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Does the speed of light slow down near heavier objects? if yes, Explain

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yes dear specially when it passes close to a very huge heavy object like sun or close to a star ..
The space fibers, as per the Einstein, is curved around the massive objects. so every thing passing near a massive object get curved as per the curvature around the object. and this is why an object revolves around other. it only changes its direction

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Other answers:

the speed doesn't change, but space is curved around massive objects, so it has to travel a longer path and therefore takes more time (which has the net effect of seeming to 'slow' it)
The speed of light will change only if the Refractive Index of the medium it's travelling in, changes. Near massive objects , I think the previous explanation is good.
agreed, speed of light remains constant till there exists no change in medium
the speed of light is actually constant in any medium, it's just that due to scattering the path it must take through different materials is very different in length.
sorry, this concept is quite alien to me and i must say, speed changes if medium is changed
If the speed didn't change we would never have been able to detect neutrinos. (Cheronkov radiation)
we must thank @kappa007 for this piece of wisdom
also, then R.I. of all media would be 1.
oh aye? well what causes light to 'slow down' in a material? magic? or is it all that gravity from all those protons and neutrons
@Algebraic! if you've got concept then only entertain else don't mislead or misguide in a funny manner
Still, you can't compare that with gravitational lensing. If we dig down to the basics of refraction.. I think the slowing down maybe because of interaction between light's electromagnetic field and the medium's e.m. field.
no kidding. absorption and re-emission. ie scattering
can you explain please.
read the wiki page or something.
The speed of light does not decrease. Anything moving at the speed of light stays at the speed of light and cannot be slowed down (unless you want to talk about "tachyons" that may or may not exist)..... When light passes through a medium the particles of light, photons, do not change speed. The wavelength of light though, does change. There is a simple relation that you can follow (wavelength in medium) = (wavelength in vacuum) / (index of refraction of medium) What is really slowing the photons down is their interaction with the particles in the medium, which shortens the wavelength. The photons themselves though still are moving at the speed of light, it is just this interaction that causes the dilation. so i don't think scattering makes any sense, it's just a blunder. or may be i am wrong
hope this help you
I'll say that, 'c' is indeed constant but not exactly the speed of light, as, the basic R.I. definition is based on such speeds only. During interaction with particles , light tends to behave as a particle. So, I think the photon-particle interaction thing must be the reason.
speed always remain constant, it's just interaction that acts as obstacle slowing down the speed learn all about it here.
nice to know, people believe in wikipedia
there's no term = 'speed' in the page suggested. I

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