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- TL;DR Summary
- In the "Neutron Star" story by Larry Niven, he says the pilot of a spaceship on a hyperbolic orbit leading to a close approach to a neutron star sees light from distant stars blueshifted in all directions. However, this appears to be incorrect. A corrected calculation is given.
In the short story "Neutron Star" by Larry Niven, the narrator, Beowulf Shaeffer, is piloting a spaceship on a hyperbolic orbit that is supposed to make a close approach to a neutron star. He says at one point that he sees incoming light from distant stars blueshifted in all directions, and gives the neutron star's gravity as the reason. Here, we will do some calculations to see if this is actually correct, and will find that it isn't.
First, we need to get some parameters from the data given in the story. The first parameter is that the neutron star's mass is 1.3 solar masses. In geometric units, in which ##G = c = 1##, this is 1.92 kilometers. The second parameter is that the ship's speed relative to the star at closest approach is half the speed of light. (There are actually conflicting statements in the story about this, which we won't attempt to analyze in detail. The half lightspeed value is reasonable and sufficient for what we're going to calculate here.) The third parameter is the distance of closest approach; we will take this to be ##R = 8M##, which is 15.36 kilometers. (Again, there are conflicting statements in the story about this; this value is reasonable and sufficient for this discussion.)
We won't need to calculate the ship's orbit in detail; we can simply make the following observations on which to base simple calculations. First, when the ship first starts to fall towards the neutron star, we can assume that it is falling radially inward, to a good approximation, and at escape velocity. Second, at the point of closest approach to the neutron star, we can assume that the ship's trajectory is exactly transverse, i.e., perpendicular to a radial line through the star's center.
The general method of calculating the observed frequency shift of incoming light from distant stars makes use of the Schwarzschild metric, which I will take to be familiar enough that it need not be written down explicitly here, and consists of two elements: first, we calculate the gravitational blueshift that would be seen by a stationary observer (i.e., an observer "hovering" at rest at constant altitude above the star) at a given radius ##R##; second, we multiply that by a Doppler shift factor due to the ship's speed relative to the stationary observer. The Doppler shift factors we will consider are those for light coming from directly to the front of the ship (where the Doppler factor is a blueshift), directly to the rear of the ship (where the Doppler factor is a redshift of the same magnitude as the front blueshift), and to the side of the ship (where the Doppler factor is a redshift due to transverse Doppler, i.e., the usual SR time dilation due to the ship's speed).
We can capture the above in the following formulas for the factor by which the observed wavelength is changed from the source wavelength:
$$
f_\text{front} = \sqrt{1 - \frac{2M}{R}} \sqrt{\frac{1 - v}{1 + v}}
$$
$$
f_\text{rear} = \sqrt{1 - \frac{2M}{R}} \sqrt{\frac{1 + v}{1 - v}}
$$
$$
f_\text{side} = \sqrt{1 - \frac{2M}{R}} \frac{1}{\sqrt{1 - v^2}}
$$
Here ##f < 1## means a blueshift, and ##f > 1## means a redshift.
All we need now is to apply these formulas to the two cases described above.
Case 1: Radial infall at escape velocity. Here ##v = \sqrt{2M / R}##, which is a nice convenient value because its square appears in the first factor in all three formulas above. So the three formulas become:
$$
f_\text{front} = \sqrt{1 - v^2 \frac{1 - v}{1 + v}} = 1 - v
$$
$$
f_\text{rear} = \sqrt{1 - v^2 \frac{1 + v}{1 - v}} = 1 + v
$$
$$
f_\text{side} = \frac{\sqrt{1 - v^2}}{\sqrt{1 - v^2}} = 1
$$
So the light coming from the front is blueshifted, the light coming from the rear is redshifted, and the light coming from the side is not shifted at all (the two effects exactly cancel).
Case 2: Transverse motion at closest approach. Here ##v = 1/2## and ##R = 8M##, and our three formulas become:
$$
f_\text{front} = \sqrt{1 - \frac{1}{4}} \sqrt{\frac{2 - 1}{2 + 1}} = \frac{1}{2}
$$
$$
f_\text{rear} = \sqrt{1 - \frac{1}{4}} \sqrt{\frac{2 + 1}{2 - 1}} = \frac{3}{2}
$$
$$
f_\text{side} = \frac{\sqrt{1 - \frac{1}{4}}}{\sqrt{1 - \frac{1}{4}}} = 1
$$
Here the results are the same qualitatively as above: light coming from the front is blueshifted, light coming from the rear is redshifted, and light coming from the side (which here means light coming radially inward) is not shifted at all. In fact, the numerical values we obtained are the same as the ones we would get from case 1 if we plugged in ##v = 1/2##. There is no deep significance to this; it is just a fortuitous consequence of choosing ##R = 8M## as the closest approach distance. However, it does imply that the qualitative behavior we have described should remain the same during the entire infall process.
