What Is A Matter Made Of?

[forests]

Hello,

 

I have no education in physics and I have a hard time understanding the subject so I need the basics. I have tried to search the internet to understand what matter is made of but I often see confusing and sometimes contradiction in different answers, so I have started this thread as a simple question. What is a matter made of?

 

The first answer that I will probably get is atoms and then sub-atomic particles I do get that, but I am not understanding beyond that, I have seen people say these are made up of energy or strings? Please explain if you can? And what does it mean by energy or strings?

[Aethelwulf]

Matter is just a form of pre-longed fluctuation of energy or a better said, a non-diffused type of energy - energy is just diffused matter. So matter is just a concentrated form of energy.

 

Strings is just another way to view fundamental energy or matter. There is no direct experimental evidence of their existence. If anything, matter performs or rather, behaves like fundamental points.

Edited July 26, 2012 by Aethelwulf

[CraigD]

I have no education in physics and I have a hard time understanding the subject so I need the basics. I have tried to search the internet to understand what matter is made of but I often see confusing and sometimes contradiction in different answers, so I have started this thread as a simple question. What is a matter made of?

This is a deceptively simple and oft vexing question, but if we can accept that a simple answer to it has (on account of all that deceptive and vexing-ness) puts one on a path with pitfalls and blinders on, I’ve got one, from the good ‘ole standard model of particle physics: Matter’s made of fermions.

 

To make any sense of this answer, you’ve got to have an inkling what a fermion is. To have an inkling of what a fermion is, you’ve got to have an inkling of what it isn’t, which is a boson. No more inklings are needed after that, because in SM particle physics, that’s all there are: fermions and bosons.

 

By defininition a Fermion is anything that follows Fermi-Dirac statistic. As you might guess by the intellectual fame of the fellows its named for, Fermi-Dirac statistic are complicated. Fortunately for folk like me, they can be summarized by a simple principle, known as the Pauli exclusion principle: (With apologies to Pauli, Fermi, Dirac, and every other mathematical physicist for my deceptive simpleton-ness) two (or more) of them can’t occupy the same space at the same time.

 

Bosons, everything that isn’t a fermion, follows Bose-Einstein statistics, which essentially mean two or more of them can occupy the same space at the same time. An example of this is that, any number of beams of light (made of lots of photons, which are bosons) can be crossed without effecting them in the least. Try that with a beam of fermions, such as electrons, and, per the exclusion principle, some will collide, limiting how “dense” where they cross can be.

 

Bosons can be described as “force carriers”, fermions as what they carry the force between.

 

Since, except for the elementary ones (such as the electron), ensembles of fermions such as protons and neutrons require forces to create and maintain their structure, matter = fermions need force = bosons to be very interesting = result in the universe we see and are part of. We might call this the first pitfall in my simple “matter’s made of fermions” answer.

 

The first answer that I will probably get is atoms and then sub-atomic particles I do get that, but I am not understanding beyond that, I have seen people say these are made up of energy or strings? Please explain if you can? And what does it mean by energy or strings?

There are 2 separate questions here.

 

“Matter is made up of energy” is a rough way of describing mass–energy equivalence, better know by the famous equation [imath]E=Mc^2[/imath]. Because, according to particle physics, particles can transform into one another (and in a very weird sense I don’t wanna get into here, must be thought of as juxpositions of many different particles at the same time), it’s handy to have this equivalence to tell what characteristic these before and after particles must be have, and essential if, for example, you want to calculate how a star or an atomic bomb works.

 

“Matter is made of strings” comes from a collection of theories called (no great revelation here) string theory. These theories attempt to explain the 16 kinds of particles (or 17 if we now include the Higgs, or 18 if we include it and ever get the graviton to fit the theory) of the Standard Model as all being made of a single kind of object: strings.

 

String theory is awfully complicated, and nobody really know if it is, in the useful sense that the SM has been, correct, or worse, whether it will ever be possible to design an experiment to show if any of its many variants are correct or wrong.

[Aethelwulf]

I posted a second post.... it has disappeared!

 

What I said was, that there is no direct experimental evidence of string theory. I'd like to warn you to take this mostly mathematical theory with a pinch of salt.

[maddog]

What is a matter made of?

The first answer that I will probably get is atoms and then sub-atomic particles I do get that, but I am not understanding beyond

that, ...

