
Re: Einstein's factor of 2 in starlight deflection
Posted:
Aug 22, 2012 11:50 AM


On Saturday, 20 August 2011 18:45:57 UTC+1, H.Jones wrote: > Einstein's original formula for deflection of light skimming the Sun's > surface breaks down to 2GM/Rc^2. This is merely the Schwarzschild > radius divided by current radius of the Sun. If all things are > considered to be happening at the rate of c and at the Schwarzschild > limit the reckoning gets easier. Then R/R=1, that is one radian, in > terms of angle and one radius in terms of length or distance. You > could say it is an equivalence of distance measured in terms of > radius. All that considered look at it this way: > If we consider that at the Schwarzschild limit, whatever the star's > mass or radius, then in that particular frame the star's mass becomes > the local timescale mass and its diameter becomes the local time > frame. Take the case of the Sun at ,say, 2x10^30kg, radius at > 2.9690906x10^3m and thus its local c at 2r, 5.9381812x10^3m. The time > unit here,then, will be 1/5.0485569x10^4 of a second. Looking at this > numerical principle, consider the following. > The formula for free fall, distance travelled, is (gt^2)/2, where g is > gravitational acceleration and t is time in local time units. If g=c > then we could look at the formula as (ct^2)/2. The formula for g is > GM/r^2, so we could look at it like this: {(GM/r^2)t^2}/2. If we > consider time to be one single unit there is no need to include it. So > our formula becomes (GM/r^2)/2. But we need to put everything in > terms of c and radius is equivalent to c/2. So our formula can be > adjusted again to: (4GM/c^2)/2 which breaks back down to 2GM/c^2. > Therefore, the formula 2GM/c^2 means the same thing as (gt^2)/2 where > g=c and refers to length or distance not angle.
Look at it another way, 2GM/c^2 is the same thing as (gt^2)/2 when g is equal to c. What must be considered when dealing with gravitational fields is that they are in two halves or hemispheres. Most classical equations deal in one hemisphere, that means radius. But the Schwarzschild radius of a timescale mass star does not equal the time unit, it is the diameter that is equal to the time unit. The time scale mass of the kilogram/second has a Schwarzschild radius equal to c/2 and a diameter equal to one light second. So in the starlight deflection procedure the time taken to reach the grazing incidence is one hemisphere of the gravitational field. This condenses, as far as the classical formula goes, to the equivalence, and remember only the equivalence, of one Schwarzschild radius of the system's star. After the grazing incidence the escape journey out travels through another hemisphere, again, equivalent to one Schwarzshild radius. This adds up to two radii. The whole trip is about a two radii or hemisphere journey. The formula for the Schwarzschild diameter is twice that for the radius and is therefore equal to 4GM/c^2. The factor of two is there already.

