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Relativity/Deflection of light

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QUESTION: I tried to do the calculation of the deflection of a beam of light passing close to the Sun
I used the formula 4*G*M / c^2*R putting all the various values ​​(found online)
assuming that M was the mass of the sun and the light beam that was deflected to pass almost touching the sun and then the R of the formula was the radius of the sun. Also in the network was also the result of what was to come from the calculation: 0.000004245 radians corresponding to 0.875 seconds of arc.
Now as I said I did the calculation, and in fact I got the result 0.000004245 correct as I found on the net.
Question: this number 0.000004245 radians are not because I knew that they were radiant, but only because I had read before.
From the formula 4GM / c^2R, how do you figure out that the number that comes as a result are then radians?
Also because in fact this number is dimensionless, so why then say that they are radians?

ANSWER: Congratulations, Nello, on your correct calculation!

Your question is easy to answer.  You know that there are 360 degrees of arc in one circle.  Why? because that defines "degrees of arc."  It is not mysterious.

Similarly, one "radian" is defined as the angle of a part of a circle from the center such that the length of the arc is equal to the radius of the circle. Since the full circumference of a circle is 2*pi, and the angle defined by the full circumference is also 360 degrees, 360 degrees = 2*pi radians, and 1 radian = 360 degrees / 2*pi.

So 1 radian = 57.3 degrees approximately.

Keep up the good work!



---------- FOLLOW-UP ----------

QUESTION: Thank you for your reply.........
Most probably my question was not clear (also because i don't speak very well English)
I try to explain better what i want know :you gave me a precise definition of what is the radian.
But my question is : i have a formula 4GM / c^2R , from this formula come out a certain number (0.000004245) which is a pure number .
If is a pure number for me it could be degrees, or seconds of arc or potatoes.....how i can say no, this number are radians ?

Answer
The fact that a number has no units does not imply that it is undefined. 57 degrees is about 1 radian with units cm/cm or m/m or miles/miles or no units at all. It is still a definite angle.

You know that a formula is in radians because the author's derivation assumes so. If the unit was assumed to be in degrees, the formula would contain a factor of about 57.

What do you think about pi?  Pi has no units, but it has a definite value. If you want to use it as an angle in radians in a paper you are writing, you just say so.

Does this help?  

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Uncle Ben

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I can answer questions regarding Einstein's Theory of Relativity, particularly in Special Relativity. I will not answer homework questions or mathematical problems that require special symbols.

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