3 ms·

I guess the fall-off is even worse. At the beginning the intensity falls off as 1/r^2, but eventually the intensity becomes so small that you're talking about i

by pontus 6y ago

I guess the fall-off is even worse. At the beginning the intensity falls off as 1/r^2, but eventually the intensity becomes so small that you're talking about individual photons. At some point the intensity will then fall from a single photon to zero. So, after some critical distance the intensity will actually drop to zero. More formally, each star emits some amount of power in each frequency band: P(f) so that \int_0^\infty P(f) df = P_total. For each frequency then, we have a total of P(f)/(hf) photon emitted per second. The total number of photons emitted per second by the star is then \int_0^\infty df P(f)/hf which is a finite number. The total number of photons received per unit area a distance r away from the star would then be \frac{1}{4\pi r^2} \int_0^\infty df P(f)/hf If your detector has an area A (e.g. your retina or some other device), you'd expect to see \frac{A}{4\pi r^2} \int_0^\infty df P(f)/hf photons per second from the star. As r gets really large, you'd see this drop arbitrarily low. Conversely, the amount of time you'd need to wait to see a single photon from that star then grows, making the star dark.

pdonis 6y ago
> after some critical distance the intensity will actually drop to zero. No, it won't. If you're going to use a quantum model of light (which you have to to use the concept of "photon"), then you have to use the quantum interpretation of "intensity". The quantum interpretation of "intensity" is the probability of detecting a photon; and this is a continuous quantity which can get smaller and smaller indefinitely without ever dropping to zero.
- pontus 6y ago
  The probability can then get arbitrarily small, meaning that the expected amount of time needed before the probability of having observed a photon would get progressively larger. My argument above is semi-classical, but it shouldn't change with a full quantum mechanical approach.
  - pdonis 6y ago
    > The probability can then get arbitrarily small, meaning that the expected amount of time needed before the probability of having observed a photon would get progressively larger. Yes, but the probability is never zero, and the expected time is never infinite. So saying "the intensity drops to zero" is never correct.
    - pontus 6y ago
      The point is not that the probability needs to hit zero, it's that it's not correct to say that you receive a quarter of the power as you move twice as far from the source. It's still true that the expected number of photons per second drops by a factor of 4, but it can drop so far as to render the source dark for an appreciable amount of time. The paradox claims that the sky should appear bright, which I take to mean that a detector should be receiving light from each point in the sky at each moment in time. It does not say that the detector will receive light from each part of the sky at some point, but that you may need to wait a million years before a particular point flickers and that, even then, there's nothing that guarantees that all points will flicker at the same time.
      - pdonis 6y ago
        > it can drop so far as to render the source dark for an appreciable amount of time Ah, I see what you mean: yes, the intensity will be 1/4, but because of quantization, you now have to draw a distinction between the time-averaged power (which behaves like the power does in the classical case--more precisely, this would be the expectation value of the power in the quantum case) and the actual power at a given time, which can vary from the average (even to the point of being zero).
        pontus 6y ago
        Yeah, exactly
      - pdonis 6y ago
        > The paradox claims that the sky should appear bright, which I take to mean that a detector should be receiving light from each point in the sky at each moment in time. I don't think this is required for the paradox. All that is required is that the average flux of radiation received from the sky as a whole should be constant, and equal, roughly speaking, to the flux corresponding to the surface brightness of a star. That will still be true, under the specified conditions of the paradox (a universe in steady state and infinitely old) even if quantization is taken into account.

by pontus 6y ago

pdonis 6y ago

> after some critical distance the intensity will actually drop to zero. No, it won't. If you're going to use a quantum model of light (which you have to to use the concept of "photon"), then you have to use the quantum interpretation of "intensity". The quantum interpretation of "intensity" is the probability of detecting a photon; and this is a continuous quantity which can get smaller and smaller indefinitely without ever dropping to zero.

pontus 6y ago
The probability can then get arbitrarily small, meaning that the expected amount of time needed before the probability of having observed a photon would get progressively larger. My argument above is semi-classical, but it shouldn't change with a full quantum mechanical approach.
- pdonis 6y ago
  > The probability can then get arbitrarily small, meaning that the expected amount of time needed before the probability of having observed a photon would get progressively larger. Yes, but the probability is never zero, and the expected time is never infinite. So saying "the intensity drops to zero" is never correct.
  - pontus 6y ago
    The point is not that the probability needs to hit zero, it's that it's not correct to say that you receive a quarter of the power as you move twice as far from the source. It's still true that the expected number of photons per second drops by a factor of 4, but it can drop so far as to render the source dark for an appreciable amount of time. The paradox claims that the sky should appear bright, which I take to mean that a detector should be receiving light from each point in the sky at each moment in time. It does not say that the detector will receive light from each part of the sky at some point, but that you may need to wait a million years before a particular point flickers and that, even then, there's nothing that guarantees that all points will flicker at the same time.
    - pdonis 6y ago
      > it can drop so far as to render the source dark for an appreciable amount of time Ah, I see what you mean: yes, the intensity will be 1/4, but because of quantization, you now have to draw a distinction between the time-averaged power (which behaves like the power does in the classical case--more precisely, this would be the expectation value of the power in the quantum case) and the actual power at a given time, which can vary from the average (even to the point of being zero).
      - pontus 6y ago
        Yeah, exactly
    - pdonis 6y ago
      > The paradox claims that the sky should appear bright, which I take to mean that a detector should be receiving light from each point in the sky at each moment in time. I don't think this is required for the paradox. All that is required is that the average flux of radiation received from the sky as a whole should be constant, and equal, roughly speaking, to the flux corresponding to the surface brightness of a star. That will still be true, under the specified conditions of the paradox (a universe in steady state and infinitely old) even if quantization is taken into account.