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In a recent thread, @sweet springs asked:
Initial conditions
The idea is that we are throwing the ball horizontally. We can use Schwarzschild coordinates, and set ##\theta=\pi/2## and work in the equatorial plane without loss of generality. That means that the initial velocity of the ball is in the ##t##-##\phi## plane. By insisting that the initial four-velocity of the ball have inner product with the four-velocity of a hovering observer of ##\gamma## and requiring normalisation, I can write $$\begin{eqnarray*}v^t&=&{{1}\over{\sqrt{1-{{{R_s}}\over{{r_0}}}}\sqrt{1-v^2}}}\\
v^\phi&=&{{v}\over{{r_0}\sqrt{1-v^2}}}\end{eqnarray*}$$Then I can plug those values into the geodesic equations (see Carroll's GR notes, equations 7.43 and 7.44) to get the constants of motion, ##E## and ##L##:
$$\begin{eqnarray*}E&=&{{\sqrt{1-{{{R_s}}\over{{r_0}}}}}\over{\sqrt{1-v^2}}}\\
L&=&{{{r_0}v}\over{\sqrt{1-v^2}}}\end{eqnarray*}$$
Equations of motion
We are interested in comparing the radial motions of balls launched at different velocities on different trajectories. So we don't want to work in terms of their proper times - rather we want to look at the coordinate velocities in some coordinate system that treats hovering as not moving. Schwarzschild fits the bill - so we want to find ##dr/dt=(dr/d\tau)\div(dt/d\tau)##. Carroll's 7.43, 7.47 and 7.48 are what we're looking for, and we get$$\left(\frac{dr}{dt}\right)^2=\frac{R_sr^2(r_0-r)(r-R_s)^2-v^2r_0(r_0^2-r^2)(r-R_s)^3}{r^5(r_0-R_s)}$$
Other constraints
We want the "acceleration due to gravity" (the proper acceleration of a hovering observer) to be a specified value, ##a##. Since this is given by $$a=\frac{R_s}{2r_0^2\sqrt{1-R_s/r_0}}$$we can solve for ##R_s## and pick the positive solution:$$R_s=2ar_0^2\left(\sqrt{a^2r_0^2+1}-ar_0\right)$$Since we're also going to take the limit as ##r_0## gets very large while considering a relatively small drop, it's helpful to write ##r=r_0-\delta##, where ##\delta## is the coordinate distance fallen.
Putting it all together
Finally, we can substitute in the expression for ##R_s## and replace ##r##. This leads to:$$\left(\frac{dr}{dt}\right)^2=\frac{P_1+P_2\sqrt{a^2r_0^2+1}}{P_3+P_4\sqrt{a^2r_0^2+1}}$$where the ##P_n## are polynomials in ##r_0##. Since they are rather lengthy, I won't write them out here - there is Maxima code below if you want to see.
Approximation
The final step is to take the limit of large ##r_0##. We can write ##\sqrt{a^2r_0^2+1}\simeq ar_0+1/2##, which reduces top and bottom to polynomials in ##r_0##, for which we can take the leading term. Doing that yields$$\left(\frac{dr}{dt}\right)^2=-16a^4r_0^3\delta\left(2v^2+1\right)$$
My problem
The problem is that all of those terms are positive - so I have a negative ##(dr/dt)^2##, which is obviously problematic. I must be doing something illegitimate, but I can't see what. I did all the algebra in Maxima, so I don't think I've made a slip there. And I think I've managed to use +--- consistently throughout. And I don't see anything wrong with the basic idea - but the answer is absurd.
Any ideas? Foot at the ready to kick myself if it's something obvious...
Maxima code that I used is hidden in the spoiler tag below.
[/spolier]
I pointed out that the claim about Newton is only strictly true in a uniform gravitational field, and proposed an approximation in GR. The idea was to take a Schwarzschild spacetime, restrict the "acceleration due to gravity" to some specified value, and take the limit as the launch ##r## coordinate grows large. I've had a go at this now, and have come out with a silly answer. I can't see what I'm doing wrong - any comments gratefully received.sweet springs said:
Initial conditions
The idea is that we are throwing the ball horizontally. We can use Schwarzschild coordinates, and set ##\theta=\pi/2## and work in the equatorial plane without loss of generality. That means that the initial velocity of the ball is in the ##t##-##\phi## plane. By insisting that the initial four-velocity of the ball have inner product with the four-velocity of a hovering observer of ##\gamma## and requiring normalisation, I can write $$\begin{eqnarray*}v^t&=&{{1}\over{\sqrt{1-{{{R_s}}\over{{r_0}}}}\sqrt{1-v^2}}}\\
v^\phi&=&{{v}\over{{r_0}\sqrt{1-v^2}}}\end{eqnarray*}$$Then I can plug those values into the geodesic equations (see Carroll's GR notes, equations 7.43 and 7.44) to get the constants of motion, ##E## and ##L##:
$$\begin{eqnarray*}E&=&{{\sqrt{1-{{{R_s}}\over{{r_0}}}}}\over{\sqrt{1-v^2}}}\\
L&=&{{{r_0}v}\over{\sqrt{1-v^2}}}\end{eqnarray*}$$
Equations of motion
We are interested in comparing the radial motions of balls launched at different velocities on different trajectories. So we don't want to work in terms of their proper times - rather we want to look at the coordinate velocities in some coordinate system that treats hovering as not moving. Schwarzschild fits the bill - so we want to find ##dr/dt=(dr/d\tau)\div(dt/d\tau)##. Carroll's 7.43, 7.47 and 7.48 are what we're looking for, and we get$$\left(\frac{dr}{dt}\right)^2=\frac{R_sr^2(r_0-r)(r-R_s)^2-v^2r_0(r_0^2-r^2)(r-R_s)^3}{r^5(r_0-R_s)}$$
Other constraints
We want the "acceleration due to gravity" (the proper acceleration of a hovering observer) to be a specified value, ##a##. Since this is given by $$a=\frac{R_s}{2r_0^2\sqrt{1-R_s/r_0}}$$we can solve for ##R_s## and pick the positive solution:$$R_s=2ar_0^2\left(\sqrt{a^2r_0^2+1}-ar_0\right)$$Since we're also going to take the limit as ##r_0## gets very large while considering a relatively small drop, it's helpful to write ##r=r_0-\delta##, where ##\delta## is the coordinate distance fallen.
