Questions About General Theory of Relativity

[Link]

There is a couple of questions that have arisen after i read about the general theory of relativity.

1. Should it be possible in principle for a photon to circle a star?

2. Do binary stars rotating around a center of mass radiate gravitation waves?

3. How are gravitational waves emitted?

4. The twin trip example of the special theory of relativity seems to contradict the general theory of relativity. According to the second stated, a clock near a massive star ticks slower than a clock on a orbiting planet. But the twin trip states that a guy in a spaceship ages more slowly when leaving earth, hence stating that the clock ticks slower than the correspondin one on earth. Isn't gravity stonger on earth?

 

[NateTG]

1. Yes, it should be possible for photons to be in orbits.

4. First off, the twin paradox has to do with general relativity, not special relativity. One of the fundamental notions of General Relativity is that accelleration due to gravity is the same as acceleration due to other forces. One result is that time passes more slowly in more accelerated reference frames, so if the twin that goes into space does space travel with an acceleration of less than 1g then the twin that was left behind would be younger at the reunification.

 

[chroot]

Originally posted by Link
1. Should it be possible in principle for a photon to circle a star?

Not a star, but a black hole. Light can orbit a black hole at a distance of

$$\frac{3 G M}{c^2}$$

2. Do binary stars rotating around a center of mass radiate gravitation waves?

Yes. Any oscillating mass will radiate gravitational waves.

3. How are gravitational waves emitted?

Gravitational waves are just "ripples" in the curvature of space that propagate away from the source of the ripples. A spherical mass sitting still produces a static, symmetric curvature around it. If you suddenly move it a bit to the left, the curvature gets deformed by the motion (since gravitational waves do not propagate infinitely fast). If you move the mass back to the right, the deformation is in the opposite direction. If you wave the mass back and forth in steady motion, the deformations continue to propagate away from the mass in a regular pattern. This is same way that electromagnetic radiation is created, by waving a charge.

4. The twin trip example of the special theory of relativity seems to contradict the general theory of relativity. According to the second stated, a clock near a massive star ticks slower than a clock on a orbiting planet. But the twin trip states that a guy in a spaceship ages more slowly when leaving earth, hence stating that the clock ticks slower than the correspondin one on earth. Isn't gravity stonger on earth?

In that context, the gravitational time dilation due to the Earth is considered negligible. You are correct, though, that it is should be included in a complete calculation.

- Warren

 

[dhris]

Originally posted by chroot
Not a star, but a black hole. Light can orbit a black hole at a distance of

$$\frac{3 G M}{c^2}$$

Why do you say this? The event horizon is at

$$\frac{2 G M }{c^2}$$

If the mass is all inside the photon sphere I don't know if that means that you are always talking about a black hole.

dhris

 

[Link]

how can a photon sphere have a mass? Isnt light massless?

 

[chroot]

Originally posted by dhris
Why do you say this? The event horizon is at

$$\frac{2 G M }{c^2}$$

Correct. Light does not orbit at the event horizon, however -- it orbits at a radius of 1.5 times the Schwarzschild radius, which is what I posted.

- Warren

 

[dhris]

Originally posted by chroot
Correct. Light does not orbit at the event horizon, however -- it orbits at a radius of 1.5 times the Schwarzschild radius, which is what I posted.

- Warren

I know. But you also said that it only happens for a black hole. My point is that since the light is not orbiting at the event horizon, why would it matter if the mass below the photon sphere has collapsed below the Schwarzschild radius?

dhris

 

[chroot]

Because it obviously can't happen with all that matter in the way:

http://www.google.com/search?num=10...+G+*+(mass+of+earth)+/+c^2&btnG=Google+Search

- Warren

 

[wolram]

may i with respect to other posters point out that gravitational
radiation has not been found to date, and is hypothetical
new detector might find it in a few years, but what if they
do not?

 

[Arcon]

Originally posted by chroot
Because it obviously can't happen with all that matter in the way:

http://www.google.com/search?num=10...+G+*+(mass+of+earth)+/+c^2&btnG=Google+Search

- Warren

Please elaborate. I see no reason that a star can't have a mass of 1.5 its Schwarzschild radius. Unless you mean that such a star must collapse?