I invite any comments from readers, particularly if anyone sees any issues with the above calculations.
First, we need to get some parameters from the data given in the story. The first parameter is that the neutron star's mass is 1.3 solar masses. In geometric units, in which ##G = c = 1##, this is 1.92 kilometers. The second parameter is that the ship's speed relative to the star at closest approach is half the speed of light. (There are actually conflicting statements in the story about this, which we won't attempt to analyze in detail. The half lightspeed value is reasonable and sufficient for what we're going to calculate here.) The third parameter is the distance of closest approach; we will take this to be ##R = 8M##, which is 15.36 kilometers. (Again, there are conflicting statements in the story about this; this value is reasonable and sufficient for this discussion.)
We won't need to calculate the ship's orbit in detail; we can simply make the following observations on which to base simple calculations. First, when the ship first starts to fall towards the neutron star, we can assume that it is falling radially inward, to a good approximation, and at escape velocity. Second, at the point of closest approach to the neutron star, we can assume that the ship's trajectory is exactly transverse, i.e., perpendicular to a radial line through the star's center.
The general method of calculating the observed frequency shift of incoming light from distant stars makes use of the Schwarzschild metric, which I will take to be familiar enough that it need not be written down explicitly here, and consists of two elements: first, we calculate the gravitational blueshift that would be seen by a stationary observer (i.e., an observer "hovering" at rest at constant altitude above the star) at a given radius ##R##; second, we multiply that by a Doppler shift factor due to the ship's speed relative to the stationary observer. The Doppler shift factors we will consider are those for light coming from directly to the front of the ship (where the Doppler factor is a blueshift), directly to the rear of the ship (where the Doppler factor is a redshift of the same magnitude as the front blueshift), and to the side of the ship (where the Doppler factor is a redshift due to transverse Doppler, i.e., the usual SR time dilation due to the ship's speed).
We can capture the above in the following formulas for the factor by which the observed wavelength is changed from the source wavelength:
$$
f_\text{front} = \sqrt{1 - \frac{2M}{R}} \sqrt{\frac{1 - v}{1 + v}}
$$
$$
f_\text{rear} = \sqrt{1 - \frac{2M}{R}} \sqrt{\frac{1 + v}{1 - v}}
$$
$$
f_\text{side} = \sqrt{1 - \frac{2M}{R}} \frac{1}{\sqrt{1 - v^2}}
$$
Here ##f < 1## means a blueshift, and ##f > 1## means a redshift.
All we need now is to apply these formulas to the two cases described above.
Case 1: Radial infall at escape velocity. Here ##v = \sqrt{2M / R}##, which is a nice convenient value because its square appears in the first factor in all three formulas above. So the three formulas become:
$$
f_\text{front} = \sqrt{1 - v^2 \frac{1 - v}{1 + v}} = 1 - v
$$
$$
f_\text{rear} = \sqrt{1 - v^2 \frac{1 + v}{1 - v}} = 1 + v
$$
$$
f_\text{side} = \frac{\sqrt{1 - v^2}}{\sqrt{1 - v^2}} = 1
$$
So the light coming from the front is blueshifted, the light coming from the rear is redshifted, and the light coming from the side is not shifted at all (the two effects exactly cancel).
Case 2: Transverse motion at closest approach. Here ##v = 1/2## and ##R = 8M##, and our three formulas become:
$$
f_\text{front} = \sqrt{1 - \frac{1}{4}} \sqrt{\frac{2 - 1}{2 + 1}} = \frac{1}{2}
$$
$$
f_\text{rear} = \sqrt{1 - \frac{1}{4}} \sqrt{\frac{2 + 1}{2 - 1}} = \frac{3}{2}
$$
$$
f_\text{side} = \frac{\sqrt{1 - \frac{1}{4}}}{\sqrt{1 - \frac{1}{4}}} = 1
$$
Here the results are the same qualitatively as above: light coming from the front is blueshifted, light coming from the rear is redshifted, and light coming from the side (which here means light coming radially inward) is not shifted at all. In fact, the numerical values we obtained are the same as the ones we would get from case 1 if we plugged in ##v = 1/2##. There is no deep significance to this; it is just a fortuitous consequence of choosing ##R = 8M## as the closest approach distance. However, it does imply that the qualitative behavior we have described should remain the same during the entire infall process.
I invite any comments from readers, particularly if anyone sees any issues with the above calculations.