Shortest Paths

[□h=-16πT]

I knew you'd say something in reply to my mention of the speed of light being constant. The very thing you're learning here is based upon this principle, so if you're going to reject it you may as well stop learning SR/GR right now. Why do you believe simultanaety is absolute: that all observers observe the same event at the same time? Don't you dare give it up though, I've spent ages teaching you this! If you do I'll punch you. Thanks for complementing my teaching ability, it's not as good as you say in person sadly .

[Johnny5]

They're called geometrized units because dealing with the geometry of space-time etc. Just as we may call the units of SR, where just c=1, "natural" because it seems natural to have such a fundamental constant, of which the entire theory is based on, just blend into the background, so to speak.

 

I'd prefer to leave them in the formulas for now, if that's ok with you.

[□h=-16πT]

I'd prefer to leave them in the formulas for now, if that's ok with you.

 

I'll be leaving them out. If you see the mention of mass anywhere in any equations I use, just place a G/c² in front of it. In the more complex cases you'll have to convert the units back to SI for yourself or just take it as it is (it changes nothing) because I'm lazy; if there's a method for making my work easier I'll use it.

 

In deriving the Riemann tensor there'll be no need for the use of geometrized units, as it's simply pure maths and no physics.

[Johnny5]

I knew you'd say something in reply to my mention of the speed of light being constant. The very thing you're learning here is based upon this principle, so if you're going to reject it you may as well stop learning SR/GR right now. Why do you believe simultanaety is absolute: that all observers observe the same event at the same time? Don't you dare give it up though, I've spent ages teaching you this! If you do I'll punch you. Thanks for complementing my teaching ability, it's not as good as you say in person sadly .

 

You say it is based on that principle, I have not seen that yet. So far, all that I have seen is notation, which is a necessary thing that must be learned, so this is perfect for a beginner, which is what I am.

 

As for the theory of General relativity, I have always understood that its fundamental postulate is the principle of equivalence (not the constancy of the speed of light in vaccum), and there are strong, and weak versions. And I understand that as either gravitational mass = inertial mass, OR a reference frame at which the center of mass of some gravitating body is at rest, and some object is at rest relative to the body, that the laws of physics where that observer is, in the gravitational frame, are the same as if the observer is uniformly accelerating in deep space.

 

Now, I have never seen anyone go from the assumption of GR (which is the principle of equivalence) directly to the field equations, and I certainly would like to see this done, because...

 

I have an argument which leads to the conclusion that the two kinds of frames are not equivalent, and that argument is based upon formulas from electrodynamics.

 

I have heard that the general theory of relativity leads to the special theory of relativity, as a special case, but this has never been proven to me.

 

Regards

[□h=-16πT]

You say it is based on that principle' date=' I have not seen that yet. So far, all that I have seen is notation, which is a necessary thing that must be learned, so this is perfect for a beginner, which is what I am.

 

Now, I have never seen anyone go from the assumption of GR (which is the principle of equivalence) directly to the field equations, and I certainly would like to see this done, because...

 

I have an argument which leads to the conclusions that the two kinds of frames are not equivalent, and that argument is based upon formulas from electrodynamics.

 

I have heard that the general theory of relativity leads to the special theory of relativity, as a special case, but this has never been proven to me.

 

Regards[/quote']

 

Yes, GR is based upon the equivilance of gravity and acceleration (the weak equivilance principle it's called, and there is also a strong but I'll discuss that later). GR is a generalisation of SR, GR still assumes c is constant in all inertial frames because of this generalisation. The SE tensor incoporates relativistic effects and you've already seen that this tensor appears in the field equations of GR. Relativistic effects are everywhere in GR, they're used to simplify equations in situations where it's not hugely relativistic and to reduce things to the Newtonian case. Yes, GR reduces to SR in cases where there is no gravity and hence the components of the metric of space-time are constant, i.e. that of SR.

 

It's important that you know the principles on which SR is based before you learn the maths. There's no point in me writing them, SR is everywhere.

[Johnny5]

I'll be leaving them out. If you see the mention of mass anywhere in any equations I use' date=' just place a G/c² in front of it. In the more complex cases you'll have to convert the units back to SI for yourself or just take it as it is (it changes nothing) because I'm lazy; if there's a method for making my work easier I'll use it.