 

[chroot]

The only mass that would affect the light is the interior mass. You are correct that, theoretically, the object would not have to be a black hole -- it would simply have to have all its mass within 3GM/c^2. This would be impossible to do in practice, however, without creating a black hole, as no normal form of matter is capable of being stable when compressed that much. Even a neutron star is a far cry from having all its mass within 3GM/c^2.

- Warren

 

[dhris]

Originally posted by chroot
The only mass that would affect the light is the interior mass. You are correct that, theoretically, the object would not have to be a black hole

That's my point. What did that google link have to do with it?

 

[chroot]

Originally posted by dhris
That's my point. What did that google link have to do with it?

It was just a calculation to show you the Schwarzshild radius of an object with the mass of the Earth -- it's centimeters. If you compressed the mass of the Earth into a volume centimeters in radius, you'd have a black hole.

- Warren

 

[NateTG]

Originally posted by chroot
It was just a calculation to show you the Schwarzshild radius of an object with the mass of the Earth -- it's centimeters. If you compressed the mass of the Earth into a volume centimeters in radius, you'd have a black hole.

- Warren

Right, but if you compresed the Earth into a sphere 1.4 thimes that size, it need not collapse, but could still have a photon orbit.

 

[Janus]

Originally posted by NateTG
One result is that time passes more slowly in more accelerated reference frames, so if the twin that goes into space does space travel with an acceleration of less than 1g then the twin that was left behind would be younger at the reunification.

It does not matter how much G-force the astronaut undergoes when it comes to whether or not he will have aged less or more than his brother, as long as he attains a high enough relative velocity and maintains it long enough. If he attains .866c by accelerating at .5g he will still age less than his brother, even though his brother experienced 1g during this period.

 

[chroot]

Well, I see your point. If you put the mass of the Earth within a sphere of radius 1.33 cm, it will certainly collapse to a black hole rather quickly. But, for a few fleeting moments, when the mass is within 3GM/c^2 but not within 2GM/c^2, you'd have a photon orbit without a black hole.

Of course, that state would last a very very short time! If the mass of the Earth were falling from 3GM/c^2 to 2GM/c^2 on a free-fall timescale, from rest, it would take only 30 milliseconds to become a black hole.

In steady state, of course, you can't have a photon orbit around anything except a black hole, because anything that dense will become a black hole very quickly.

- Warren

 

[Arcon]

Originally posted by chroot
Well, I see your point. If you put the mass of the Earth within a sphere of radius 1.33 cm, it will certainly collapse to a black hole rather quickly. But, for a few fleeting moments, when the mass is within 3GM/c^2 but not within 2GM/c^2, you'd have a photon orbit without a black hole.

Of course, that state would last a very very short time! If the mass of the Earth were falling from 3GM/c^2 to 2GM/c^2 on a free-fall timescale, from rest, it would take only 30 milliseconds to become a black hole.

In steady state, of course, you can't have a photon orbit around anything except a black hole, because anything that dense will become a black hole very quickly.

- Warren

For a naive calculation suppose you assume that the largest density that can exist is that of a neutron star where rho ~ 3x10¹⁷kg/m³. Set R = 2.5GM/c². Then R ~ 3 million kilometers. It seems reasonable to assume that crushing the Earth to within the photonsphere will force it to collapse to a black hole. Although this was a rough calculation. It left out the contribution of gravitational forces due to pressure.

 

[chroot]

Arcon,

I don't know what you're talking about. A typical neutron star is about the mass of the Earth and is about 7 km in radius. The photonsphere for an object with the mass of the Earth is 1.33 centimeters. I have no idea where you got 3 million kilometers from.

- Warren

 

[Arcon]

Originally posted by chroot
Arcon,

I don't know what you're talking about. A typical neutron star is about the mass of the Earth and is about 7 km in radius. The photonsphere for an object with the mass of the Earth is 1.33 centimeters. I have no idea where you got 3 million kilometers from.

- Warren

Neutron stars do not have a mass ever close to that of the Earth. A neutron star is what is left after a high mass star goes supernova and has a mass of about the same as our sun.

