EVENT HORIZON
IS NOT
SCHWARZSCHILD RADIUS
MIRCEA
BIDIAN
Independent
Researcher
This year marks the 100th anniversary of the solar eclipse that made Albert Einstein famous, thanks to the experiment conceived in 1917 by Sir Frank Watson Dyson, the Astronomer Royal of Britain, Sir Arthur Eddington, who led the experiment, and Andrew Crommelin, who led the Brazilian team.
From the very beginning of modern physics, there were suspicions that light could be distorted in a strong gravitational field. At the end of his treatise Opticks (1704), Isaac Newton pondered the question:
Do not Bodies act upon Light at a distance, and by their action bend its rays, and is not this action (cæteris paribus) strongest at the least distance?
In 1783, English philosopher John Mitchell argued that light—which, according to Isaac Newton, should be made up of tiny corpuscles—does not always travel in straight lines but can be deflected by the gravitational pull of a body. Around 1784, Henry Cavendish calculated the deflection of light by a massive body, assuming Newton's corpuscular theory of light and the law of gravity. He never published his result, but a rough draft of his calculations remains as proof. John Mitchell went even further. He described a massive star whose gravity was so great that the light corpuscles emitted by the star would fall back onto its surface. John Mitchell effectively described what we now call a black hole, a term first used in written literature by Ann Ewing in 1963 and popularized by John Wheeler in 1967.
Just a few years later, in 1796, French mathematician Pierre-Simon Laplace, independently of John Mitchell, suggested that there could be bodies with such a strong gravitational force that light could not escape from them. Laplace calculated such an object and found that the total mass of the Sun would have to be contracted into a sphere just six kilometers in diameter. In 1801, astronomer J. Soldner published an article titled Über die Ablenkung eines Lichtstrahls von Seiner Geradlinigen Bewegung durch die Attraktion eines Weltkörpers, where he investigated the error in determining the angular positions of stars due to the deflection of light. He calculated the orbit (which was, in fact, a hyperbola) of a body moving at the speed of light \(c\) as it passed near a spherical mass \(M\) with radius \(r\), obtaining a deflection angle of 0.875 arcseconds.
In 1911, Albert Einstein performed similar calculations to find the deflection angle, obtaining a value of 0.83 arcseconds, and the equation below:
\[\varphi=\frac{2GM}{c^2 r}[m]\]
which brought him very close to Soldner's result, calculated 110 years earlier. However, the solar eclipses of 1912 in Brazil and August 21, 1914, in Kiev and Crimea were doomed to failure due to bad weather and the onset of World War I. Looking back at these experiments, one could say that it was fortunate for Einstein, as the results would not have supported his theory if the eclipse photography had succeeded.
On December 20, 1915, Einstein received a letter from Karl Schwarzschild, containing a mathematical solution in polar coordinates that would later be known as the Schwarzschild solution or Event Horizon for black holes. Einstein himself presented this solution to the Prussian Academy of Sciences. The equation's solution is:
\[r_S=\frac{2GM}{c^2}[m]\]
where \(r_S\) is the Schwarzschild radius, \(G\) is the gravitational constant, \(M\) is the mass of the black hole, and \(c\) is the speed of light in a vacuum. If the radius of the central body is smaller than the Schwarzschild radius, this represents the point beyond which all massive objects, even light, must inevitably fall into the central body (ignoring quantum tunneling effects near the boundary). When the mass density of this central body exceeds a certain threshold, it triggers a gravitational collapse, which, if spherical symmetry is preserved, produces what we call a Schwarzschild black hole. Of course, this solution applies to a black hole with zero rotational spin.
In May 1916, Einstein published his essay The Foundation of the General Theory of Relativity in Annalen der Physik, where the value of the deflection angle was 1.75 arcseconds, double what he had calculated in 1911 and what Soldner had obtained in 1801. On March 8, 1919, two expeditions set out for different destinations: Sir Arthur Eddington to the island of Príncipe in Africa and Andrew Crommelin to Sobral in Brazil, for the solar eclipse on May 29, 1919. Sir Eddington managed to take 16 photographs, of which only two were usable, and Andrew Crommelin took 8 usable photographs.
Once the teams returned to England, work began on evaluating the photographic plates. A preliminary result from Sir Eddington in September 1919 was followed by Crommelin's final result on November 6, where the values from one telescope were 1.98 ± 0.18 arcseconds and 1.60 ± 0.31 arcseconds from the second telescope. All the newspapers of the time marked the event on November 7. Due to persistent doubts over time about Sir Eddington's result, the photographic plates were remeasured in 1979 using modern instruments, and the result of 1.90 ± 0.11 arcseconds confirmed the value announced in 1919. Although these early measurements were not very precise (with errors of around 30% for the deflection angle), Newtonian values were clearly ruled out. The general theory of relativity had passed the test.
The deflection angle of light near the Sun has a value of 1.75 arcseconds and can be calculated using the formula below:
\[\varphi=\frac{4GM}{c^2 r}[m]\]
