Black Holes Help (page 2)

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By — McGraw-Hill Professional
Updated on Sep 17, 2011

The Schwarzchild Radius

There is a formula, based on the mass of an object, that allows us to calculate the radius to which any spherical object would have to be squashed in order for the surface to reach the event horizon. This radius is called the Schwarzchild radius , named after the astrophysicist Karl Schwarzchild who first came up with the formula in 1916. He based the derivation of his formula on the supposition that if an object got small enough, the energy of photons emitted from the surface of that object would not be sufficient to propel them out of its gravitational field. Sometimes the Schwarzchild radius is called the gravitational radius .

Suppose that an object has a mass M (in kilograms). Its Schwarzchild radius r (in meters) would depend directly on the mass. The greater the mass, the greater is the Schwarzchild radius. The formula is remarkably simple:

r = 2 GM / c 2

where c 2 is the speed of light squared (approximately 8.9875 × 10 16 m 2 /s 2 ) and G is a constant known as the gravitational constant , a characteristic of the Cosmos. The value of G has been measured by painstaking experimentation and has been found to be approximately 6.6739 × 10 –11 .

You are invited to perform some calculations with this formula if you like. Do you want to figure out the gravitational radius of your body? Remember, you must use your mass in kilograms to get the answer. You’ll come up with an exceedingly small value. You would not survive being crushed to such a submicroscopic size.

The Earth would have to be compressed to the size of a grape in order to fall within its Schwarzchild radius. The Sun would have to collapse to a radius of approximately 2.9 km (1.8 mi). It is believed that a star must originally have at least three solar masses to generate enough gravitational power, on its demise, to collapse into a black hole.

Apparently, our parent star, the Sun, will never achieve the dubious distinction of withdrawing from the Universe in a cocoon of its own gravitation. It’s not massive enough. This might bring to mind a new meaning of the old adage, “The bigger they are, the harder they fall.” Size does matter!

Warping Of Space

We ordinarily think of space as having three dimensions (height, width, and depth, for example). According to Einstein, however, three-space , as it is called, can be curved with respect to some fourth spatial dimension that we cannot see. We expect that rays of light in space should obey the rules of Euclidean geometry. One of the fundamental rules of light behavior, according to Newtonian physics, dictates that rays of light always travel in straight lines. Such a continuum is called Euclidean space . However, what if space is non-Euclidean ?

According to the theory of general relativity, space is distorted, or bent, by gravitation. The extent of this bending is insignificant at the Earth’s surface. It takes an enormously powerful gravitational field to cause enough bending of space for its effects to be observed as an apparent bending of light rays. However, the effect has been seen and measured in starlight passing close to the Sun during total solar eclipses, when the Sun’s disk is darkened enough so that stars almost behind it can be seen through telescopes. The light from certain distant quasi-stellar sources ( quasars ), when passing near closer objects having intense gravitational fields, has been seen to produce multiple images as the rays are bent.

The bending of space in the vicinity of a strong gravitational field has been likened to the stretching of a rubber membrane when a mass is placed on it. Suppose that a thin rubber sheet is placed horizontally in a room, attached to the walls at a height of, say, 1.5 m above the floor. This sheet is stretched until it is essentially flat. Then a small but dense object such as a ball of solid iron is dropped right in the middle of the sheet. It will be bent in a funnel-like shape. The curvature of the sheet will be greatest near the ball and will diminish with increasing distance from the ball. This is how astronomers believe that space is warped by intense gravitation (Fig. 14-7).

Extreme Objects in Our Galaxy Matter And
Antimatter Warping Of Space

Figure 14-7. Space is warped in the vicinity of an object having an intense gravitational field.

It’s easy to see how objects traveling through warped space can’t possibly go in straight lines: There are no straight paths within such a continuum. According to the theory of general relativity, real space is non-Euclidean every where because there are gravitational fields everywhere. In some regions of space, gravitation is weak, and in other places, it is strong. But there is no place in the Universe that entirely escapes the influence of this all-pervasive force.

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