Saturday Night Science: Orbits in Strongly Curved Spacetime

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Angular momentum: Mass: Max radius:
    Click the title of this post to see the interactive simulation.

Introduction

The display above shows, from three different physical perspectives, the orbit of a low-mass test particle, the small red circle, around a non-rotating black hole (represented by a grey circle in the panel at the right), where the radius of the circle is the black hole’s gravitational radius, or event horizon. Kepler’s laws of planetary motion, grounded in Newton’s theory of gravity, state that the orbit of a test particle around a massive object is an ellipse with one focus at the centre of the massive object. But when gravitational fields are strong, as is the case for collapsed objects like neutron stars and black holes, Newton’s theory is inaccurate; calculations must be done using Einstein’s theory of General Relativity.

In Newtonian gravitation, an orbit is always an ellipse. As the gravitating body becomes more massive and the test particle orbits it more closely, the speed of the particle in its orbit increases without bound, always balancing the gravitational force. For a black hole, Newton’s theory predicts orbital velocities greater than the speed of light, but according to Einstein’s Special Theory of Relativity, no material object can achieve or exceed the speed of light. In strong gravitational fields, General Relativity predicts orbits drastically different from the ellipses of Kepler’s laws. This article allows you to explore them.

The Orbit Plot

Orbit PlotThe panel at the right of the animation shows the test mass orbiting the black hole, viewed perpendicular to the plane of its orbit. The path of the orbit is traced by the green line. After a large number of orbits the display will get cluttered; just click the mouse anywhere in the right panel to erase the path and start over. When the test mass reaches its greatest distance from the black hole, a yellow line is plotted from the centre of the black hole to that point, the apastron of the orbit. In Newtonian gravity, the apastron remains fixed in space. The effects of General Relativity cause it to precess. You can see the degree of precession in the displacement of successive yellow lines (precession can be more than 360°; the yellow line only shows precession modulo one revolution).

The Gravitational Effective-Potential

Effective potentialThe two panels at the left of the animation display the orbit in more abstract ways. The Effective Potential plot at the top shows the position of the test mass on the gravitational energy curve as it orbits in and out. The summit on the left side of the curve is unique to General Relativity—in Newtonian gravitation the curve rises without bound as the radius decreases, approaching infinity at zero. In Einstein’s theory, the inability of the particle to orbit at or above the speed of light creates a “pit in the potential” near the black hole. As the test mass approaches this summit, falling in from larger radii with greater and greater velocity, it will linger near the energy peak for an increasingly long time, while its continued angular motion will result in more and more precession. If the particle passes the energy peak and continues to lesser radii, toward the left, its fate is sealed—it will fall into the black hole and be captured.

The Gravity Well

Gravity wellSpacetime around an isolated spherical non-rotating uncharged gravitating body is described by Schwarzschild Geometry, in which spacetime can be thought of as being bent by the presence of mass. This creates a gravity well which extends to the surface of the body or, in the case of a black hole, to oblivion. The gravity well has the shape of a four-dimensional paraboloid of revolution, symmetrical about the central mass. Since few Web browsers are presently equipped with four-dimensional display capability, I’ve presented a two-dimensional slice through the gravity well in the panel at the bottom left of the animation. Like the energy plot above, the left side of the panel represents the centre of the black hole and the radius increases to the right. Notice that the test mass radius moves in lockstep on the Effective-Potential and Gravity Well charts, as the radius varies on the orbit plot to their right.

The gravity well of a Schwarzschild black hole has a throat at a radius determined solely by its mass—that is the location of the hole’s event horizon; any matter or energy which crosses the horizon is captured. The throat is the leftmost point on the gravity well curve, where the slope of the paraboloidal geometry becomes infinite (vertical). With sufficient angular momentum, a particle can approach the event horizon as closely as it wishes (assuming it is small enough so it isn’t torn apart by tidal forces), but it can never cross the event horizon and return.

Hands On

Orbits in Strongly Curved Spacetime control panel

By clicking in the various windows and changing values in the controls at the bottom of the window you can explore different scenarios. To pause the simulation, press the Pause button; pressing it again resumes the simulation. Click anywhere in the orbit plot at the right to clear the orbital trail and apastron markers when the screen becomes too cluttered. You can re-launch the test particle at any given radius from the black hole (with the same angular momentum) by clicking at the desired radius in either the Effective Potential or Gravity Well windows. The green line in the Effective Potential plot indicates the energy minimum at which a stable circular orbit exists for a particle of the given angular momentum.

