Time dialation effect on observed speed of light

[rcgldr]

Assume observer is located on the surface of the Earth with a small amount of time dilation due to gravity. There are two beams of light moving approximately parallel to the surface of the Earth at some moment in time and the reflection of the beam off some objects back to the observer is being used to determine the speed of light. One of the beams is very close to the observer and affected by the time dilation effect of gravity, while the other beam is far from the Earth (at a known distance), with almost no time dilation effect due to gravity. So is the observed speed of light the same for both beams?

Or for an extreme case, imaging the distant beam is approaching the event horizon of a black hole, does it's speed appear to be slowing (perhaps approaching zero?) as it approaches the event horizon to the observer back on earth?

Or is the requirment for an observered constant speed of light require that the light have the same time dilation effects as the observer (both experience the same gravitational intensity)?

 

[Matterwave]

In general relativity, the notion of an observed speed is a local notion. A distant object's speed is not very well defined because there are may be no global coordinate systems available.

Relativity says, all local measurements of the speed of light will yield c, it does not make such a claim for any global speed of light, if any such notion even exists.

 

[rcgldr]

Thanks, that's what I thought, but I one of my kids was asking about this.