I have seen people say these are made up of energy or strings? Please explain if you can? And what does it mean by energy or strings?

I can not speak for others on what was said (correctly or not). For the moment lets leave strings aside for now.

 

As you imply, almost all matter we know of is made of compounds which are themselves collections of elements which are a collection of three elementary particles (or so we used to think) of electrons, protons and neutrons. Electrons are in the family of leptons and protons and neutrons are in the family of baryons. All are considered fermions (leptons/baryons - being of the class spin 1/2). This is the static picture (simplest).

 

More dynamic is how these elements/compounds interact. In physics we describe four major forces Gravity, Electromagnetism (EM), Weak force, and the Strong force, with gravity being the weakest. It is thought a particle of spin 1 represent each force. Gravity is represented by the graviton which not yet been observed yet as a particle (or as a wave). EM is represented by the photon or light. Weak force is represented by three particles W, Z+, Z-. Strong force is represented by gluons (now I getting ahead of myself - I will come back to this). Each of these four forces are quantized forms of energy and they interact with the fermions mentioned earlier above. This class is called Bosons (spin 1, with the exception of graviton which is spin 2, weak force which are spin 0 -- just think integral spin vs half spin.

 

Back to the baryons: In the late 60s it was discovered that both protons and neutrons had internal structure (they themselves were made of stuff). This formed a new field called Quantum Chromodynamics or QCD. It was determined that baryons were made up of 3 quarks and a gluon to hold them together. Each quark was a fractional charge +/- 1/3 or +/- 2/3 by which you can get all the combinations of +1 for a proton, 0 for neutron or -1 for antiproton (antimatter). In addition to baryons were mesons which were made up of 1 quark and 1 anti-quark. Mesons were highly unstable particles with short half-lifes found in cosmic rays or inside particle accelerators like LHC for example. In addition to this, for both leptons and hadrons had generations (3) of similarities. One other particle I have so far neglected to mention until now is the neutrino which is also a lepton. The generations for the electron is in addition to itself is also the muon, and the tau. So to complete this neutrinos have also these three flavors which have recently found to oscillate between them while traveling through matter.

 

Back to QCD: Remember how I spoke of baryons (protons, neutrons, etc) being made of more stuff? Well these quarks also had generations. Before I screw this up go look up QCD (spelled out) or quarks on wiki.com to get the table. All of these quarks and anti-quarks make up all the baryons and mesons I spoke earlier with one additional tweak. That is color. There are Red, Blue, Green added to each (and their respective gluons which have to balance out.

 

Getting back to strings, remember I said I would(?) In around the early 70's (a little earlier Venaziano - may have mispelled his name) came up with an idea patterning after the notion of a simple harmonic oscillator (think of a string tied on one to something and vibrate the other end). The information embedded in the vibration of the string could describe all the quantum information to make up all the quantum numbers of charge, quark, color, etc. In 1974 Schwarz and Green figured out the lowest vibration which was odd because it had integral spin (remember bosons) with a value of spin 2. This was later resolved to be the graviton. Later (I forget what year) all the bosons could be worked out if more dimensions were added. In particular if 26 dimensional space was used this would be complete. At the same time another group was studying fermions (spin 1/2) and found out if 10 dimensions were used it would be perfect. We don't see these dimensions so we roll them up to be smaller than we can see. An early version this was discover by Kaluza who wrote to Einstein in 1919 of a theory that in five dimensions could make a unification of gravity and EM. Later in 1925 a Klein thought by rolling up this fifth dimension could get it work out. At this time the other two forces were still yet to be discovered.

 

Another problem that was kicking around at the time (late 70's) was this notion of Supersymmetry which basically says for every fermion or boson there is a supersymmentric partner of the opposite class.

 

SS(Boson) = Fermion; SS(Fermion) = Boson, etc.

 

By the 90's there were five separate string theories that didn't agree with each other. About this time, Ed Witten discovered by thinking of these string theories embedded on the surface of a even larger space could make all five theories be isomorphic to each other (think of this being like equivalent to). He reformulated this into a new theory called M-theory (what the M refers to, get 5 physicist in a room and you will get 5 answers!).

 

The sad fact of all this there no experiments so far that can be measured (with maybe one exception - I will have to look it up - there is one where they are trying to discover if more dimensions exist than we currently observe 3 + 1 (3 space, 1 time).

