# Centrifugal force from GR perspective

Garth
Gold Member
Let me reiterate:- Whether inertial forces such as centrifugal forces are "real-to-you" or not depends on your frame of reference, that is your perspective on the world around you. Looked at from a fully geometric spacetime perspective inertial forces do not exist, not even gravity, they can be transformed away and are therefore artefacts of geometry.

Garth

pervect
Staff Emeritus
So far I've been busy on some of my own projects and haven't had a chance to look at the article you mentioned, but I can hopefully shed a little light on the centrifugal force "paradox" in GR.

It's well known that a massive body can orbit a black hole only outside the photon sphere, at 1.5 times the Schwarzschild radius. (3GM/c^2).

At the photon sphere, the orbital velocity is the speed of light.

Inside the photon sphere, a body can't orbit the BH, no matter how fast it moves.

Unless I've made a big error in the calculations (sadly, it's all too possible) what happens is that when you try to orbit a BH inside the photon sphere, you actually increase the acceleration (d^2 r/dtau^2) at which you fall into the black hole!

This is based on the equation for geodesic motion in geometric units

(dr/dtau)^2 = (E/m)^2 - (1-2M/r)(1+L^2/(r^2 m^2))

You can convert dr/dtau to d^2r / dtau^2 by taking

dr/dtau = f(r), the square root of the above quation

d^2r/dtau^2 = d(dr/dtau)/dr * dr/dtau = df/dr*f

When you do this you get

dr^2 / dtau^2 = -M/r^2 + L^2(3M-r)/m^2*r^4

Here M is the mass of the black hole, r is the distance away from the black hole, L is the angular momentum, and m is the mass of the small test body.

Getting away from the math, this means that inside the photon sphere, you _always_ have to thrust away from the black hole to avoid falling in. You can't just orbit it.

If you try to orbit the black hole (you have a non-zero angular momentum), you have to thrust even _harder_ inside the photon sphere than if you stayed still (no angular momentum).

Mathematically, this is because the second term above changes sign when r=3M, so L goes from helping you hold position to hurting you when you move inside the photon sphere.

I view this intuitively as the black hole's gravity becoming stronger when one tries to orbit it, rather than the direction of centrifugal force changing.

This is perhaps a somewhat problematic intepretation, so I'll say something hopefully less controversial.

If you look at the tidal force on a body orbiting a black hole, it is a stronger tidal force than a body "hovering" at the same radius. The tidal force isn't dpendent on the global coordinate system, it's something an observer can measure locally.

(Note: I've only seen this analysis performed outside the photon sphere, where orbits exist. It's fairly easy to do the analysis if you ignore frame dragging induced rotation, but it's not particularly easy if you do include this effect).

Where I saw the analysis done is

here

Andrew Mason
Homework Helper
pervect said:
If you try to orbit the black hole (you have a non-zero angular momentum), you have to thrust even _harder_ inside the photon sphere than if you stayed still (no angular momentum).
Is this because orbital motion requires such enormous energy that it increases the mass of the orbiting body, and therefore the gravitational attraction?

AM

pervect
Staff Emeritus
No, the best anology is with the electrostatic case of what happens to the field of a charge when it moves relative to you (or when you move relative to it). The electric field in this case increased by a factor of gamma in the transverse direction. Something roughly similar happens with the gravitational field.

One of my projects has been to work out _exactly_ what happens to the gravitational field in this situation (actually, I'm interested in the situation of a moving mass, but of course a body moving near a mass is very similar). Unfortunately, when I take different approaches to the problem, I've been coming up with different answers. Thus I don't have a result I trust yet on any but a qualitiative level. (The results at least agree qualitiatively, which is better than nothing I suppose).

So far I've learned that Maple displays the Christoffel symbols backwards, and that there are some additional terms in non-coordinate bases when calculating geodesic deviation that aren't present in a coordinate basis. But I still don't have a total resolution I'm happy with.

CHI Meson - you might want to read Feynman's short expose' on the subject in Volume 1 of his lectures on Physics. He questions whether gravity is a pseudo force like inertia since these forces are always proportional to mass. He ruminates that perhaps gravity is only a pseudo force that results from the fact we do not live in a Newtonian inertial frame.

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Homework Helper
yogi said:
CHI Meson - you might want to read Feynman's short expose' on the subject in Volume 1 of his lectures on Physics. He questions whether gravity is a pseudo force like inertia since these forces are always proportional to mass. He ruminates that perhaps gravity is only a pseudo force that results from the fact we do not live in a Newtonian inertial frame.
Thanks, I have read it. This is what my initial viewpoint was primarily based on, and which was countered by the Wikepedia definition that has since been shown to be at fault.

I always turn to Feynman, but it has been over thirty years and a new position could have been solidified since then. I'm glad to know I'm not in the wrong tree (not very high up that tree mind you, but at least it's the right one).

I also am a Feynman fan - some years ago I wrote an article based upon Feynman's idea showing that Hubble expansion leads to a divergent volumetric acceleration of magnitude equal to the gravitational constant.

Fictious or inertial forces are dangerous. They can kill you! They differ from real forces in that they do not occur in action/reaction pairs. F=ma in an inertial frame only. If body A has an acceleration a relative to an inertial frame and body A' has a mass m and an acceleration a' relative to body A. Then for body A' in the inertial frame one has F= m(a+a') . In the accelerated frame, riding A, one has F-ma =ma'. The fictitious force -ma has no reaction partner and is in the opposite direction to the acceleration a. In relativity the acceleration is the covariant derivative of velocity and is made of two pieces: the usual coordinate partials plus a Christoffel piece which accounts for how the coordinate directions change from point to point. As others have said fictitious forces like centrifugal, coriolus, and the Newtonian mg arise from this second piece.