 

In deriving the Riemann tensor there'll be no need for the use of geometrized units, as it's simply pure maths and no physics.[/quote']

 

If you are going to leave out G,c; then could you briefly explain this notion of geometrized units, I would like to check the concept for errors first.

 

Let me think about something for a moment too.

 

In non-geomtrized units the Einstein field equation is:

 

[math] G_\alpha_\beta = \frac{8 \pi \G}{c^4} T_\alpha_\beta [/math]

 

In geometrized units, the same formula is to be this:

 

[math] G_\alpha_\beta = 8 \pi T_\alpha_\beta [/math]

 

Right??

 

So where is the "mass" term in the formula above?

 

Regards

[Johnny5]

Yes' date=' GR is based upon the equivilance of gravity and acceleration (the weak equivilance principle it's called, and there is also a strong but I'll discuss that later). GR is a generalisation of SR, GR still assumes c is constant in all inertial frames because of this generalisation. The SE tensor incoporates relativistic effects and you've already seen that this tensor appears in the field equations of GR. Relativistic effects are everywhere in GR, they're used to simplify equations in situations where it's not hugely relativistic and to reduce things to the Newtonian case. Yes, GR reduces to SR in cases where there is no gravity and hence the components of the metric of space-time are constant, i.e. that of SR.

 

It's important that you know the principles on which SR is based before you learn the maths. There's no point in me writing them, SR is everywhere.[/quote']

 

I understand the principles upon which SR is based, you do not have to explain them.

 

Furthermore, I can follow the logic, by seeing your statements as 'if' such and such then such and such. So the fact that I know simultaneity is absolute, will not hinder my following your logic.

 

Lets see...

 

Oh, ok, I have a question...

 

Is there any modified version of GR, in which the speed of light isn't the same in all inertial frames? And if so, what is it called?

 

Thank you

[□h=-16πT]

If you are going to leave out G' date='c; then could you briefly explain this notion of geometrized units, I would like to check the concept for errors first.

 

Let me think about something for a moment too.

 

In non-geomtrized units the Einstein field equation is:

 

[math'] G_\alpha_\beta = \frac{8 \pi \G}{c^4} T_\alpha_\beta [/math]

 

In geometrized units, the same formula is to be this:

 

[math] G_\alpha_\beta = 8 \pi T_\alpha_\beta [/math]

 

Right??

 

So where is the "mass" term in the formula above?

 

Regards

 

The mass term is in the SE tensor, and the extra 1/c² bit comes from other units that need changing back into SI. Ok, geometrized units.

 

c=3x10^8 m/s

 

if we take one second to be equal to 3x10^8 metres then we can normalise c and make it dimensionless. These are just units we're playing with, it doesn't matter if we start changing them, as long as we change them back when we compare to experiment etc. and besides we're taking time to be a dimension so why not give it dimensions of length...? Next: normalising G

 

G=6x10^-11m³/kg/s² in SI

 

If we slap in s=c m we have

 

G=6x10^-11m/kg/c² (the metres cancel)

 

So if we make one Kg equal to G/c² metres we can normalise G and make it dimensionless as well. Thats all ther is to it. Don't start debating the validity of this, they're just units and can easily be converted back into SI. It makes sense to have such fundamental constants unified and at the end of the day it's a labour saving device not a method of confusing students.

 

To your last question: no there isn't. Just accept this principle and read up on the Michelson-Morely experiment. Whether you believe it or not is irrelevant if you want to learn a theory in which it is true.

[Johnny5]

The mass term is in the SE tensor' date=' and the extra 1/c² bit comes from other units that need changing back into SI. Ok, geometrized units.

 

c=3x10^8 m/s

 

if we take one second to be equal to 3x10^8 metres then we can normalise c and make it dimensionless.[/quote']

 

How in the world, can anyone justify taking a unit of distance, and equating it to a unit of time?

[□h=-16πT]

How in the world, can anyone justify taking a unit of distance, and equating it to a unit of time?

 

They're just units! Physicists change them because they're lazy and one can easily change the units back into SI. If you'd rather not convert them because it seems convoluted then so be it. To me it seems perfectly rational.

[Johnny5]

They're just units! Physicists change them because they're lazy and one can easily change the units back into SI. If you'd rather not convert them because it seems convoluted then so be it. To me it seems perfectly rational.

 

By setting the constants of nature to 1, you are losing information.

 

Cant you see that?

