Black Holes

As you compress a given mass into a smaller and smaller radius, the surface gravity grows. Eventually, it will be so strong that the escape velocity will exceed the speed of light. Such an object would appear dark, since no light could escape from its surface.

The escape velocity is the velocity at which a projectile (or particle) would have to be fired straight up so that it will eventually (infinitely far in the future) escape the gravity (come to rest at zero velocity infintely far away). The escape velocity can be calculated from the energy equation:

For escape, v=0 at R=infinity, so therefore in such an orbit E=0. Therefore, at the surface (or any radius R), the escape velocity is given by:

Note that this velocity is higher than the (circular) orbital speed given by the centripetal velocity:

by a factor SQRT(2). If you increase the speed of the Earth in its orbit by more than the factor 1.414, then it would no longer be bound in orbit about the Sun and would be free to fly about the galaxy. Q: What is the escape velocity from the surface of the Earth?

Therefore, if a mass M is compressed to a radius

or smaller, then the escape velocity at the radius GM/c^2 will equal the speed of light. This radius is called the Schwarzschild Radius for the astrophysicist Karl Schwarzschild who calculated it soon after the publication of Einstein's theory in 1916.

An object with a radius equal to or less than the Schwarzschild Radius GM/c^2 is called a black hole. Light, nor anything else, can ever escape the surface of such an object, and it will appear dark. Note that this calculation uses only Newton's theory for gravity. In fact, the possibility for the existence of "dark stars" was postulated as early as 1783.

The Schwarzschild radius for 1 Msun is about 3km - if the Sun were to suddenly (and inexplicably) collapse to this radius it would become a black hole - though our orbit would remain unchanged since the gravitational force depends only on the mass and distance, not the size of the mass!

The effective radius of a black hole, the Schwarzschild radius, depends only on the mass itself, not on the actual density the the mass has (beyond the fact that it must be within its own Schwarzschild radius. As you increse the mass, the radius of the black hole increases proportionally to the mass. Furthermore, since nothing can escape, even light, the mass and size of a black hole can only increase with time.

The spherical "surface" surrounding a black hole of mass M at distance of the Schwarzschild radius GM/c^2 is called the event horizon. Once within the event horizon, matter (or radiation) is lost forever from contact with the universe outside the event horizon. The event horizon is the boundary between what we can know about and what we cannot at outside the horizon. Of course, someone unlucky to be inside the event horizon of the black hole can receive news of the outside world in a one-way information transfer.

Note that degenerate objects such as white dwarfs and neutron stars shrink as mass is added, as the increased gravity compresses the matter. The radius of a 1 Msun neutron star is around 10km, while the radius of a 3 Msun neutron star is about 9km. However, the Schwarzschild radius for 1 Msun is 3 km, and thus for 3 Msun it is 9km. Thus, a 3 Msun neutron star would be within its own Schwarzschild radius and would be a black hole (and we would never know whether it had a neutron star structure). This is the Schwarzschild limit for neutron stars. Calculations show masses greater than around 3 Msun will collpase indefinitely, as neutron degeneracy cannot withstand gravity. This is similar to the Chadrasekhar limit for white dwarfs. Barring any other sort of degeneracy or quantum effect, such a mass could collapese to infinite density, but we would never know since it is within the event horizon (unless you would wish to journey there).

Black Hole Effects

The angle of gravitational deflection of light rays passing the limb of an object:

The presence of a strong gravitational field, like near the surface of a compact object like a white dwarf, neutron star, and black hole causes the wavelength of light to increase, just like the Doppler effect for a source moving away from the observer. Light coming from a source at the event horizon of the black hole to an observer far from the black hole is redshifted to infinite wavelength.

Because gravity causes the warping of space and time, a light ray passing near a compact object is time-delayed. That is, it takes longer for the photon to travel a path passing near a large mass, than it does for the same path were the mass not there. When we discuss general relativity below, we will see that this is not because light is slowed (the speed of light is a constant in vacuum) but because space-time is curved. A light ray passing the limb of a black hole (the event horizon) is infinitely delayed, though not completely captured.