Note that I didn't say that the calculation was for a neutron star. I said assume that the maximum density was that of a neutron star and then set R = 2.5GM/c². That means assume that you have a spherical object whose mass density is that of the center of the nucleus of an atom and assume that the radius allowed for a photonsphere. What is that radius? Again - assume a naive calculation by leaving out pressure - just to give a rouch idea of the size of such an object.

The density of the nucleus = density of a neutron star = 3x10¹⁷kg/m³.

$$M = \rho V = \rho (\frac {4}{3}\pi R^{3})$$

Substitute that into R = 2.5GM/c² and solve for R. You'll get about 2.2x10⁹ meters.

The whole point was to illustrate that such an object is unlikely to exist.

 

[chroot]

If you're asking:

What would be the radius of an object, with density 3 * 10^17 kg/m^3, whose photonsphere is right at its edge?

The answer is 18,901 meters.

I still have no idea where you're getting 2.2 * 10^9 meters, and I have no idea why you're using a factor 2.5 instead of 3.

- Warren

 

[Arcon]

Originally posted by chroot
If you're asking:

What would be the radius of an object, with density 3 * 10^17 kg/m^3, whose photonsphere is right at its edge?

The answer is 18,901 meters.

I still have no idea where you're getting 2.2 * 10^9 meters, and I have no idea why you're using a factor 2.5 instead of 3.

- Warren

I see what happened. I did the arithmetic wrong. I used 2.5 to be somewhere betweeen a black hold radius and a photonsphere radius. Otherwise you'd have the photons riding on the surface.

 

[treat2]

Originally posted by wolram
may i with respect to other posters point out that gravitational
radiation has not been found to date, and is hypothetical
new detector might find it in a few years, but what if they
do not?

Wolf, that is no longer true. Check out the BBC's Web Site, and look in the Science Section.

Some scientists have recently created a "Gravitational Wave Detector",
and have (to their satisfaction) proved the existence of a gravity wave, and obtained additional information about it.

 

[treat2]

Originally posted by Link
how can a photon sphere have a mass? Isnt light massless?

Link, despite what I read in every physics book, photons have been found to NOT be massless particles.

If I recall correcly, the mass of a photon is estimated at 10^-48th
of the mass of an electron.

This is published on a reputable Web Site, which I appologize for not being able to provide, BUT I can provide other links that
can tell you the estimated mass of a photon:

"What is the Mass of a Photon?" See:
http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/photon_mass.html

For a discussion of it, in another String Theory Forum, See:
http://superstringtheory.com/forum/partboard/messages10/23.html

If you want more, Google: photon mass
(without quotes, using both words)

 

[selfAdjoint]

Treat, please stop posting this, you have misinterpreted Baez' site (a good one), and the superstring site is by a crank, kx21.

Baez says, as I replied to you on another thread, that those small numbers are limits. They are the unavoidable experimantal error limits of the measurements; they are NOT the mass of the photon.

 

[GRQC]

Originally posted by treat2
Wolf, that is no longer true. Check out the BBC's Web Site, and look in the Science Section.

Some scientists have recently created a "Gravitational Wave Detector",
and have (to their satisfaction) proved the existence of a gravity wave, and obtained additional information about it.

No, they haven't. If gravity waves had been detected, then more than just the BBC would be reporting it (specifically, every physics-related publication in the world).

Also note that there are more than one "gravity wave detector". The most commonly known is LIGO, but there are many others (LISA, etc...).

 

[Sammywu]

It seems to me nowadays people just believe in the rest mass. I remember seeing a news in yahoo.com sometimes earlier and Arcon mentioned that. The experiment was performed in measuring possible gravity effect between two light waves. I remember it did say the experiment shows positive result suggesting attracting force between two light waves.

What is Modern Physics's opinion on this experiment?

 

[Stingray]

Gravity waves have been "detected" indirectly for a long time now. Hulse and Taylor won a Nobel prize for finding that a system of binary pulsars was losing energy at exactly the rate predicted by GR due to gravitational radiation. Recent observations have narrowed down the error bars dramatically on these types of systems, and GR still passes.

I agree though that seeing gravitational waves through LIGO et al would be huge news.

Anyway Link, gravitational waves are "created" similarly to electromagnetic waves. You accelerate charges to get light, and you accelerate masses to get gravity waves (more or less).