 

Regards

[Johnny5]

 

To your last question: no there isn't. Just accept this principle and read up on the Michelson-Morely experiment. Whether you believe it or not is irrelevant if you want to learn a theory in which it is true.

 

 

Agreed.

 

From memory:

 

Michelson Morely experiment' date=' late 1800's, Albert Michelson wanted to measure the speed of light. He used a device which was a modified version of the device used by Fizeau to measure the speed of light.

 

It involved a spinning mirror. The rate of revolution was known. Through some, perhaps specious line of reasoning, there was to be a difference in the diffraction pattern being observed.

 

I remember this too...

 

That it was thought, by Michelson, that the speed of light would be c in some special reference frame, but that in others it would be different.

 

Using Michelson's train of thought, if the earth were at rest in the special frame, then the speed of a particular electromagnetic wave would be c.

 

But...

 

If instead, the earth was moving relative to that same photon, parallel to its direction of motion relative to earth, then if v denoted the relative speed between the rest frame of the earth, and the rest frame of the photon, then

 

1. if the earth was moving at the photon, the speed of the photon at impact would be v+c, and

 

2. if the earth was moving away from the photon before impact, then the photon would strike whatever measured it with speed c-v.

 

Albert Michelson thought, like others, that light was an electromagnetic wave (I suppose this was due in part to Maxwell's theory, in which the solution to Maxwells equations lead to wave equations, where the speed of an EM wave is [math'] 1/\sqrt{\epsilon_0 \mu_0}) [/math]. The photon idea came later, with theoretical physicist Albert Einstein's explanation of the photoelectric effect.

 

So Michelson performed his experiment at a time of year when the earth was moving in one direction relative to our sun, which emits "light".

 

And, six months later, he repeated the measurement.

 

Now the earth travels a roughly circular path around the sun.

 

So therefore, six months later, the earth was moving in the opposite direction it was moving six months earlier (in the rest frame of the sun).

 

So...

 

 

Let v denote the tangential speed of the center of inertia of earth in the sun's rest frame.

 

Look at the solar system from above. Let the center of inertia of the earth, and the center of inertia of the sun be located in the XY plane of a three dimensional rectangular coordinate system. Let the origin be the center of inertia of the sun.

 

Thus, the earth is moving circularly, about the origin of the frame, and the motion is constrained to be in the XY plane.

 

At one moment in time, the velocity of the earth is given by:

 

[math] \vec v = -v \hat i [/math]

 

And six months later, its velocity is

 

[math] \vec v = v \hat i [/math]

 

So its tangential speed has been presumed to be constant, roughly so, since the gravitational force is approximately constant, and the earth is travelling through a vacuum, which we assume does not impede the foward motion of the earth in any way.

 

Now, I don't perfectly remember the device, because it isn't ever explained clearly, and furthermore there was some kind of error in the derivation. I remember there was, something strange about the S and S` thing in the derivation. The book was Serway, Moses, Moyer, modern physics.

 

At any rate, I will re-analyze the Michelson experiment, and show you where the error is.

 

Moving on...

 

 

Oh yeah...

 

Suppose Michelson's basic idea was right, and that he should be able to detect the earth's motion through the luminiferous ether...

 

We get something like this...

 

In order for the Michelson experiment to work, the experiment must be carried out during the daytime.

 

Now the earth is spinning on its axis... we all know it's spinning, because we don't all burn up. 12 hours of day, followed by 12 hours of night.

 

And the sun regularly traces out the pretty much the same path in the sky, day after day... rises in the east, and sets in the west.

 

So lets see what part of the earth the Michelson experiment took place in, I want to say New York...

 

No, I was incorrect, it was Annapolis, so Maryland.

 

The experiment took place in 1879.

 

So, lets think about the earth's motion in the XY frame.

 

Lets stipulate that the earth is moving counterclockwise in the frame.

[□h=-16πT]

By setting the constants of nature to 1' date=' you are losing information.

 

Cant you see that?

 

Regards[/quote']

 

I can see what you're getting at. But the geometrized units system is just a system, just as SI is and imperial, metric etc. G and c are of extreme importance to relativistic theories, so why not create a new system in which these simply "blend into the background" of the theory? You aren't losing information because the constants are being "absorbed" into the theory, changing all other units (such as pressure etc.) and one can just as easily attain a physical interpretation of an equation in relativity by considering or ignoring the units, or converting back to SI, as you can in good ol' mechanics. We use light metres and light years to measure times and distances repectively, so why not take it one step further and make time a length?