 

The gist of why all this crazy investment into string theory and the like is because General Relativity (GR) of Einstein is found not to agree in small dimensions with Quantum Mechanics (QM). String theories (M-theories what have you) are still theoretical and require all the parameters to be plugged in. Nothing is deduced. String theories still can not agree in how many dimensions (last I hear 12 was being considered - F12 group).

 

So there is a flip side to string theory. It is called Loop Quantum Gravity (LQG) with an institute in Toronto, being chaired by Lee Smolin.

 

Then there is the resurrection of an old idea by Roger Penrose called Twister theory. My avatar is a twistor representation of a photon. In the last few years I have heard that Ed Witten has considered combining Twistor theory with M-theory. I have not heard of the outcome.

 

I do think think a proper representation of reality can only be made by reconciling M-theory (in whatever form) with LQD and possibly adding Twistor theory.

 

One other thought I have is to complexify our coordinate system. In particular to complexify time. This would allow tachyons (faster than light particles) even though theoretical are thought not to exist. They are allowed in QM when not observed as virtual particles. This is a start on a whole new thread/concept, so I will only comment that string theories already have them, called ghosts. No one has satisfactorily been able to show that ghost states are truly prohibited.

 

I apologize for jumping around. It just that what matter is made of is really convoluted. ;-)

 

maddog

Edited July 26, 2012 by maddog

[maddog]

By defininition a Fermion is anything that follows Fermi-Dirac statistic. As you might guess by the intellectual fame of the fellows its named for, Fermi-Dirac statistic are complicated. Fortunately for folk like me, they can be summarized by a simple principle, known as the Pauli exclusion principle: (With apologies to Pauli, Fermi, Dirac, and every other mathematical physicist for my deceptive simpleton-ness) two (or more) of them can’t occupy the same space at the same time.

I find it interesting thing about the Pauli Exclusion principle when the electron itself is considered a point particle! However this can be explained in that they interact with each other by the concentration in their Electric field (or E field) and when this electron is in motion which is always the case, it induces a Magnetic field (or B field). This directs its motion around another electron that gets to close.

 

maddog

[Pmb]

I have no education in physics and I have a hard time understanding the subject so I need the basics. I have tried to search the internet to understand what matter is made of but I often see confusing and sometimes contradiction in different answers, so I have started this thread as a simple question. What is a matter made of?

Matter is a vauage term and has had varying definition over the years. Einstein defined matter as anything where the stress-energy-momentum tensor doesn't vanish. That's a general classification. Specifically matter is composed of atoms and molecules and a large number of elementary particles. Under Einstein's definition, which is the one I go by, anything that has energy has matter and anything that has matter has both energy adn mass. This means that a static electric/magnetic field has matter as do photons. According to Einstein's definition, the gravitational field doesn't have matter.

 

The first answer that I will probably get is atoms and then sub-atomic particles I do get that, but I am not understanding beyond that, I have seen people say these are made up of energy or strings? Please explain if you can? And what does it mean by energy or strings?

You're right. Everything is made up as you state. This is the nature of nature. You can always ask "What is X" made of?" When you get the answer "X is made of Y" then your next question might be "What is Y made of?" whereing you'd get the answer "Y is made of Z." and it keeps on going forever.

[Pmb]

This is a deceptively simple and oft vexing question, but if we can accept that a simple answer to it has (on account of all that deceptive and vexing-ness) puts one on a path with pitfalls and blinders on, I’ve got one, from the good ‘ole standard model of particle physics: Matter’s made of fermions.

That was a very strange reply. All I could make out was Matter is made of fermions so that’s what I’ll respond to. Fermions are merely one of two classes of particles. A fermion is any particle characterized by Fermi-Dirac statistics. Some examples are electrons, protons and neutrons are fermions. The other class of particles is called bosons, which are particles characterized by Bose-Einstein statistics. Some examples are pi mesons, alpha particles, He+ atoms and deuterons. To say that matter is made of fermions is to say that alpha particles are not matter. And I see no valid reason why alpha particles shouldn’t be thought of as matter.

 

Bosons can be described as “force carriers”, fermions as what they carry the force between.

I disagree. Just because some bosons are force carriers it doesn’t mean that all of them are. Alpha particles certainly aren’t force carriers. Bosons and fermions are defined in terms of the symmetry or antismmetry of their wave functions. They are also defined by the eigenvalue of the exchange operator.

 

“Matter is made up of energy” is a rough way of describing mass–energy equivalence,..