The angular momentum is specified by the box at left in terms of the angular momentum per unit mass of the black hole, all in geometric units—all of this is explained in detail below. What’s important to note is that for orbits like those of planets in the Solar System, this number is huge; only in strong gravitational fields does it approach small values. If the angular momentum is smaller than a critical value (\(2\sqrt 3\), about 3.464 for a black hole of mass 1, measured in the same units), no stable orbits exist; the particle lacks the angular momentum to avoid being swallowed. When you enter a value smaller than this, notice how the trough in the energy curve and the green line marking the stable circular orbit disappear. Regardless of the radius, any particle you launch is doomed to fall into the hole.

The Mass box allows you to change the mass of the black hole, increasing the radius of its event horizon. Since the shape of the orbit is determined by the ratio of the angular momentum to the mass, it’s just as easy to leave the mass as 1 and change the angular momentum. You can change the scale of all the panels by entering a new value for the maximum radius; this value becomes the rightmost point in the effective potential and gravity well plots and the distance from the centre of the black hole to the edge of the orbit plot. When you change the angular momentum or mass, the radius scale is automatically adjusted so the stable circular orbit (if any) is on screen.

Kepler, Newton, and Beyond

In the early 17th century, after years of tedious calculation and false starts, Johannes Kepler published his three laws of planetary motion:

  • First law (1605): A planet’s orbit about the Sun is an ellipse, with the Sun at one focus.
  • Second law (1604): A line from the Sun to a planet sweeps out equal areas in equal times.
  • Third law (1618): The square of the orbital period of a planet is proportional to the cube of the major axis of the orbit.

Kepler’s discoveries about the behaviour of planets in their orbits played an essential rôle in Isaac Newton’s formulation of the law of universal gravitation in 1687. Newton’s theory showed the celestial bodies were governed by the same laws as objects on Earth. The philosophical implications of this played as key a part in the Enlightenment as did the theory itself in the subsequent development of physics and astronomy.

While Kepler’s laws applied only to the Sun and planets, Newton’s universal theory allowed one to calculate the gravitational force and motion of any bodies whatsoever. To be sure, when many bodies were involved and great accuracy was required, the calculations were horrifically complicated and tedious—so much so that those reared in the computer age may find it difficult to imagine embarking upon them armed with nothing but a table of logarithms, pencil and paper, and the human mind. But performed they were, with ever greater precision as astronomers made increasingly accurate observations. And those observations agreed perfectly with the predictions of Newton’s theory.

Well,… almost perfectly. After painstaking observations of the planets and extensive calculation, astronomer Simon Newcomb concluded in 1898 that the orbit of Mercury was precessing 43 arc-seconds per century more than could be explained by the influence of the other planets. This is a tiny discrepancy, but further observations and calculations confirmed Newcomb’s—the discrepancy was real. Some suggested a still undiscovered planet closer to the Sun than Mercury (and went so far as to name it, sight unseen, “Vulcan”), but no such planet was ever found, nor any other plausible explanation advanced. For nearly twenty years Mercury’s precession or “perihelion advance” remained one of those nagging anomalies in the body of scientific data that’s trying to tell us something, if only we knew what.

In 1915, Albert Einstein’s General Theory of Relativity extended Newtonian gravitation theory, revealing previously unanticipated subtleties of nature. And Einstein’s theory explained the perihelion advance of Mercury. That tiny discrepancy in the orbit of Mercury was actually the first evidence for what lay beyond Newtonian gravitation, the first step down a road that would lead to understanding black holes, gravitational radiation, and the source of inertia, which remains a fertile ground for theoretical and experimental physics a century thereafter.

If we’re interested in the domain where general relativistic effects are substantial, we’re better off calculating with units scaled to the problem. A particularly convenient and elegant choice is the system of geometric units, obtained by setting Newton’s gravitational constant G, the speed of light c, and Boltzmann’s constant k all equal to 1. We can then express any of the following units as a length in centimetres by multiplying by the following conversion factors.

Geometric units

The enormous exponents make it evident that these units are far removed from our everyday experience. It would be absurd to tell somebody, “I’ll call you back in \(1.08\times 10^{14}\) centimetres”, but it is a perfectly valid way of saying “one hour”. The discussion that follows uses geometric units throughout, allowing us to treat mass, time, length, and energy without conversion factors. To express a value calculated in geometric units back to conventional units, just divide by the value in the table above.

The Gravitational Effective-Potential

Effective potential

The gravitational effective-potential for a test particle orbiting in a Schwarzschild geometry is:

V²(L,r)=1-2M/r)(1+L²/r²)

where \(\tilde{L}\) is the angular momentum per unit rest mass expressed in geometric units, M is the mass of the gravitating body, and r is the radius of the test particle from the centre of the body.