Putting it all together
Finally, we can substitute in the expression for ##R_s## and replace ##r##. This leads to:$$\left(\frac{dr}{dt}\right)^2=\frac{P_1+P_2\sqrt{a^2r_0^2+1}}{P_3+P_4\sqrt{a^2r_0^2+1}}$$where the ##P_n## are polynomials in ##r_0##. Since they are rather lengthy, I won't write them out here - there is Maxima code below if you want to see.
Approximation
The final step is to take the limit of large ##r_0##. We can write ##\sqrt{a^2r_0^2+1}\simeq ar_0+1/2##, which reduces top and bottom to polynomials in ##r_0##, for which we can take the leading term. Doing that yields$$\left(\frac{dr}{dt}\right)^2=-16a^4r_0^3\delta\left(2v^2+1\right)$$
My problem
The problem is that all of those terms are positive - so I have a negative ##(dr/dt)^2##, which is obviously problematic. I must be doing something illegitimate, but I can't see what. I did all the algebra in Maxima, so I don't think I've made a slip there. And I think I've managed to use +--- consistently throughout. And I don't see anything wrong with the basic idea - but the answer is absurd.
Any ideas? Foot at the ready to kick myself if it's something obvious...
Maxima code that I used is hidden in the spoiler tag below.
Code:
/* sweet springs asked if a ball dropped next to a ball thrown */
/* horizontally hit the ground simultaneously in GR as they do */
/* in Newton. */
/* Initial conditions: U is 4-velocity of a hovering observer and */
/* V is the 4-velocity of the ball. c=1, work in equatorial plane */
/* so theta=pi/2 and dTheta=0, r=r0, and use g_ab U^a V^b = gamma */
/* and g_ab V^a V^b=1 */
assume(v<1,v^2<1);
gamma:1/sqrt(1-v^2);
gtt:1-Rs/r0;
gpp:-r0^2;
Ut:1/sqrt(gtt);
Vt:gamma/(gtt*Ut);
Vp:rhs(solve(1=gtt*Vt^2+gpp*Vp^2,Vp)[2]);
/* Plug into Carroll's 7.43 and 7.44 to get conserved quantities, L and E */
E:Vt*(1-Rs/r0);
L:substitute(-gamma*v,sqrt(gamma^2-1),r0^2*Vp); /* (gamma v)^2 = 1 - gamma^2 */
/* Plug back into 7.43 to get dt/dtau at arbitrary r, and into */
/* 7.47 and 7.48 to get (dr/dtau)^2 at arbitrary r. epsilon=1 */
/* and lambda=tau for a massive particle. Use Phi for effective */
/* potential, so as not to confuse with four-velocity V. */
dtdtau:E/(1-Rs/r);
Phi:(1-Rs/r)*(1+L^2/r^2)/2;
drdtau2:ratsimp(E^2-2*Phi);
/* We want to compare dr/dt for various gamma. Calculate */
/* (dr/dtau)^2 / (dt/dtau)^2 */
drdt2:ratsimp(drdtau2/dtdtau^2);
/* We want to choose our "acceleration due to gravity" rather */
/* than Rs so write down the proper acceleration of a hovering */
/* observer, solve for Rs, and substitute. */
a=(Rs/(2*r0^2*sqrt(gtt)));
Rs:rhs(solve(lhs(%)^2=rhs(%)^2,Rs)[2]);
/* Put it all together */
substitute(r0-delta,r,drdt2);
substitute(Rs,'Rs,%);
drdt2final:ratsimp(%);
/* Approximate and find leading order terms */
drdt2approx:substitute(a*r0+1/2,sqrt(a^2*r0^2+1),drdt2final);
drdt2num:collectterms(expand(num(drdt2approx)),r0);
drdt2numHi:hipow(drdt2num,r0);
drdt2num:r0^drdt2numHi*coeff(drdt2num,r0,drdt2numHi);
drdt2denom:collectterms(expand(denom(drdt2approx)),r0);
drdt2denomHi:hipow(drdt2denom,r0);
drdt2denom:r0^drdt2denomHi*coeff(drdt2denom,r0,drdt2denomHi);
drdt2LeadingOnly:ratsimp(drdt2num/drdt2denom);