I disagree. In my opinion one should never think of matter as being made up of energy. There is merely a relationship between mass and energy. That relationship has to do with two things (1) if a body decreases its energy content by [math]\Delta m[/math] then its mass decreases by the amount [math]\Delta E[/math] where [math]\Delta E = \Delta m c^2[/math]. (2) If a particle has proper mass m then there it has proper energy E where [math]E = mc^2[/math]. This comes into place in energy conservation equations where particles are being created and/or destroyed.

[Aethelwulf]

I know I've said that energy is just diffused matter, and matter is just concentrated energy. But I guess in a sense pmb is right - matter isn't exactly energy. I guess there are two ways to view it:

 

1) Matter is a property of an interaction between a particle and the Higgs Field

 

or

 

2) Mass is a different form of energy - which is the same in your Higgs Model as saying your goldstone Boson has fluctuated away from the ground state and absorbed by a particle. (The appearance of mass).

 

 

Mass for 1) appears from a Higgs Field through an interaction. Such an interaction term might look like [math]\psi g\psi^{\dagger}[/math]. Here, [math]M[/math] the mass is replaced by yukawa mass coupling [math]g[/math]. Mass is actually a very interesting thing when one considers a Dirac Equation and its respective Langrangian. Antiparticles for instance, are describes as left movers while normal particles are considered the ''other handedness'' - your normal every day matter. What a mass does is very interesting, it mixes up the left movers and right movers

 

[math]-i(\alpha \hat{p})c\psi + \beta_s M_s c^2 \psi = i\hbar \partial_t \psi[/math]

 

and move everything to the left

 

[math]i\hbar \partial_t \psi -( i(\alpha \hat{p})c + \beta_s M_s c^2) \psi = 0[/math]

 

multiply this by psi dagger:

 

[math]\psi^{\dagger}((i\hbar \partial_t - i(\alpha \hat{p})c + \beta_s M_s c^2 )\psi) = \mathcal{L}[/math]

 

And simplifying gives the Dirac Langrangian

 

[math]\bar{\psi}\gamma^{0} (i \hbar \partial_t) \psi - ((\gamma^{i} \cdot \hat{p})c + M_s)\bar{\psi}\psi = \mathcal{L}[/math]

 

the mass term in this next equation shows you how it mixes up terms.

 

[math]\gamma^{0} \cdot i\hbar \partial_t(\psi^{\dagger}_L \psi_R + \psi^{\dagger}_R \psi_L) - \gamma^{i} \cdot kc(\psi^{\dagger}_L \psi_R + \psi^{\dagger}_R \psi_L) - M_s(\psi^{\dagger}_L \psi_R + \psi^{\dagger}_R \psi_L) = \mathcal{L}[/math]

 

So mass indeed is an interesting property, but is it energy itself? I guess it can be argued that it is more than just that - mass appears to be a very simple but elegant interaction in quantum field theory where your Boson is gobbled up. Of course fundamentally speaking, I suppose mass is still the presence of some kind of energy flux.

 

This is all presuming the Higgs Boson is indeed how mass appears.

Edited July 29, 2012 by Aethelwulf

[Aethelwulf]

Of course one might ask the question, ''well, if mass isn't energy, why is energy made equal to mass times celeritas squared?''

 

One needs to remember, [math]E=Mc^2[/math] is a very simplified equation of mass - it describes the change of energy if one wanted and how it would effect the inertia of a system. Then again, this is why it led Einstein to question whether the inertia of a system (it's mass) is really determined by it's energy content.

[Pmb]

Of course one might ask the question, ''well, if mass isn't energy, why is energy made equal to mass times celeritas squared?''

 

One needs to remember, [math]E=Mc^2[/math] is a very simplified equation of mass - it describes the change of energy if one wanted and how it would effect the inertia of a system. Then again, this is why it led Einstein to question whether the inertia of a system (it's mass) is really determined by it's energy content.

(let c = 1) The inertial mass density of a perfect fluid given by [math]\rho = u + p[/math] where [math]\rho[/math] is the inertial mass density, u is the energy density and p is the fluid pressure. If mass and energy were the same thing we'd have [math]\rho = u[/math]. Since we don't then one can't say that energy and mass are the same thing. Although many relativists do start our by saying that mass and energy are the same thing and then arrive at [math]\rho = u + p[/math] and call it the inertial mass density. I think that they're being hypocritical when they do that.

