Black Holes

Australia: The Land Where Time Began

Black Holes

There is a limit to the mass a white dwarf (the Chandrasekhar limit) can have, beyond which the star cannot support the weight of the martial composing it. There is also a limit to the mass a neutron star can support, the current estimates putting this figure at about 2-3 M_ʘ. The most massive outer layers may not have been dispersed into space during the explosion in some supernovae, with matter possibly falling back onto the core that is already dense. The core of the neutron star may have been pushed above its limit by this extra material, and neutron degeneracy pressure will not be able to fend off gravity.

The core will continue to collapse catastrophically and not even the increasing temperature and pressure can stop the inevitable result. As is seen from Einstein’s equation E = mc², energy is equivalent to mass, therefore the energy associated with the very extreme temperatures and pressure that is concentrated the core, that is now tiny, acts like additional mass, thereby hastening the collapse.

Nothing is known of that is able to stop the crush of gravity. The core collapses without end, forming a black hole, at the end of the life of the star.

A rigorous description of a black hole requires a thorough background in tensor calculus and general relativity, though the broad description of such an exotic object is quite simple, and according to Inglis it is very easy to calculate a few of its basic characteristics.

If the star core contains about 3 or more solar masses, nothing will stop it collapsing even beyond the neutron degeneracy stage. The core collapses to an object that has zero radius. Something with zero radius has no physical size, it has no size at all. The core collapses and therefore its density and surface gravity increases. If it collapses to zero size, the gravity becomes infinite. This entity, that has no physical size yet has infinite gravity, is the singularity.

The escape velocity is the velocity required by an object to allow it to escape from the pull of gravity of a celestial object. E.g. a spacecraft needs to achieve a velocity of about 11 km/second to escape from Earth’s gravity. The mass of the celestial body and the distance of the escaping object from the centre of the celestial body both determine the value of the escape velocity that is required to actually escape the gravitational field of the celestial body. Thus the escape velocity would increase if the escaping object was very massive or very small.

There may be a point at which not even light, the fastest-moving thing in the universe, can escape as it can’t achieve the escape velocity, so light would never escape and the object could never be seen. There will be an area of space, usually spherical, that surrounds the singularity where the escape velocity will be so high that not even light can escape. The entity called a black hole is the sphere of space within which the escape velocity is equal to, or greater than, the escape velocity of light, and so could never be seen. Rotating black holes are oblate, and so not spherical. Therefore their description is far more complicated. Where there is no central pulsar or neutron star in a supernova remnant there may be a black hole.

Therefore, an object inside a black hole would need to travel faster than light to escape the gravitational pull of the black hole; outside the black hole this need for speed is much less. The event horizon is the boundary between these 2 regions; any event occurring within the event horizon can never been seen by an observer outside the event horizon.

Albert Einstein was the first person to combine space and time into a single entity, space-time, and it was his theories that showed that gravity could be portrayed as a curvature of spacetime. Karl Schwarzschild solved Einstein’s equations to give the first ever relativistic description of a black hole. Non-rotating black holes are now called ‘Schwarzschild black holes, to distinguish them from rotating black holes.

Schwarzschild showed that if the conditions are right, such as if matter is packed into a small enough volume of space, then spacetime can curve back on itself. This means that an object, or light, can follow a path, aka a geodesic, into a black hole, though inside the black hole the curvature of spacetime is so extreme that there is no path that leads out.

The event horizon is the boundary that separates the universe from region inside the event horizon which is forever isolated, a region of extreme curved spacetime, the black hole. The radius of the black hole, i.e. the distance from the singularity to the event horizon, is called the Schwarzschild radius, R_Sch.

The size of a black hole

To determine the approximate radius of a black hole, the Schwarzschild radius, R_Sch, the progenitor mass, in relation to the Sun’s mass M_ʘ, is needed.

The radius is given by the formula:

R_Sch~ 3M_ʘ

where R_Sch is in kilometres.

Example:

Χ² Orionis is a B-type supergiant star in the constellation Orion, which has an estimated mass of 42 M_ʘ. Determine the radius of a black hole that may form when the star eventually dies as a supernova, leaving a remnant core of mass ~5 M_ʘ.

R_Sch~ 5M_ʘ

R_Sch~ 3x 5

~ 15 km

Therefore, a star with a mass of 42 M_ʘ could form a black hole with a diameter of 30 km.

Where does Einsteinian gravity take over from Newtonian gravity?

At a large distance a black hole exerts a gravitation force according to Newton’s Law. However, there is a point that is reached where Newton’s laws are no longer valid, beyond which the gravitation effects are now explained by using general relativity of Einstein. The change occurs at a distance of about 3 R_Sch.

Example

Χ² Orionis could form a black hole that has a radius of about 15 km. At what distance from the black hole does the gravity increase from what Newton’s Law predicts?

Using the above formula this distance is about 3 R_Sch, thus:

3 R_Sch ~ 3 x 15

~ 45 km.

The gravitation force will increase to considerably more than is predicted by Newton’s Law as the black hole is approached past 45 km.

Sources & Further reading