starthaus said:
If you don't know how differential equations describe the equations of motion, that's ok.
I was talking about something else and you unfortunately AGAIN couldn’t get the point as you never can when it comes to creating "Physical mold" of basic ODEs. Your problem is that you don't understand basic calculus and this got exposed itself to me after I asked you a simple question in
https://www.physicsforums.com/showpost.php?p=2742039&postcount=306") about 300 posts earlier was in a void attempt at responding to me ignored by you twice in that thread.
If you want to learn, then it is not necessary to add your name to the list of trollers, wait a little for espen180 to use the tools I gave him and you'll learn.
Unfortunately you have nothing to teach me!
No, I'm not missing anything, I am just pointing out that several of you are blissfully basking in the same elementary mistake. Instead of trolling, can you try deriving the equation of motion? It is really simple, you know.
You're not missing because you can't probably see clearly. Yeah I know and of course I'm going to show which "brother" has been trolling all along since the beginning of this thread.
I assume that the observer who measures the orbital speed of a particle near a gravitating body is hovering so that his 4-velocity is given by
u^a=(\gamma_g,0,0,0),
where \gamma_g=\frac{1}{\sqrt{1-2m/r}}. The orbiting particle itself has the following 4-velocity:
u_a=(E,0,0,L),
with E being the conserved energy of particle per unit mass and L is the orbital angular momentum of the particle per unit mass, again. But what are the precise expressions corresponding to each of these quantities? From the Newtonian theory of gravitation, we remember that the angular momentum per unit mass of a particle in an orbit at r is the simple equation
L=(mr)^{1/2},
where m stands for the gravitating body's mass. Now for the Schwarzschild spacetime, one from the Euler-Lagrange equations would get:
L=r^2\dot{\phi},
for a particle in a circular orbit at r and again with a unit mass. Here the over-dot represents differentiation wrt the parameter of geodesics, e.g. s. For a circular motion, dr/ds=0, and considering motion taking place in the plane for which \theta=\pi/2 so that the metric
ds^2=(1-2m/r)dt^2-1/(1-2m/r)dr^2-r^2(d\theta^2+\sin^2(\theta)d\phi^2)
gives
1=(1-2m/r)\dot{t}^2-r^2d\dot{\phi}^2.
On the other hand, from the first Euler-Lagrange equation and dr/ds=0, it is obvious that
\ddot{t}=0;
thus yielding
\dot{t}=k=const.*
The second of Euler-Lagrange equations (the equation for the radial component) with the same assumptions would lead to the following expression for \dot{\phi} (which is very incorrectly given a name like "proper angular velocity" or stuff like that by non-experts):
\dot{\phi}^2=\frac{m}{r^3}k^2.**
Introducing * and ** into the relationship derived from the Schwarzschild metric gives
1=(1-\frac{2m}{r})k^2-\frac{m}{r}k^2\rightarrow
k^2=\frac{1}{1-3m/r}.
Now the expression for L reads
L=\sqrt{\frac{mr}{1-3m/r}}.
The first Euler-Lagrange equation,
\frac{d}{ds}[(1-2m/r)\dot{t}]=0
if integrated would have teh following simple solution:
E=\frac{1-2m/r}{\sqrt{1-3m/r}}=\frac{1}{\gamma_g^2\sqrt{1-3m/r}}=const.
To wit, the energy of the particle is a conserved quantity. Now what about the energy per unit mass of particle with respect to the hovering observer? Let such energy be denoted by\gamma (with c=1), then projecting the 4-momentum of the particle onto the 4-velocity of the observer gives
\gamma=u^au_a={E}\gamma_g.
Recall that the theory of special relativity portrays \gamma=1/\sqrt{1-v^2} to hold, when c=1, between any two inertial frames. So using the equation for E above and solving this for the orbital velocity v yields
v=(m/r)^{1/2}\gamma_g.
And we are done. Clearly, putting r=3m leads to v=1=c which stands for the orbital speed of photons.
It is not necessary to resort to your hacks about "momentary" and "instantaneous" motion. If you knew how, you could have derived the general equation of motion, applicable for any t. How about you tried that instead on spending so much energy in ranting? Feel free to use the hints that I gave out in this thread.
How about you now to think about the idea that says "if you don't know what is meant by something, then stop being nonsense when answering?” You're not supposed to give us your nonsense prolongated and boring hints that are much of a pain to a student with a really basic knowledge of calculus and algebra. In the meanwhile, try to learn something about the difference between "proper" and "coordinate" quantities in GR! You can use my post over PF on this topic.
AB