The radius of a particle from the centre of attraction evolves in proper time τ (time measured by a clock moving along with the particle) according to:

(dr/dTAU)² + V(L,r) = E²

where \(\tilde{E}\) is the potential energy of the test mass at infinity per rest mass.

Angular motion about the centre of attraction is then:

dPHI/dTAU = L/r²

while time, as measured by a distant observer advances according to:

dt/dTAU = E / (1 - 2M/r)

and can be seen to slow down as the event horizon at the gravitational radius is approached. At the gravitational radius of 2M time, as measured from far away, stops entirely so the particle never seems to reach the event horizon. Proper time on the particle continues to advance unabated; an observer on-board sails through the event horizon without a bump (or maybe not) and continues toward the doom which awaits at the central singularity.

Circular Orbits

Circular orbits are possible at maxima and minima of the effective-potential. Orbits at minima are stable, since a small displacement increases the energy and thus creates a restoring force in the opposite direction. Orbits at maxima are unstable; the slightest displacement causes the particle to either be sucked into the black hole or enter a highly elliptical orbit around it.

To find the radius of possible circular orbits, differentiate the gravitational effective-potential with respect to the radius r:

DV²/dr = (2 (3L²M - L²r + Mr²)) / (r^4)

The minima and maxima of a function are at the zero crossings of its derivative, so a little algebra gives the radii of possible circular orbits as:

(L(L ± sqrt(L² - 12M²)) / 2M

The larger of these solutions is the innermost stable circular orbit, while the smaller is the unstable orbit at the maximum. For a black hole, this radius will be outside the gravitational radius at 2M, while for any other object the radius will be less than the diameter of the body, indicating no such orbit exists. If the angular momentum L² is less than 12M², no stable orbit exists; the object will impact the surface or, in the case of a black hole, fall past the event horizon and be swallowed.

References

Gallmeier, Jonathan, Mark Loewe, and Donald W. Olson. “Precession and the Pulsar.” Sky & Telescope (September 1995): 86–88.
A BASIC program which plots orbital paths in Schwarzschild geometry. The program uses different parameters to describe the orbit than those used here, and the program does not simulate orbits which result in capture or escape. This program can be downloaded from the Sky & Telescope Web site.
Misner, Charles W., Kip S. Thorne, and John Archibald Wheeler. Gravitation. San Francisco: W. H. Freeman, 1973. ISBN 978-0-7167-0334-1.
Chapter 25 thoroughly covers all aspects of motion in Schwarzschild geometry, both for test particles with mass and massless particles such as photons.
Wheeler, John Archibald. A Journey into Gravity and Spacetime. New York: W. H. Freeman, 1990. ISBN 978-0-7167-5016-1.
This book, part of the Scientific American Library series (but available separately), devotes chapter 10 to a less technical discussion of orbits in Schwarzschild spacetime. The “energy hill” on page 173 and the orbits plotted on page 176 provided the inspiration for this page.

Here is a short video about orbiting a black hole:

This is a 45 minute lecture on black holes and the effects they produce.

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Author: John Walker

Founder of Ratburger.org, Autodesk, Inc., and Marinchip Systems. Author of The Hacker's Diet. Creator of www.fourmilab.ch.

8 thoughts on “Saturday Night Science: Orbits in Strongly Curved Spacetime”

  1. Bryan G. Stephens:
    I knew the basics on our innermost planet, but not the math.

    The interesting thing is that while the math of Schwarzschild geometry can be intimidating, the visualisation of it as the effective-potential and gravity well plots (which I encountered in the Wheeler book cited in the post—I don’t know who invented it) makes the behaviour of orbits deep in the gravity well very intuitive, and animating it makes it even more instructive.  Most people are amazed at what has come to be called the “zoom-whirl” nature of the orbits (many loops near the throat, then a zoom out to apoapsis), but when you see the body lingering near the inner peak of the effective-potential it’s obvious what’s going on.

    Aesthetically, running the simulation for many orbits often produces intriguing Spirograph-like patterns in the orbit plot.

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  2. John Walker:
    Since few Web browsers are presently equipped with four-dimensional display capability, I’ve presented a two-dimensional slice through the gravity well in the panel at the bottom left of the animation.

    Heh.  Just casually toss that off.  I’m looking forward to your patches adding this feature to Firefox…. (-;

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  3. So… I will venture that this is still an elliptical orbit if expressed in the units given above.

    I noticed a long time ago, playing some gravity-simulator-orbits game, that if something went 180° around without reversing sign on radial velocity, it was gone one way or the other.  This was, of course, a Newtonian game.

