Variable
Speed of Light:
Recalling the very famous second postulate of Special Relativity declared by Einstein (1905): "The velocity c of light in vacuum is the same in all inertial frames of reference in all directions and depend neither on the velocity of the source nor on the velocity of the observer". Einstein's theory of special relativity says that the speed of light in vacuum is always the same (at 299,792.458 km/s) however this is only true locally for systems that are inertial, which means not accelerating. From Newton's second law: if forces exist implies acceleration exists; this means that if you are in a spaceship and fire your rockets then you are not inertial. The other factor besides acceleration is gravity. Albert Einstein himself emphasized in his paper in 1917: "The results of the special relativity hold only so long as we are able to disregard the influence of gravitational fields on the phenomena".
In 1915 (10 years after Special Relativity) Einstein developed another theory called General Relativity that deals with gravitational fields and according to this latest theory the velocity of light appears to vary with the intensity of the gravitational field. For example, an observer outside gravitational fields measures the speed of light locally (in his location) at 299792.458 km/s but when he looks towards a black hole he sees the speed of light there to be as slow as a few meters/sec. At the same time an observer freefalling into that black hole (zero-g) measures the speed of light locally (in his location) at 299792.458 km/s; when he looks towards the black hole he sees the speed of light there much slower; when he looks away from the black hole he sees the speed of light there much faster. If he tries to resist his freefall into that black hole (by firing his rockets for example) he will not measure the speed of light locally anymore at 299792.458 km/s; instead the stronger the g-force that he feels the faster light appears to him. Again when he looks towards the black hole he sees the speed of light there much slower; when he looks away from the black hole he sees the speed of light there much faster. In any case, freefalling or not, he will never see the speed of light outside gravitational fields at 299792.458 km/s. Finally, there is no difference between the effects of g-forces experienced from these rockets and the effects of g-forces experienced when standing on planets, stars... hence an observer standing on a black hole measures the speed of light locally (in his location) much faster than 299792.458 km/s; when he looks towards outside gravitational fields he sees the speed of light there a zillion km/s.
To see the steps how Einstein theorized
that the speed of light in a gravitational field is actually not a constant, but rather a
variable depending upon the reference frame of the observer:
'On the Influence of Gravitation on the Propagation of Light', Annalen der
Physik, 35, 1911.
Einstein wrote this paper in 1911 in German (download from:
http://www.physik.uni-augsburg.de/annalen/history/papers/1911_35_898-908.pdf). It predated the full formal
development of general relativity by about four years. You can find an English
translation of this
paper in the Dover book 'The Principle of Relativity' beginning on page 99;
you will find in
section 3 of that paper Einstein's derivation of the variable speed of light
in a gravitational potential, eqn (3). The result is:
Where
is the gravitational potential relative to the point where the speed of light
co
is measured. Simply put: Light appears to travel slower in stronger
gravitational fields (near bigger mass).
You can find a more sophisticated
derivation later by Einstein (1955) from the full theory of general
relativity in the weak field
approximation:
'The Meaning of Relativity', A. Einstein, Princeton University Press (1955).
See pages 92-93, eqn (107); the variable velocity of light expressed in
coordinates is:
Simply put: Light appears to
travel slower near bigger mass (in stronger gravitational fields). A non-mathematical discussion of this can be found in:
'The Riddle of Gravitation', Peter G. Bergmann, Charles Scribner's Sons, NY
(1987).
See pages 65-66. Bergmann takes the deflection of light by the gravitational field
of a star as evidence of the decreased speed of light in a gravitational field.
You can
also find modern direct derivations that lead to
the same results by Einstein:
'Relativity, Gravitation, and Cosmology', T. Cheng, Oxford University Press
(2005).
For the 1911 results see pages 48-49, eqn (3.39):
For the 1955 results but not in coordinates see page 93, eqn (6.28):
Namely the 1955 approximation shows a variation in km/sec twice as much as first predicted in 1911.
In conclusion, the velocity of light in a gravitational field is not a constant, but rather a variable depending upon the reference frame of the observer; what one observer sees as true another observer sees as false. The only observers that can actually agree that the speed of light outside gravitational fields is 299792.458 km/s are those who are themselves outside gravitational fields.
By using classical orbital mechanics we discovered that outside the gravitational field of the sun 12000 Lunar Orbits/Earth Day becomes equivalent to the local speed of light. An observer near a black hole for example sees the speed of light outside gravitational fields a zillion km/s, but still equal to 12000 Lunar Orbits/Earth Day! This means that if the local speed of light (or the local speed of any object) were defined in km/sec then it will appear to vary for observers in different gravitational fields; however if this speed is defined in Lunar Orbits/Earth Day then it will never appear to vary to anyone because 12000 Lunar Orbits/Earth Day is common to all observers. It also turned out to be a constant forever (Learn more).
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