Edited July 29, 2012 by Pmb

[Aethelwulf]

(let c = 1) The inertial mass density of a perfect fluid given by [math]\rho = u + p[/math] where [math]\rho[/math] is the inertial mass density, u is the energy density and p is the fluid pressure. If mass and energy were the same thing we'd have [math]\rho = u[/math]. Since we don't then one can't say that energy and mass are the same thing. Although many relativists do start our by saying that mass and energy are the same thing and then arrive at [math]\rho = u + p[/math] and call it the inertial mass density. I think that they're being hypocritical when they do that.

 

 

True.

 

When I think of mass, I think of a few things. You may have followed my own thread on ''forgotten theories of mass'' on the previous forum we once attended. I like to think of mass like Lloyd Motz had: that being mass is a type of charge on the system.

 

Similar how an electron picks up an electric charge by moving in an electromagnetic field, the mass of obtained by a particle is in the same idea, picked up by some kind of symmetry breaking moving in a Higgs Field.

 

The gravitational charge is simply given as[math]\sqrt{G}M[/math]. I did discover however, that our usual contention that the classical electron radius is

 

[math]\frac{e^2}{Mc^2} = r[/math]

 

There is also a gravitational analogue in terms of the gravitational charge:

 

[math]\frac{GM^2}{Mc^2} = r_s[/math]

 

But it was a complete analogue, derived by setting the Compton wavelength equal to the Schwarzschild radius

 

[math]\frac{\hbar c}{Mc^2} = \frac{GM^2}{Mc^2}[/math]

 

I actually believed this was an important realization when it came to seeing ''mass as a charge,'' if not a discovery. Since you are on the discussion of density, I also realized that the proper density of a particle would therefore have the relationship as well

 

[math]8\pi \rho_0 (\frac{G}{c^2}) = \frac{GM^2}{Mc^2}[/math]

 

(Which came to me when reading how Motz equated [math]8\pi \rho_0 (\frac{G}{c^2}) = 6(\frac{\hbar}{Mc})^{-2})[/math]

 

and keeping in mind that [math]\frac{\hbar}{Mc} = \frac{GM^2}{Mc^2}[/math]

 

where [math]GM^2[/math] can be seen as the square of the charge similar to [math]e^2[/math]. I also realized how the energy in such an equation might determine inertial changes, similar how you might expect inertia to change in Einstein's equation [math]\Delta E \propto \Delta Mc^2[/math]. Doing so, I derived a new form of the equation above to satisfy

 

[math]8\pi \rho_0 (\frac{GM}{\Delta E}) = \frac{\Delta GM^2}{Mc^2}[/math]

 

So that if you have a variation of energy on the left (totally consistent with your units [math]\frac{G}{c^2} = \frac{GM}{E}[/math]) then the gravitational charge will also vary according to either an increase or decrease in the energy content of your system.

 

aided by work provided by Motz http://www.gravityresearchfoundation.org/pdf/awarded/1971/motz.pdf

Edited July 29, 2012 by Aethelwulf

[Aethelwulf]

So consider the equation you presented,

 

[math]\rho = u + p[/math]

 

rho is your inertial mass density which and u is your energy density and p is your pressure. Take my equation

 

[math]8\pi \rho_0 (\frac{GM}{\Delta E}) = \frac{\Delta GM^2}{Mc^2}[/math]

 

rho may be seen as the inertial mass density but the energy is located or wrapped up in the denominators on both sides of the equation but are not explicitly defined as densities. What is considered densities in this equation is the mass not the energy. The energy effects the inertial properties therego.

[Aethelwulf]

It's also funny how you may think of density. Density in our dimensions we use everyday has dimensions of [math]\frac{M}{\ell^3}[/math] which is mass over some length cubed. But you never see an energy density have the form [math]\frac{E}{\ell^3}[/math] because that has the wrong dimensions. So when physicists talk about an energy density over some volume, we surely don't mean it to have the same kind of Newtonian density.

Edited July 29, 2012 by Aethelwulf

[Aethelwulf]

If one employs some relativistic relationships, you can simplify

 

[math]8\pi \rho_0 (\frac{G}{c^2}) = \frac{GM^2}{Mc^2}[/math]

 

this equation drastically with [math]k = 8\pi \rho_0[/math] . Also if you let [math]\frac{GM^2}{Mc^2} = r_s[/math] then

 

[math]k(\frac{G}{c^2}) = r_s[/math]

 

Takes all the interesting dynamics out of it at last glance though.

