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enter image description hereImagine that the Sun is not rotating. It also has a tunnel throughout its body exactly through the core. Please disregard any other effect then gravity. From a far away point A (far from the Sun) I would shoot two entangled photons, one through the tunnel, and one a little bit next to the Sun (so just to pass next to the Sun). Point B is so that the line from point A to point B is on a straight (in 3D) line through the tunnel.

Now I am shooting the photons at an angle, such that both entangled photons will pass through both point A and B. (given GR/SR bending effects, the photon#2's path will be bent just right to pass through point B.)

Questions:

  1. Which photon would according to GR/SR arrive at point B first?
  2. My question's answer could sound at first obvious, but considering that the photon travelling inside the tunnel would be affected by gravity too, the answer is not obvious.
  3. What if I use neutrinos or electrons(or anything fast enough with some rest mass ) instead of photons?
Shing
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Árpád Szendrei
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  • What about a nice picture to make it easier to understand? –  Oct 26 '16 at 19:35
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    Why is it relevant that the photons are entangled? – tparker Oct 26 '16 at 23:28
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    https://en.wikipedia.org/wiki/Shapiro_delay – Count Iblis Oct 27 '16 at 00:54
  • I understand the shapiro delay I believe. Both photons have to travel at speed c. In our 3D world, the path around is longer in length. So if we would exist in 3D the photon around would arrive later. But we exist in 4D, and a cornerstone to this question would be, since the one traveling through the tunnel is closer to the center of gravity, it has to be affected by time delation? So although its 3D path is shorter, its 4D path would be exactly the same as the path around? – Árpád Szendrei Oct 27 '16 at 16:57
  • Iam not sure if entanglement is important but there were two reasons: #1 I added the entanglement, because I wanted to eliminate any kind of difference in the photons attributes, and the position and time they would be launched, and wanted to keep this equality interactively, so that when the photon travels through the tunnel, its getting red/blueshifted more, and that has to redshift the other photon the same way too I believe. I am not sure if that is important or not in this experiment though to keep their red/blueshifting the same all the time. – Árpád Szendrei Oct 27 '16 at 17:04
  • #2 And entanglement would help I believe to decide easier whether one of them arrives earlier, because to decide if any of them was faster, we would just have to see either one photon arriving, (since the other one's wave function would collapse the same time before that arrives). And the angle would tell us which way it came from. – Árpád Szendrei Oct 27 '16 at 17:05
  • OK for neutrinos (they have rest mass, and must be affected by SR time dilation) I think I understand the answer would be they'd arrive at the same time. The reason is that the neutrino traveling through the tunnel travels a shorter distance in 3D. Its clock though would tick slower due to being closer to the center of gravity. But speed is distance/time. So it would travel a shorter distance in shorter (relative) time. While the one going around travels a longer path, but in longer time (its clock ticks faster). So they would arrive at the same time. – Árpád Szendrei Oct 27 '16 at 17:35
  • And photons? Are they affected by SR time dilation? – Árpád Szendrei Oct 27 '16 at 17:35
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    once the photons interact, i.e. exchange a graviton, with the gravitational field of the sun (and your bending photon does that) the entanglement is broken anyway. This might enlighten you http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters_2017_Jan_1/general_relativity/index.html#Free – anna v Dec 27 '16 at 07:13

3 Answers3

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It seems to me that your central question is: Does the particular lensed path light travels on affect its arrival time? If so, the answer is a resounding yes, with evidence from astronomy! For example, the Twin Quasar (https://en.wikipedia.org/wiki/Twin_Quasar) is a system consisting of two images of the same quasar, which traveled different paths to the observer and hence generate two different images. The interesting part of this is that one of the images is 400 days behind the other! Researchers identify this difference with a path length difference of 1.1 ly. So it seems that the photon with shorter path length (i.e. photon #1) would win.

To anticipate some attacks involving the fact that space is curved around a massive object and so maybe photon #2 wins, I say that the problem never provides dimensions, in particular the vertical separation between points A and B. Take those two points far enough away from each other, and it becomes extremely clear that, at least in this limit, photon 1 wins. At that point the total relative alteration to photon 1's path becomes miniscule, but photon 2 still has to travel in quite a large arc. My argument here is that since the Sun is (hopefully) not (yet) a compact object, but instead has a rather low density, significant curvature will not be seen within its body. Therefore, the same gentle-curvature limit should apply in all reasonable cases.

probably_someone
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I think I have half of an answer for you. Following my old question about the reactionless drive (deleted now), I've worked out a formula for Shapiro delay in a straight line, through a tunnel in the mass.

The delay caused by mass is $t = (\ln(X/R)+1/3)2GM/c^3$ where $X$ is the distance between the point $A$ and the center of mass of the Sun

$R$ is the radius of the Sun

$M$ is the mass of the Sun

$c$ is the speed of light

The delay is from $X$ (in your drawing is $A$) to the center of the mass, so if you want to go to point B you have to take twice the time. After that, you have to get the Shapiro delay around the mass (actually, it is better to start with the path around so you can get the $A$ and $B$ points) and compare it with my formula.

My formula assumes that the density of the mass is constant (it matters when the photon goes inside the mass; when you get around it the density does not matter).

From my experience there is no straight answer to your question - it depends on the density variation of mass and the distance from which you launch the photon. You can have a winning photon either way.

xray0
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Misuser
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Extreme value observations are a good way of making estimates.

If the star is a black hole, the photon passing by laterally clearly wins. If the star is not present, the photon flying at an angle to the observer never arrives.

As Misuser has already stated, it depends on the concrete parameters.

HolgerFiedler
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