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### Scalar Variance of Speed of Light

Posted: Tue Feb 05, 2008 8:03 am
Km/sec is a scalar, however gravitational length contraction and time dilation make it impossible to represent the speed of light by a scalar. There is a difference between the radial speed of light and the tangential speed of light. The effects of gravitation can only be accurately represented by a tensor field. You can find an online solution Reflections on Relativity, Chapter 9:
http://www.mathpages.com/rr/s6-01/6-01.htm wrote:Specifically, Einstein wrote in 1911 that the speed of light at a place with the gravitational potential φ would be c0 (1 + φ/c0²), where c0 is the nominal speed of light in the absence of gravity. In geometrical units we define c0 = 1, so Einstein's 1911 formula can be written simply as c = 1 + φ. However, this formula for the speed of light (not to mention this whole approach to gravity) turned out to be incorrect, as Einstein realized during the years leading up to 1915 and the completion of the general theory. In the general theory of relativity the speed of light in a gravitational field cannot be represented by a simple scalar field of c values. Instead, the "speed of light" at a each point depends on the direction of the light ray through that point – and also on the choice of coordinate systems – so we can't generally talk about the value of c at a given point in a non-vanishing gravitational field. However, if we consider just radial light rays near a spherically symmetrical (and non- rotating) mass, and if we agree to use a specific set of coordinates, namely those in which the metric coefficients are independent of t, then we can read a formula analogous to Einstein's 1911 formula directly from the Schwarzschild metric. The result differs from the 1911 formula by a factor of 2...

...

Now that we have derived the Schwarzschild metric, we can easily correct the "speed of light" formula that Einstein gave in 1911. A ray of light always travels along a null trajectory, i.e., with dt = 0, and for a radial ray we have dθ and dπ both equal to zero, so the equation for the light ray trajectory through spacetime, in Schwarzschild coordinates (which are the only spherically symmetrical ones in which the metric is independent of t) is simply:

from which we get:

where the ± sign just indicates that the light can be going radially inward or outward. (Note that we're using geometric units, so c = 1.) In the Newtonian limit the classical gravitational potential at a distance r from mass m is φ = -m/r, so if we let cr = dr/dt denote the radial speed of light in Schwarzschild coordinates, we have:

which corresponds to Einstein's 1911 equation, except that we have a factor of 2 instead of 1 on the potential term. Thus, as φ becomes increasingly negative (i.e., as the magnitude of the potential increases), the radial "speed of light" cr defined in terms of the Schwarzschild parameters t and r is reduced to less than the nominal value of c. The factor of 2 relative to the equation of 1911 arises because in the full theory there is gravitational length contraction as well as time dilation. Of course, the length contraction doesn’t affect the gravitational redshift, which is purely a function of the time dilation, so the redshift prediction of 1911 remains valid. Only the radial speed of light (in terms of Schwarzschild coordinates) is changed.

On the other hand, if we define the tangential speed of light at a distance r from a gravitating mass center in the equatorial plane (θ = π/2) in terms of the Schwarzschild coordinates as ct = r(dθ/dt), then the metric divided by (dt)² immediately gives:

Thus, we again find that the "velocity of light" is reduced in a region with a strong gravitational field, but this speed is the square root of the radial speed at the same point, and to the first order in m/r this is the same as Einstein's 1911 formula, although it is understood now to signify just the tangential speed. This illustrates the fact that the general theory doesn't lead to a simple scalar field of c values. The effects of gravitation can only be accurately represented by a tensor field.

### Re: Scalar Variance of Speed of Light

Posted: Fri Dec 31, 2010 5:20 am
You can also find another online solution: Variable Speed of Light in General Relativity

http://www.d1heidorn.homepage.t-online.de/Physik/VSL/VSL.html wrote:
In two works from 1907 and 1911 Einstein introduces a variable speed of light. Sometimes this is taken as a contradiction to the constancy of the speed of light, which was postulated in the foundation of Special Relativity in 1905. However there is no contradiction at all - even if in the fully developed GR from 1916 there is a variable speed of light. In fact all seeming contradictions vanish, if the difference between local and non-local is taken into account.

...

The speed of light at A results as follows:

* measured by the observer at A:

That means (as stated before): In local inertial frames of reference the speed of light is constant. Local measurements always result in c = 2,99792458 ⋅ 10^8 m/s.

* measured by the observer at B:

The interpretation of variable speed of light as locally observable in Einstein’s 1911 work was preliminary. The meaning of the equation cA = c ⋅ (1916) is different: this equation is non-local, it describes the speed of light at a point distant from the observer in the observers non-local system of co-rdinates.

Of course the observer at A measures no local variability in his local frame of reference. Thus he cannot use the non-local equation for cA for his local speed of light.

The difference between

and

is the factor 2 with the gravitational potential.

### Re: Scalar Variance of Speed of Light

Posted: Tue Mar 15, 2011 3:42 pm
Raef Fanous wrote:
You can also find another online treatment: Variable Speed of Light in General Relativity

http://www.d1heidorn.homepage.t-online.de/Physik/VSL/VSL.html wrote:
In two works from 1907 and 1911 Einstein introduces a variable speed of light. Sometimes this is taken as a contradiction to the constancy of the speed of light, which was postulated in the foundation of Special Relativity in 1905. However there is no contradiction at all - even if in the fully developed GR from 1916 there is a variable speed of light. In fact all seeming contradictions vanish, if the difference between local and non-local is taken into account.

...

The speed of light at A results as follows:

* measured by the observer at A:

That means (as stated before): In local inertial frames of reference the speed of light is constant. Local measurements always result in c = 2,99792458 ⋅ 10^8 m/s.

* measured by the observer at B:

The interpretation of variable speed of light as locally observable in Einstein’s 1911 work was preliminary. The meaning of the equation cA = c ⋅ (1916) is different: this equation is non-local, it describes the speed of light at a point distant from the observer in the observers non-local system of co-rdinates.

Of course the observer at A measures no local variability in his local frame of reference. Thus he cannot use the non-local equation for cA for his local speed of light.

The difference between

and

is the factor 2 with the gravitational potential.

Raef,

I am a high school student working on a project regarding the speed of light and different light waves. I used my father's information to sign up for your site just to make sure I got in Based on what you are saying right here I am led to believe that light waves have the potential to travel at varying speeds based on direction. Is this true? Could you explain this in terms a young student might have a better chance of understanding? Thank you.

### Re: Scalar Variance of Speed of Light

Posted: Wed Mar 16, 2011 5:10 am
Well AstroPop for high-school it is like this: Suppose that you have a clock and a ruler (which is not rotating with respect to stars) and that you are not accelerating (inertial). Locally (where you are) you will always measure the speed of light at 299792.458 km/sec. However in the presence of gravity if I am at a different location than yours then I could measure the speed of light at your location to be any value smaller or greater than 299792.458 km/sec. It depends on where I am and where you are (it depends on locations). So in the presence of gravity the speed of light becomes relative (variable depending on the reference frame of the observer). This does not mean that photons accelerate or decelerate. This is just gravity causing clocks to run slower and rulers to shrink. (This might sound crazy however it is true.)

### Re: Scalar Variance of Speed of Light

Posted: Fri Apr 20, 2012 5:17 am