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If I live on a planet that is heavy enough, would the CMB get blue shifted enough in the atmosphere of this heavy planet, due to gravitational blue-shifting, that the CMB would be in the visible spectrum?

How heavy does this planet need to be relative to Earth?

Community
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bubakazouba
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    Did you try to do a back-of-an-envelope-calculation? – Qmechanic Mar 26 '21 at 07:13
  • I like the idea of seeing the cosmic radiation with unaided human eye, but the gravitational method is less practical than the Doppler method. Your best bet is to jump aboard a rocket and accelerate up to $\gamma = 1000$. – Andrew Steane Mar 27 '21 at 12:42

3 Answers3

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The cosmic background radiation has a wavelength about 2000 times longer than visible light. So you would need to be sitting deep in a gravitational well such that local time progresses 2000 times more slowly than distant time. Gravitational time dilation is given by the formula $\sqrt{1-r_\mathrm{S}/r}$ where $r$ is the distance from the center of the (spherical) mass and $r_\mathrm{S} = 2GM/c^2$ is its Schwarzchild radius.

For a time-dilation factor of 1/2000, one would have to be at a radius that exceeds the Schwarzchild radius by one part in 4,000,000. In other words, the object could only be a black hole and you would be barely outside its event horizon. Assuming you are somehow held stationary above the event horizon, you would not survive the gravitational field. The actual mass of the black hole is not specified, it could be large or small.

Buzz
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Roger Wood
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    The redshift $z$ of the CMB is ~1089 so you "only" need half the time dilation specified above to blueshift the CMB back to the orange glow shown here. Of course, that doesn't make too much difference. According to this answer, a maximally compressed neutron star has $\frac{r_s}{r}\le 0.71$, which equates to $\gamma\approx 1.85$ – PM 2Ring Mar 26 '21 at 07:43
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    I thought only tidal forces were survivable or not? Although in this case, you may not survive the acceleration required to stop you dropping into the black hole. – user253751 Mar 26 '21 at 12:17
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    You don't need to be held stationary; "pretend" you're in a stable orbit. Yes, I know you'll have to be a 2-dimensional creature projected onto the sphere of the orbital path to avoid tidal force disaster. – Carl Witthoft Mar 26 '21 at 14:12
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    What you're decribing kind of feels like lockdown, to be honest. – Strawberry Mar 26 '21 at 16:10
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    You're doing this the hard way. Put the planet in a star system just outside a center-of-the-galaxy black hole. – Joshua Mar 26 '21 at 18:48
  • @WaterMolecule how do you make subscripts? – Roger Wood Mar 26 '21 at 22:45
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    @RogerWood Subscripts are made with an underscore. $r_s$ renders as $r_s$. You can find a tutorial on basic MathJax syntax here. – J. Murray Mar 27 '21 at 02:16
  • @J.Murray thank you! Let me see if it works in comments too: $sub_script$ and $super^script$ - Ah! I see one needs to do it for each letter $ex_{ample}$ Got it! thanks – Roger Wood Mar 27 '21 at 02:51
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    @RogerWood If you want more than a single character, you must use curly braces: $sub_{script}$ $\rightarrow sub_{script}$. – J. Murray Mar 27 '21 at 02:52
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    Here's a brief beginner's guide to MathJax: https://math.meta.stackexchange.com/q/33179/207316 There's a bunch of useful MathJax links here: https://math.meta.stackexchange.com/q/1773/207316 including Carol Fisher's Alphabetical List of TEX Commands available in MathJax, which gives examples of all MathJax commands. – PM 2Ring Mar 27 '21 at 04:29
  • @Joshua Good idea, but unfortunately it's unlikely that any real BH has enough spin to get that much time dilation in a stable orbit. Kip Thorne ignored that inconvenience in his calculations for Interstellar. See https://physics.stackexchange.com/a/146683/123208 for details. It seems that the best we can get in a stable orbit is a time dilation factor of $\gamma\approx10$. You could get a much higher $\gamma$ by hovering just above the event horizon of a SMBH, but nothing could cope with the insanely high acceleration. – PM 2Ring Mar 27 '21 at 05:03
  • @user253751 Just to reply to remark about tidal forces: if you are standing still on a planet then you experience the upward force from the ground (e.g. electromagnetic force) and this pushes your feet more than your head so it squashes you. To avoid this you need to be in free fall, only then is the tidal force the dominant force. – Andrew Steane Mar 27 '21 at 14:26
  • @PM2Ring: So I punched in 4,000,000,000 (astronomical units) * c*c / 2 / G into wolfram alpha to get the size of a black hole big enough to have a star system in that orbit and got 20x the mass of the local supercluster. Trying to put the division by the mass of the milky way into the equation was enough for wolfram alpha to go off into the weeds. It's going to be the mass of a few hundred spiral galaxies, which is indeed huge but perhaps not completely outrageous for a quasar remnant. – Joshua Mar 27 '21 at 19:48
  • The idea being the star system is nowhere near its own roche limit while orbiting the black hole near the lowest stable orbit. – Joshua Mar 27 '21 at 19:49
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The extreme time dilation proposed can only come if you were on a planet that was orbiting a black hole.

The time dilation for a circular orbit around a Schwarzschild black hole is given by $(1 - 3r_s/2)^{1/2}$, where $r_s$ is the Schwarzschild radius.

Thus arbitrarily large time dilations can be achieved as an orbit approaches $1.5 r_s$, and the orbital speed approaches the speed of light.

Such orbits are not stable around a Schwarzschild black hole and in fact there would be no stable circular orbit that could give you the time dilation required. However, stable prograde orbits at $1.5r_s$ could be stable around a spinning black hole and this is the situation envisaged for "Miller's World" in the film "Interstellar". However, the stable orbit needs to be even closer to the event horizon and the black hole needs to be spinning at almost it's maximal value to get to the required time dilation factor.

There is no issue with the "gravity" crushing you, because an orbiting object is in free fall. There could be an issue with tidal forces, but these are not necessarily so extreme if the orbit is around a supermassive black hole exceeding $10^8M_\odot$.

Note that this just considers time dilation and not the Doppler shift. The specific intensity and blueshift of the CMB would be highly beamed in the direction of orbital motion. This additional blueshift means that the orbit need not be quite so extreme to see a spot of visible CMB (though the orbital speed would need to be highly relativistic)

ProfRob
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Going by Buchdahl's Theorem.

If the as the radius of a body approaches $R=\frac{9R_s}{8}$ The pressure inside diverges to infinity.

Since the formula for gravitational Time dilation is $\sqrt{1-r_\mathrm{S}/r}$ This gives a maximum of $\sqrt{1-8/9}=1/3$

In comparisoon, the CMB has wavelengths 2000 longer then visible light.

blademan9999
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