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If we observe light that originates X light years away, but it passes between black holes X/2 light years away, will it be normal or red shifted or blue shifted? What if the black holes were X/4 or 3X/4 light years away?

I have always wondered if the apparent red shift of distant galaxies must be due to relative velocity with us.

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Why is it important that there be two or more black holes? Is there something you expect to happen in those cases that wouldn't affect light that passes near a single, isolated black hole? – David Z Jan 14 '11 at 2:15

There are two ways that light frequencies can be shifted. One is if the source of the light is moving toward or away from the observer. The other is if the light has fallen into, or climbed out of a gravitational potential on its way.

If light passes near a black hole, it will be blue-shifted since it is falling into a gravitational well. However, if the light makes it back out of the gravitational well and makes it to Earth, it will have to climb out of the same well, getting red-shifted an equal amount. So, on average there is no effect from having passed near the black hole.

The energy will be the same, but the direction may change since the black holes can act like a lens.

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there are three ways that light frequencies can be shifted. Third is due to the space expansion. – voix Jan 14 '11 at 17:19

If you assume that two equally heavy black holes are at rest relatively to one another (and you may need some arrangement to prevent them from attracting one another if they're too close), then the light emitted at frequency $f$ at a distance $R$ from the first black hole's center will be observed as $f$ at the same distance $R$ from the second black hole's center.

It's because the light will be redshifted and then blueshifted by the same factor - assuming that the source and the observer are sitting at fixed distances from the black hole centers (and they may need jets to be saved from the attraction towards the black holes). Only the ratios of $g_{00}$ influences the frequency ratio.

The red shift is also caused by the Doppler effect, the relative motion of the source and the observer. The Doppler effect exists even in the flat space, in special relativity, i.e. in the absence of gravity.

We observe the distant galaxies to be "redder" than the nearby galaxies. It means that the same kinds of sources - the same stars - produce "redder" light if the stars sit in a distant galaxy. Even if you assumed that a gravitational red shift from the stars etc. influences the light coming from the other galaxies, the red shift (expressed as a ratio of frequencies) would still have to be equal for the nearby and distance galaxies.

It's not equal which is why the hypothesis that all the "Hubble" red shift is due to the gravity of the massive objects themselves is ruled out. In reality, we know that it comes from the relative motion. But it is true that in general relativity, one can't "qualitatively" distinguish these two causes of red shift. In particular, the relative motion of galaxies away from each other may also be interpreted as a gravitational field - as the increasing size of the component of the metric tensor $g_{xx}$ in the direction $x$ connecting them, using coordinates in which both galaxies sit at time-independent values of $x$.

When these "unusual coordinates" and interpretations are allowed, it is true that all red shift in general relativity may be interpreted as a (generalized) gravitational red shift.

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As the other posts say, there will be no net redshift. But there will be a net time delay, called the Shapiro Delay, which should be observable if the light source is variable, and multiple images are resolved.

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