Thinking of a Prince
What the mind thinks of atop a volcano. We've all read Antoine de Saint-Exupery's THE LITTLE PRINCE. Saint-Exupery was an aviator before becoming a writer too. The Little Prince lived on a planet so tiny you could walk around it. His planet had many volcanoes which he frequently had to clean out. Could such a place exist?
A typical primordial singularity has the mass of about 10^11 kg. If you trapped it in a box one meter wide and stood on the box, would you get sucked in? Think about it: gravitational acceleration is GM/r^2. Newton's G is 6.67 x 10^{-11} and r^2 is just 1 in MKS units. The acceleration you feel from the Black Hole one meter away is just 6.67 m/sec^2, or only 2/3 of what you feel from Earth!
Suppose you are suicidal, open the box and reach in. As soon as your fingertip touches the singularity, the first gram of you is converted into the energy of a small atomic explosion. The rest of you is blown sky high, easily reaching escape velocity and never seeing the Black Hole again. You couldn't get sucked up if you wanted to.
The Little Prince's planet has a radius on the order of 10 metres, made of about 10^7 kg of rock gathered around a tiny singularity at the centre. It has enough surface gravity that the Little Prince can walk around it without falling off. Radiation from the singularity would create many volcanoes for him to tend to. The infant Earth was once a tiny planetesimal about this size. Perhaps our world was like that of the Little Prince.
A typical primordial singularity has the mass of about 10^11 kg. If you trapped it in a box one meter wide and stood on the box, would you get sucked in? Think about it: gravitational acceleration is GM/r^2. Newton's G is 6.67 x 10^{-11} and r^2 is just 1 in MKS units. The acceleration you feel from the Black Hole one meter away is just 6.67 m/sec^2, or only 2/3 of what you feel from Earth!
Suppose you are suicidal, open the box and reach in. As soon as your fingertip touches the singularity, the first gram of you is converted into the energy of a small atomic explosion. The rest of you is blown sky high, easily reaching escape velocity and never seeing the Black Hole again. You couldn't get sucked up if you wanted to.
The Little Prince's planet has a radius on the order of 10 metres, made of about 10^7 kg of rock gathered around a tiny singularity at the centre. It has enough surface gravity that the Little Prince can walk around it without falling off. Radiation from the singularity would create many volcanoes for him to tend to. The infant Earth was once a tiny planetesimal about this size. Perhaps our world was like that of the Little Prince.
posted by L. Riofrio at 9:45 AM
9 Comments:
Hi Babe,
Nice post... and nice blog. I just linked you :)
Cheers,
T.
Newton's gravitation says the acceleration field a = MG/r^2.
Let's go right through the derivation of the Einstein-Hilbert field equation in a non-obfuscating way.
To start with, the classical analogue of general relativity's field equation is Poisson's equation
(div.^2)E = 4*Pi*Rho*G
The square of the divergence of E is just the Laplacian operator (well known in heat diffusion) acting on E and implies for radial symmetry (r = x = y = z) of a field:
(div.^2)E
= (d^2)E/dx^2 + (d^2)E/dy^2 + (d^2)E/dz^2
= 3*(d^2)E/dr^2
To derive Poisson's equation in a simple way (not mathematically rigorous), observe that for non-relativistic situations
E = (1/2)mv^2 = MG/r
(Kinetic energy gained by a test particle falling to distance r from mass M is simply the gravitational potential energy gained at that distance by the fall! Simple)
Now, observe for spherical geometry and uniform density (where density Rho = M/[(4/3)*Pi*r^3]),
4*Pi*Rho*G = 3MG/r^3 = 3[MG/r]/r^2
So, since E = (1/2)mv^2 = MG/r,
4*Pi*Rho*G = 3[(1/2)mv^2]/r^2 = (3/2)m(v/r)^2
Here, the ratio v/r = dv/dr when translating to a differential equation, and as already shown (div.^2)E = 3*(d^2)E/dr^2 for radial symmetry, so
4*Pi*Rho*G = (3/2)m(dv/dr)^2 = (div.^2)E
Hence proof of Poisson's gravity field equation:
(div.^2)E = 4*Pi*Rho*G.