Edited July 29, 2012 by Aethelwulf

[Aethelwulf]

Here's a chicken egg scenario... is mass the presence of curvature or does curvature create mass?

 

This is a whole new way to view the question of what mass is instead of thinking of it purely as a type of energy. Instead, could matter be a fluctuation created from the presence of a type of intrinsic geometry of spacetime? There is believe it or not, already existing tantalizing evidence for this:

 

Taylor and Wheeler:

 

''The concept of 'relativistic mass' is subject to misunderstanding. That's why we don't use it. First, it applies the name mass - belonging to the magnitude of a 4-vector - to a very different concept, the time component of a 4-vector. Second, it makes increase of energy of an object with velocity or momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of space-time itself.''

 

If increase of energy of a system originates from the properties of space and time, then perhaps mass is a type of geometrical property as well; keep in mind that mass is the presence of curvature within the standard equations of relativity. So what does my own equations have to say about this?

 

Let's go back to them:

 

[math]8\pi \rho_0 (\frac{G}{c^2}) = \frac{GM^2}{Mc^2}[/math] [1]

 

and it's simplified brother

 

[math]k(\frac{G}{c^2}) = r_s[/math] [2]

 

The radius of curvature [math]R[/math] (related to quantum mechanics is)

 

[math]RMc = \hbar[/math]

 

Now if in eq [1]

 

[math]\frac{GM^2}{Mc^2} = \frac{\hbar}{Mc}[/math]

 

Then we plug [math]RMc = \hbar[/math] this into this equation we have

 

[math]\frac{GM^2}{Mc^2} = \frac{RMc}{Mc}[/math]

 

Obviously the [math]mc[/math]'s cancel out and what you are left with is the square of the gravitational charge over the inertial energy equals the radius of curvature. So simple, but easily appreciated in relativity since mass is honestly related to geometric interpretation of the vacuum. But here is a bit more interesting. Before you cancel everything, just rearrange the equation to give

 

[math]\frac{GM^3c}{Mc^2} = RMc[/math]

 

This simplifies to

 

[math]\frac{GM^2}{c} = RMc[/math]

 

This means the radius of curvature is related to the gravitational charge. This is true since

 

[math]GM^2 = \hbar c[/math]

 

and

 

[math]\frac{GM^2}{c} = \hbar[/math].

 

Curvature then is written into the equations. Instead of seeing mass as simply energy, it might be best to think of energy and mass as the geometric properties of the vacuum.

Edited July 29, 2012 by Aethelwulf

[Aethelwulf]

Now say you solve for the definition of mass from the equation

 

[math]\frac{GM^2}{c} = RMc[/math]

 

You should get

 

[math]M = R(\frac{c^2}{G})[/math]

 

Does the right hand side even look slightly familiar? It might to someone, but I will point out my equation again

 

[math]8\pi \rho_0 (\frac{G}{c^2}) = \frac{GM^2}{Mc^2}[/math]

 

It has a factor similar to my equation on the left hand side but instead is the inverse. This means, that it will satisfy [math]\frac{E}{GM}[/math] which is just the inverse energy solution I plugged into my equation.

 

And so the radius can be obtained in a new relationship.

 

Since

 

[math]M = R(\frac{c^2}{G})[/math]

 

rearrange

 

[math]\frac{M}{R} = (\frac{c^2}{G})[/math]

 

then take its inverse

 

[math]\frac{R}{M} = (\frac{G}{c^2})[/math]

 

Plug this into my equation

 

[math]8\pi \rho_0 (\frac{G}{c^2}) = \frac{GM^2}{Mc^2}[/math]

 

to give

 

[math]8\pi \rho_0 (\frac{R}{M}) = \frac{GM^2}{Mc^2}[/math]

 

multiply through by M and simplify

 

[math]8\pi \rho_0 R = \frac{GM^2}{c^2}[/math]

 

Then multiply by c squared

 

[math]8\pi \rho_0 R c^2 = GM^2[/math]

 

which yields the gravitational charge charge again (mass)

 

then solve for the radius of curvature

 

[math]R = \frac{GM^2}{8\pi \rho_0 c^2}[/math]

 

If I have done this right.

Edited July 29, 2012 by Aethelwulf