Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: LeeE on 12/09/2010 12:44:30
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The formula for an approximation of orbital velocity around a mass M (where the mass of the orbiting 'body' is negligible) is:
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2F6%2F8%2F9%2F689f11381e4a870a9e229a4d1c1fef31.png&hash=4ce2728eaa89a115c971828bd86c05e6)
ref: http://en.wikipedia.org/wiki/Orbital_velocity (http://en.wikipedia.org/wiki/Orbital_velocity)
...and the formula for the Photon Sphere radius is:
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2F0%2F0%2F4%2F004bc9f2fab3b3515b1af9206858c465.png&hash=f4d3cfe16a5ad17305e99d91d29782c7)
ref: http://en.wikipedia.org/wiki/Photon_sphere (http://en.wikipedia.org/wiki/Photon_sphere)
...and finally, the formula for the Schwarzchild Radius (of a non-rotating BH) is:
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2F1%2F3%2F6%2F1367343b8711a257d90f36e56cdfa773.png&hash=620bdb7639ae0dae690c66ddd307e5af)
ref: http://en.wikipedia.org/wiki/Schwarzschild_radius (http://en.wikipedia.org/wiki/Schwarzschild_radius)
Where (using mks units):
G = the Gravitational constant = 6.674 x 10-11
M/m = the mass of the orbited body (we'll use the mass of the Earth here, just for fun) = 5.9736 × 1024
c = the speed of light = 299792458
Using these values, the Schwarzchild Radius for an Earth sized BH is: 0.0088718 (m) i.e. a little under 9 mm
...and using this as the radius to find the orbital velocity we get 211985280 (m/s), which is < 'c'
However, the radius of the Photon sphere, where the orbital velocity must be 'c', turns out to be: 0.01330768 (m), which is well outside the Schwarzchild radius.
So can anyone explain why a negligible mass object orbiting a BH just outside its event horizon will be travelling more slowly than the light in the Photon Sphere, which is orbiting it quite a bit further away?
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First, the concept of a "photon sphere" is the most speculative nature of Black Hole mechannics. In fact the wiki page that you quote is asking for more references and proof of the photon sphere concept.
For example the Event Horizon defines the "non rotating" Spheroid Black Hole and the Schwarzschild Radius (r = 2mG/c²)
Modeled around this Event Horizon an additional Roy Kerr rotating Oblate Spheroid Black Hole that is rotating at the same Schwarzschild Radius (r = 2mG/c²). The escape velocity for this Kerr rotating orbit is equal to the speed of light. The rate at which spacetime or space is force to rotate in orbit dragging space around the black hole is equal to the speed of light reduced by the square root of 2.(v = c/sqrt(2)).
At some distance (d) away from the rotating black hole and far out into space the (d > r) the typical Newtonian orbital velocity applies (v = sqrt(mG/d)).
Now, the "Photon Spheroid" is theorized to exist at a distance of either three times Schwarzschild Radius (rp = 3r = 6mG/c²) or three halves the Schwarzschild Radius (rp = 3r/2 = 3mG/c²). This article is stating that it is one and a half times the Schwarzschild Radius.
Next, the "Photon Spheroid" does not obey the typical Newtonian orbital velocity applies(v = sqrt(mG/d)) it always has the fixed speed of light value. It is possible that there are a new series of orbits associated with this "Photon Spheroid" where the velocity of those orbits are diminished with increasing distance from the center. I think that the jury is still out on this?
Now the question becomes what produces this "Photon Spheroid?" And why is it only produced at some distance away from the Schwarzschild Radius? My speculation is that separate Gravitation mechanism is creating this photon spheroid pressure.
As Black Hole mechanics become more and more popular, I believe you will see different interpretations for this phenomenon.
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You should take in account relativistic effects for the orbital velocity. Furthermore, between the photon sphere and the event horizon, a photon going outbound could still escape from the black hole. The Event Horizon should not be defined as the radius at which light crossing it could not escape, but the radius beyond which light cannot escape, whatever is his direction. So if a photon going outbound is created at the event horizon, it could not escape it...
Remarkably, as a photon going outbound perpendicularly to the event horizon is created nearer the event horizon, the less energy any observer in the Universe will measure from it (Doppler effect) until it is 0 at the event horizon. In order not to have a discontinuity at the event horizon, if we could measure the energy of the photons inside the event horizon, we would measure a negative energy... But we can't measure it... Or maybe... [:0]
You have to read this: crazy...
http://en.wikipedia.org/wiki/Exotic_matter
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http://en.wikipedia.org/wiki/Tachyon ---> singularity could be resolved... string theory...
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I wondered about relativistic effects but then if, instead of locating our orbiting body just outside the EH, we put it in orbit at the same radius as the photon sphere then the relativistic conditions for both will be the same, except that the orbiting body will now be travelling even slower than it was when just outside the EH.
If we say that both the orbiting body and the photons in the photon sphere are orbiting the BH due to the curvature of space-time then we end up with two different things in the same orbit but travelling at different speeds, which doesn't make sense.
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look at this: http://www.fourmilab.ch/gravitation/orbits/
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That's a nice page CPT, but it doesn't quite answer everything.
The existence of the photon sphere implies that a negligible mass particle in the same orbit will be travelling at 'c', but this in turn suggests that anything passing closer should be moving > 'c', which it can't (and which raises additional problems re objects approaching the EH). Now although we might expect to see relativistic mass increase, to account for the energy discrepancy, the mass of an orbiting body doesn't make any difference to the orbital velocity, and hence the orbital radius - it just shifts the mutual center of rotation of the two bodies - so it still doesn't seem to add up.
Oh well, something to put on the back-burner for a while.
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I understand what you mean, because i wonder why light would bend so much. I did not find any real useful calculations about the bending of light in a gravitational field. But what i understand is that photons have a very small angular momentum so that can explain why it bend so much. Light has no mass so you cannot use the same equations that those used for matter. The photon sphere is only good for light and not for matter, of what i understand...
Now, i understand why, in theory, there is a finite time for the fall of an object from the event horizon to the singularity. It is not because we are not seeing anything that there is anything. According to General relativity, matter will fall at the event horizon with speed lower than the speed of light so to have a continuity, there is a finite time to the fall to the singularity.
The problem i have is that the speed of light is linked to the fabric of spacetime in order to be constant for any referential frame. So if light is trapped beyond the event horizon, this principle can be false inside the event horizon... The general relativity cannot explain the singularity... Spacetime could be very strange or simply nonexistent inside the event horizon... Maybe it is a negative spacetime... Maybe only the singularity is a negative spacetime...
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I don't think it's possible for stable orbits to exist within the photon sphere, although it is still possible to escape from the black hole by accelerating away from it. A spaceship passing within the photon sphere could escape by turning on its engines, but an asteroid would get sucked in, since it has no mechanism to escape. The event horizon is the place where no matter how much you try to accelerate, there's no way to escape.
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Doesn't it imply that spacetime is infinitely stretched beyond the event horizon? If you have a rocket with infinite power - 1???
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I don't think it's infinitely stretched. It's just that it's so stretched that no matter how much you accelerate, you can only move closer to the center. The only mathematical infinity involved is at the center of the black hole, I believe.
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Here's a couple of other tidbits about it:
http://www.spacetimetravel.org/expeditionsl/erklaerung3.html
http://en.wikipedia.org/wiki/Black_hole#Photon_sphere
Interestingly, the Wikipedia article points out that you could have a neutron star that has a radius in between its Schwarzschild radius and its photon sphere radius so that it would appear to suck in light similar to a black hole.