    This trivial observation suggests that these are still elliptical orbits, but expressed in a space which is wrapped around the central object apparently in the direction of travel of the orbiting object.

    Imagine a grid square pattern thrown down around our solar system.  Rubber sheet analogy and all that.  Consider the orbit of a comet which shuttles back and forth between the orbits of Jupiter at aphelion and Earth at perihelion, so that it never really goes too far, or gets too close.  It’s in a well-behaved, supposedly Newtonian orbit.  Watch it leave a paint stripe along the rubber sheet, cutting across several (zillions) of these grid squares.  The comet, orbiting around the sun when it is out at Jupiter’s distance (but not near Jupiter hisself) has insufficient energy to stay that far out (“up” in the gravity well) and begins to angle in toward the sun.  It picks up speed and zooms in toward Earth’s orbit, where it levels off at its minimum distance from the sun and its maximum speed.  This will happen 180° forward (“later”) than the maximum distance, minimum speed occurred, when it was at Jupiter’s distance.

    Now if we analyze that paint stripe as it cuts across all of those grid squares, we can count the crossings, and record the angle at which each line was crossed.  Let’s see what happens in a relativistic orbit.

    Imagine that we now have some ridiculously heavy thing at the center of the solar system.  Say that its event horizon is much reasonably close to the orbit of Earth.  Mercury and Venus are already inside this thing.  There are already several things blatantly wrong about this, but go with it.   To convert our normal space into the stuff underlying John’s magnificent illustration, we grab hold of the center of the “rubber sheet” at the center, and twist mightily so that the sheet “wraps up” like a dragster tire at launch.

    In a space of two dimensions, supposedly right-angled grid squares begin to “lean over” and squash inward, stretching around the orbited point.  To what degree?  To the point that crossing the same number of grid lines in a Newtonian orbit traveling 180° from outer max to inner minimum, will also get you from apastron (aphelion) to periastron(?) (perihelion).  Or, to run the problem the other way, “unwind” the twisted-up orbits in John’s illustration above, but drag all of the pixels on your screen with the orbit lines as they move, and you will see a Newtonian orbit, presented in a twisted mass of pixels on your now unserviceable screen.

    Finally, why do I not say space-time?  Because I suspect that this twisting and leaning is in fact the time component of space-time becoming visible.  A parting supposition is that the crew of the red dot in John’s illustration feels that they travel in a plain elliptical, Newtonian orbit.  We think it takes longer for them to go several times around the star as they complete a single orbit, but they think they execute a single ellipse orbit in the usual Newtonian time.

    Okay, just thinking out loud here.  Now I’ll go snoop this out.

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  4. Haakon Dahl:
    I will venture that this is still an elliptical orbit if expressed in the units given above.

    In the language of general relativity, the path of any object not subject to an external force is a straight line (geodesic) in spacetime.  This is true regardless of the strength of the gravitational field(s) the object encounters or how fast it is moving.  When field strengths are low and the object is moving slowly, the geodesic, plotted in Euclidian space, is a Keplerian ellipse.  The reason astronomers only detected the precession due to general relativity in the orbit of Mercury is that only it orbits close enough to the Sun and moves fast enough in its orbit that the effect becomes large enough to observe (and even there, it’s tiny: 43 arc-seconds per century).

    There’s certainly a difference between a straight line and an ellipse (or an ellipse and a zoom-whirl trajectory), but that’s only because you’re projecting them onto flat three dimensional space.  In curved four-dimensional space time, the path is a geodesic, which is defined as the path which has the greatest proper time for a co-moving observer.  (“Proper time” is the time measured by a clock on the moving object.)

    Now, for observers on board the test particle, their experience will be radically different than those on an orbit in a Keplerian ellipse.  Looking at the distant stars, they will see them whip around in space rapidly, often multiple full circuits, while in the whirl phase close the central body, and then slow down dramatically during the zoom phase,  This is entirely different from the observation on an elliptical orbit, where there is a single periapsis and the change in velocity and radial distance is smooth.  Also, the flow of time will be altered, slowing down when in the deep gravity well and with the rapid motion close to the body.

    All of this discussion assumes the test particle has negligible mass and size compared to the central body and spacetime curvature.  An extended object, such as a planet, spaceship, or human, will be subject to tidal forces because every atom of its own substance is in its own orbit and wants to follow a geodesic path independent of the path of the centre of mass.  If the tidal forces exceed those binding the object together, it will be disrupted (usually stretched into spaghetti) as described in Larry Niven’s story “Neutron Star”.

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