 

I know exactly where you are coming from and I knew you'd question the logic of geometrized units, but at the end of the day it is, as I have already said, a labour saving device not a method of confusing people and making physics more convoluted than it already is.

 

The experiment was originally designed to attain the absolute velocity of the earth by measuring its motion relative to the ether (which was theorised to be an absolute rest frame). The idea being that if the ether existed light should travel through it at a constant speed and that different frames would attain different values of the speed of light, hence determining the absolute velocity of the earth. However, when the experiment was conducted light was found to be travelling at the same speed predicted by electromagnetism, by an observer at rest, despite the motion of the earth. We already know that Earth moves so, upon further experiments in different frames, one can postulate that the speed of light remains constant for all inertial frames.

[□h=-16πT]

I can see this simple thing getting in the way here. So lets forget about it.

[Johnny5]

I can see this simple thing getting in the way here. So lets forget about it.

 

No, I feel I need to fully understand not only the Michelson Morely experiment, which took place around 1879, but also the General theory of relativity. I would be interested in any version of it, which incorporates particle exchange.

 

Thank you, and regards

[□h=-16πT]

No' date=' I feel I need to fully understand not only the Michelson Morely experiment, which took place around 1879, but also the General theory of relativity. I would be interested in any version of it, which incorporates particle exchange.

 

Thank you, and regards[/quote']

 

I ment geometrized units, not the MM experiment. Geometrized units can be left out with no change to the theory. I'm going to finish that thing on th Riemann tensor later this evening, so it'll be posted tomorrow evening (I have school you see and this site is blocked there, for some reason).

[Johnny5]

I ment geometrized units, not the MM experiment. Geometrized units can be left out with no change to the theory. I'm going to finish that thing on th Riemann tensor later this evening, so it'll be posted tomorrow evening (I have school you see and this site is blocked there, for some reason).

 

Riemann tensor... ok I still haven't seen that yet.

 

My linear algebra review is going well by the way.

 

Also, do the formulas for the general theory of relativity, connect to the concept of path of least resistance?

 

Like suppose that the sun is a giant magnetic field source.

 

Sending out magnetic monopoles that interact with the earth's Van Allen Belt.

 

Then the earth is moving through a swarm of particles.

 

Can the formulas be extended to this idea? "Path of least resistance?"

[□h=-16πT]

Riemann tensor... ok I still haven't seen that yet.

 

My linear algebra review is going well by the way.

 

Also' date=' do the formulas for the general theory of relativity, connect to the concept of path of least resistance?

 

Like suppose that the sun is a giant magnetic field source.

 

Sending out magnetic monopoles that interact with the earth's Van Allen Belt.

 

Then the earth is moving through a swarm of particles.

 

Can the formulas be extended to this idea? "Path of least resistance?"[/quote']

 

My education in electromagnetism is more or less nonexistant so I can't help. My self education lacks structure and I pretty much learn what I want to learn, a case of learn what takes my fancy. Initially there was some structure: I'd always wanted to learn GR and SR so I taught myself all the prerequisits and got an intro book on GR and topology/diff. geometry. Now my education is all over the place; the other month I read something on Lebesgue integral and measure and now I'm educating myself in that. I'm reading books on classical mechanics and a book covering all the maths in most physics/engineering courses as well as loads other. I'll do some reading up on EM theory whenever I find a decent looking book on it in the library. So, until then, I'm of no help.

[Johnny5]

My education in electromagnetism is more or less nonexistant so I can't help. My self education lacks structure and I pretty much learn what I want to learn, a case of learn what takes my fancy. Initially there was some structure: I'd always wanted to learn GR and SR so I taught myself all the prerequisits and got an intro book on GR and topology/diff. geometry. Now my education is all over the place; the other month I read something on Lebesgue integral and measure and now I'm educating myself in that. I'm reading books on classical mechanics and a book covering all the maths in most physics/engineering courses as well as loads other. I'll do some reading up on EM theory whenever I find a decent looking book on it in the library. So, until then, I'm of no help.

 

Well I know EM theory, but it sounds like you have your hands full. But I am enjoying learning the general theory of relativity from you, so any way that you want to teach it, is ok by me.

