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This comes from a discussion forum, where a friend of mine asked the following:

We can see objects in space billion of light years away, right? I started wondering about that.

If you take 2 objects in space, the other should be able to see the other no matter what angle in degrees you position it at. That would almost seem to imply that light is being sent out in an infinite number of degrees/angles from the source. But that cannot be true because energy cannot be infinite.

If the observer goes out far enough from the source, would there be gaps in the light? Could you pick a viewing angular degree (of extremely high angular precision) where there's no light?

I'm actually quite curious about this question myself and really have no answer, and the discussion hasn't really yielded a satisfying answer. So I figured I would bring it here on behalf of my friend and to sate my own curiosity.

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3 Answers 3

up vote 6 down vote accepted

Well, and in the two wave answers nobody considers the quantum mechanical picture.

Photons compose these classical waves.

Photons carry momentum equal to energy if we set c=1, p=hnu

When the distance from the source becomes large enough that individual photons can be counted in a counter, there will be a point where gaps will exist and no photons will be counted.

Taking this solution from Yahoo answers one can see that for a given wavelength and intensity, a delta(x) between two detected photons can be found where individual photons will be very rare.

So the answer depends on the original intensity,which falls with the distance as 1/r**2 ,the distance, the wavelength observed and the time available for the detection. If one waited an infinite time, the answer "there is no gap would hold probabilistically. For any reasonable delta(t) there will be gaps that cannot be predicted in r,theta,phi because they will depend on the probability function of the photons.

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Indeed, the Hubble Ultra Deep Field received only a photon per minute from the weakest (and farthest) objects see: hubblesite.org/newscenter/archive/releases/2004/07 –  Frédéric Grosshans Jun 22 '11 at 19:03

If the observer goes out far enough from the source, would there be gaps in the light? Could you pick a viewing angular degree (of extremely high angular precision) where there's no light?

No, you can't. Light is a wave, and accordingly it spreads out in all directions and fills all of space (if you assume there are no other objects that might cast shadows). So no matter how far out you go, you will never fully escape the light wave.

Of course, the intensity of the light (energy per unit detector area per unit time) does become smaller as you move further away. So for any given intensity $I_0 > 0$, you can put your detector far enough away that it will detect an intensity less than $I_0$. But the intensity never actually drops to zero.

That would almost seem to imply that light is being sent out in an infinite number of degrees/angles from the source. But that cannot be true because energy cannot be infinite.

In short, your friend's misconception is that it would require an infinite amount of energy for light to be sent out in an infinite number of directions. That is not the case.

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You're looking at this the wrong way. The total energy being sent out is finite. And the total energy contained within a solid angle $d\Omega$ is proportional to $d\Omega$. So the energy contained in a very tiny solid angle gets tinier as the angle is decreased and approaches zero as the angle approaches zero. In other words, the smaller a cone you count photons in, the lesser photons you'll find. This ensures that the total energy (obtained by integrating over all solid angles) is finite as we started out.

This is how almost any integral you'll find in physics works.

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