These strong forces, along with the tremendous gravitational energy gained by matter falling from a large distance, cause gas swept up by the black hole to form an accretion disk. Like the accretion disk of a neutron star, this gas would be heated to extemely high temperatures and would emit large numbers of X-rays and Gamma Rays from the thermal emission alone.

Black Hole Lore

Black holes do not suck up matter like a cosmic vacuum cleaner. The gravitational force at a given distance from a black hole is the same as any spherical body of the same mass. It is just that you can get so much closer relative to where the mass is concentrated.

What enters a black hole event horizon appears to be lost from the outside world forever. Does that mean that information sent into the black hole (like matter, quantum numbers, etc.) are lost forever?

Special Relativity

1. The postulate of relativity: The laws of physics are the same for all inertial (uniformly moving) observers.
2. The constancy of the speed of light: The speed of light is measured to be the same for all inertial observers, regardless of their motion relative to the source of the radiation. (This was new, and the crucial new ingredient.)

The relativity postulate is the same as Galileo's. These uniformly moving frames of reference are called inertial or "Galilean" because Galileo's law of inertia holds in them.

The postulate of the constancy of the speed of light for all observers turns out to be deep. Suppose I turn on a flashlight while flying my spaceship past you at 0.5c. You might suppose that the speed of the photons will be 1.5c, but this is not the case, the photons move at c. Furthermore, we both measure the velocity of the photons to be the same: 3x10^5 km/s!

See the Special Lecture on the speed of light for more details.

If the speed of light is the same for all observers, then some thing must be different to compensate for the different relative velocities. It turns out that our measurements of lengths and time intervals that change.

To understand this, we must think not of three dimensional space, and time, as separate entities, but of a four-dimensional spacetime.

The relativistic effects on the length, time, mass, wavelength get extreme as the velocity approaches the speed of light. The common factor in all these is the Lorentz factor (this is supposed to be the greek letter gamma):

As v approaches c, then gamma approaches infinity as the denominator goes to zero

General Relativity

Einstein's General Theory of Relativity (1916) deals with the laws of physics for accelerated observers, just as the Special Theory did for uniformly moving observers. The cornerstone of this theory is the equivalence principle: the laws of physics are the same for all freely falling observers. A freely falling observer is an accelerated observer who is in motion as defined by a gravitational field (without fighting against it). A spaceship or planet in orbit is freely falling (though not falling in the colloquial sense), as is a plummeting airplane. Someone standing on the ground is not, because the ground is exerting a counter-force to the gravitational force on us.

There is no gravitational force on an body infinitely far from all masses in the universe. This body would be un-accelerated, and would remain in uniform motion forever, and would be in all senses "weightless" (though not massless or inertia-less). We know from experience (or at least watching astronauts experience it) that things in a spaceship orbiting the Earth appear also to be "weightless". However, the fact that they are in orbit shows that there is gravitational force (not too much less than on the Earth's surface since an orbit at 500km is only a fraction more than at the Earth's radius of 6400km). It is just that everything in the shuttle feels the same acceleration and is accelerated together. Likewise, a diving airplane (or falling elevator) can provide the experience of weightlessness even in the presence of gravity - this is how the film Apollo 13 was shot.

What Einstein realised is that this was not just an illusion, but that it was a deep principle (like the constancy of the speed of light). He postulated that the laws of physics were the same for all freely falling observers, and furthermore an accelerating observer is equivalent to being in a gravitational field --- there is no difference! Artificial gravity from an accelerating rocket (or an accelerating elevator, car, or jet) is equivalent to real gravity!

Light (a photon) is always freely falling. It should always follow straight line path even in the presence of gravity. But light is deflected by gravity, and appears to follow a curved path. Therefore, it must be that spacetime is curved in the presence of mass - this is the manifestation of gravity. Gravity is the curvature of spacetime in the presence of mass.

In (possibly curved) spacetime, light follows geodesics (straight lines). An analog is drawing a straight line on a sphere, like the surface of the Earth. We can draw a straight line, like a long long road, but eventually we will find our selves back where we started! A straight line on the curved surface of the sphere is a great circle (like the equator, or any meridian of longitude). But is, as far as someone confined to the surface is concerned, straight. This is the same sort of curvature that mass can have on spacetime.

Steven T. Myers - Last revised 09May96