- #1
johne1618
- 371
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Hi,
I was wondering what happens if one has a photon that is so energetic that it gets trapped in its own gravitational field.
Imagine a mass m orbitting a large mass M in a circular orbit with radius r and with transverse velocity v. By equating the gravitational force on m with its centripetal acceleration we have:
G M m / r^2 = m v^2 / r
Now let velocity v approach the speed of light c. The mass m will be boosted by a Lorentz factor but I think the equation should still hold so that we have:
G M m / r^2 = m c^2 / r
Thus cancelling m we have:
M = (c^2 / G) r
or in terms of Energy E = M c^2 we have
E = (c^4 / G) r (*)
This is the energy that a photon must have to get trapped in its own gravitational field so
that it orbits in a circle with radius r. The quantitiy c^4 / G is also the string tension so maybe this is also a model of a string.
Now let us suppose that the photon has angular momentum hbar. Then we have
r * p = hbar
where p is the linear momentum of the photon.
Now we know that p = E / c for photons so that we have:
r * E / c = hbar
E = hbar * c / r
E = h * c / 2.pi.r
E = h / t (**)
where t is the time period of the photon orbit.
If we substitute E=h/t into equation (*) we get:
h / t = (c^4 / G) * r
Rearranging we get:
r * t = h G / c^4
Thus we find that the spacetime area occupied by this bound photon, r * t, is quantized.
I was wondering if this is similar to the result that the world sheet area swept out by strings is quantized.
I was wondering what happens if one has a photon that is so energetic that it gets trapped in its own gravitational field.
Imagine a mass m orbitting a large mass M in a circular orbit with radius r and with transverse velocity v. By equating the gravitational force on m with its centripetal acceleration we have:
G M m / r^2 = m v^2 / r
Now let velocity v approach the speed of light c. The mass m will be boosted by a Lorentz factor but I think the equation should still hold so that we have:
G M m / r^2 = m c^2 / r
Thus cancelling m we have:
M = (c^2 / G) r
or in terms of Energy E = M c^2 we have
E = (c^4 / G) r (*)
This is the energy that a photon must have to get trapped in its own gravitational field so
that it orbits in a circle with radius r. The quantitiy c^4 / G is also the string tension so maybe this is also a model of a string.
Now let us suppose that the photon has angular momentum hbar. Then we have
r * p = hbar
where p is the linear momentum of the photon.
Now we know that p = E / c for photons so that we have:
r * E / c = hbar
E = hbar * c / r
E = h * c / 2.pi.r
E = h / t (**)
where t is the time period of the photon orbit.
If we substitute E=h/t into equation (*) we get:
h / t = (c^4 / G) * r
Rearranging we get:
r * t = h G / c^4
Thus we find that the spacetime area occupied by this bound photon, r * t, is quantized.
I was wondering if this is similar to the result that the world sheet area swept out by strings is quantized.