 

What is it i should study between now and tomorrow... the Riemann tensor?

[fuhrerkeebs]

Can the formulas be extended to this idea? "Path of least resistance?"

 

Why would it take the path of least resistance? The way I see it, it would be taking the path of most resistance, because matter is attracted to matter, and the path that contains the most matter would have the most resistance.

[Johnny5]

Why would it take the path of least resistance? The way I see it, it would be taking the path of most resistance, because matter is attracted to matter, and the path that contains the most matter would have the most resistance.

 

I don't get this in the least.

 

I was thinking of a stream.

 

Imagining earth to be traveling through a swarm of magnetic particles, some passing through the earth, others interacting with the earth, to give it its elliptical path around the sun.

 

Suppose that there is something in the way of the path of the earth.

 

The earth has some momentum, its barelling through space... following the law of inertia.

 

Particles that get in the way, earth just pushes aside.

 

But in the process, during the magnetic/graviational interaction, the center of mass of earth changes position slightly, in the solar system rest frame.

 

The net effect must lead to an elliptical orbit, with the center of mass of the solar system at one foci.

 

The formula to be obtained is:

 

Link to Wolfram on the ellipse

 

[math] \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 [/math]

 

The equation above was the one I was thinking of.

 

Notice, that in the discussion of the ellipse at Wolfram, the center of the ellipse is defined to be the point midway between the two foci of the ellipse.

 

Wolfram also has a picture of how to construct an ellipse, using two tacks, and a string, and your pencil. Professor Richard Feynman discussed this method for constructing an ellipse in his now famous Feynman lectures. In the lecture I am thinking of, which took place in California, I think Caltech, professor Feynman was talking about gravity. The whole lecture was about Kepler's laws. What Professor Feynman was attempting to do, was use primitive geometry to derive Kepler's laws of planetary motion.

 

The equation above is not true for the center of an ellipse at some arbitrary point in a frame, the center of the ellipse has to be located at the origin of the frame in order for that definition to be true.

 

But given that it is, you can interpret a,b just as they do at wolfram.

 

The key thing to remember about the ellipse is that the sum of the distances from foci to point which is in orbit, is a constant. You can see this fact from the Wolfram diagram.

 

 

A special case of an ellipse is a circle. For a circle a=b=R, where R is the radius of the circle.

 

And in this case, the two foci are located at the origin.

 

Now, if you look at the orbit of the point in the Wolfram diagram, you should notice that there is an angle defined right where the two lines meet.

 

As a later excercise we are going to compute that angle.

 

For right now, it suffices to just learn what a,b denote, and the answer is in the figure at Wolfram.

[□h=-16πT]

Well I know EM theory' date=' but it sounds like you have your hands full. But I am enjoying learning the general theory of relativity from you, so any way that you want to teach it, is ok by me.

 

What is it i should study between now and tomorrow... the Riemann tensor?[/quote']

 

What's your email? I have a good .PDF file I can send you that takes SR and GR up to a really high level. It's about 1MB and around 200 pages, and I think they're lecture notes for the University of California.

[Johnny5]

What's your email? I have a good .PDF file I can send you that takes SR and GR up to a really high level. It's about 1MB and around 200 pages, and I think they're lecture notes for the University of California.

 

I'd prefer to learn it from you, but if you're too busy don't worry about it.

 

 

Riemann tensor...

 

here is a wikipedia link to Reimann tensor.

 

 

I can't make any sense out of that.

[□h=-16πT]

I can't make any sense out of that.

 

It's not a very good discussion. The upside down triangle is the notation for the covariant derivative.

 

Look up parallel transport and geodesics. Then consider the parallel transport of some vector around a closed loop on a general Riemannian manifold and then compute the change in the vector upon traversing fully this loop. Use the equation of geodesic and some of the definitions I gave in my other post. If your up for a challenge that is; if not I'll derive it for you and give a decent discussion of it ASAP.

[Johnny5]

It's not a very good discussion. The upside down triangle is the notation for the covariant derivative.

 

Look up parallel transport and geodesics. Then consider the parallel transport of some vector around a closed loop on a general Riemannian manifold and then compute the change in the vector upon traversing fully this loop. Use the equation of geodesic and some of the definitions I gave in my other post. If your up for a challenge that is; if not I'll derive it for you and give a decent discussion of it ASAP.

 

Do you think I can handle your discussion?