To get this expressed as tensors you define a Ricci tensor R_uv for curvature (this is a shortened Riemann tensor). The crucial insight begins with a tensor version of this with a contraction:
R_uv = 4*Pi*G*T_uv,
where T_uv is the energy-momentum tensor which includes potential energy from pressures and field energy, but is analogous to the density term Rho in Poisson's equation. (The density of energy can be converted into mass density simply by E=mc^2.)
However, this equation R_uv = 4*Pi*G*T_uv was found by Einstein to be a failure because the divergence of T_uv should be zero if energy is conserved. (A uniform energy density will have zero divergence, and T_uv is of course a density-type parameter. The energy potential of a gravitational field doesn't have zero divergence, because it diverges - falls off - with distance, but uniform density has zero divergence simply because it doesn't fall with distance!)
The only way Einstein could correct the equation (so that the divergence of T_uv is zero) was by replacing T_uv with T_uv - (1/2)(g_uv)T, where R is the trace of the Ricci tensor.
R_uv = 4*Pi*G*[T_uv - (1/2)(g_uv)T]
which is equivalent to
R_uv - (1/2)Rg_uv = 8*Pi*G*T_uv
Which is the full general relativity field equation (ignoring the cosmological constant and dark energy, which is incompatible with any Yang-Mills quantum gravity because to use an over-simplified argument, the redshift of gravity-causing exchange radiation between receding masses over long ranges cuts off gravity, negating the need for dark energy to explain observations).
I'm quite serious as I said to Prof. Distler that tensors have their place but aren't the way forward with gravity because people have been getting tangled with that since 1915 and have a landscape of cosmologies being fiddled with cosmological constants (dark energy) to fit observation, and that isn't physics.
For practical purposes, we can use Poisson's equation as a good approximation and modify it to behave like the Einstein' field equation. Example: (div.^2)E = 4*Pi*Rho*G becomes (div.^2)E = 8*Pi*Rho*G when dealing with light that is transversely crossing gravitational field lines (hence light falls twice as much towards the sun than Newton's law predicts).
When gravity deflects an object with rest mass that is moving perpendicularly to the gravitational field lines, it speeds up the object as well as deflecting its direction. But because light is already travelling at its maximum speed (light speed), it simply cannot be speeded up at all by falling. Therefore, that half of the gravitational potential energy that normally goes into speeding up an object with rest mass cannot do so in the case of light, and must go instead into causing additional directional change (downward acceleration). This is the mathematical physics reasoning for why light is deflected by precisely twice the amount suggested by Newton’s law.
That's the physics behind the maths of general relativity! It's merely very simple ENERGY CONSERVATION, obfuscated with technical details of tensors. Why can't Prof. Distler grasp this stuff? I'm going to try to rewrite my material to make it crystal clear even to stringers who don't like to understand the physical dynamics!
I've put a slightly clearer version of my comment above at http://electrogravity.blogspot.com/2006/12/newtons-gravitation-says-acceleration.html. Basically the string theorists are just trying to obfuscate. You can do some things far more simply using simple maths. It's partly their self-imposed constraints to use only the most generalized mathematical tools which has created such an insoluble stringy mess in the mainstream.
‘It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of spacetime is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities.’
- R. P. Feynman, Character of Physical Law, November 1964 Cornell Lectures, broadcast and published in 1965 by BBC, pp. 57-8.
Thanks anon and qds, every little bit helps.
Nigel, thank you for an extremely informative comment. That gets to the heart of what they were arguing over at asymptotia. It can be frustrating, but your work shows that there is something to all this.
That's great, Tommaso! I feel like we have a nice community here now.
I'm honoured to hear from you too, Tomasso. I'll be updating my links soon. The singularity is near, very near!
Nigel, I read your comments again and they are right on target! That puts the lie to those who say that T_uv should have certain units. It is other people who need to study more physics.
Great site, I am bookmarking it!Keep it up!
With the best regards!
Frank
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