I'll just have to stick with
$space-time \neq fluid$
Lolz, the math you guys are getting into is way beyond what I'm familiar with. In particular, I have about zero exposure to what the different symbols mean when it comes to general relativity and higher level physics stuff like that. Since I haven't had exposure to those equations, I don't expect to understand them the same way I would the Navier Stokes equations or the thermal diffusion equations, etc.

I need to get a good starter book for someone coming from an engineering background. As is right now, I have a limited understanding of the fundamental equations that drive general relativity.