Gravity and Advance Particle Physics; research and discussion

Schneibster

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OK, I've done some quick research and have some results to report. In addition, I've been threatening (heh) to start a conversation with Yllanes and Ben Tilly about physics. So I think I'll make a combination of the two, rather than be a hog and make two threads. Moderators can decide if I was wrong and split them later if they like.

So, here's the research status:
1. LIGO, the LASER Interferometer Gravity-wave Observatory experiment, a pair of mile-plus-long orthogonal LASER interferometers in Louisiana and Washington State intended to detect and measure gravity waves generated by astronomical events. The initial data run in 2005 has not been replicated; no identifiable gravity waves were detected during this run. A proposal has been floated to upgrade the detection systems; the first funding is expected in 2008 or 2009, and completion of the project and first light (an astronomer's slang term referring to the first use of a telescope to look at the sky, as opposed to calibration of the instrumentation) is expected around 2013. The sensitivity is expected to be increased by something like an order of magnitude.
2. Gravity Probe B, a satellite experiment which used four gyroscopes in orbit to plot gravity differences in Earth orbit. The primary mission is to examine the possibility of "frame dragging," a prediction of GRT that would result in the creation around spinning black holes of a region called the "ergosphere" and the possibility of the existence of a so-called "naked singularity." The data-gathering portion of the experiment has been completed. The first two phases of data analysis, which calibrate the gyros and then establish baseline measurements with which the results can be compared to see what happened, are complete; the third and final stage of analysis is underway, and is expected to complete sometime in February of 2007.

Both of these experiments have the capability to confirm GRT; Gravity Probe B has the possibility to at least question it, and perhaps show that it is incorrect or incomplete, if evidence of frame dragging is not found.
 
This is a recent article by John Baez on LIGO. If anyone is interested, I suggest they Google for LISA, a proposed more sophisticated set up (a huge Michelson interferometer, 5 million km long, formed by three satellites).

Gravitational waves, unlike atomic spectra, are very difficult to interpret, even if they are detected. The main reason is that atomic spectra are linear phenomena, while GR is highly non linear. This means that to identify a certain detection of gravitational waves, we must compare it with some sort of 'catalogue', which should be compiled using numerical calculations. This is no easy task: people working in numerical relativity have to model the collisions of black holes or neutron stars and lots of things go wrong very fast when this is attempted outside very simple cases. There are many opportunities for interesting results there, both in the theoretical calculations and in the detection.

A related interesting problem is that Einstein's equations admit several mathematically equivalent formulations. These formulations, when discretised and plugged into a computer, give different results... It is very important to understand why this happens because it is very likely some physics is behind it.

Both of these experiments have the capability to confirm GRT; Gravity Probe B has the possibility to at least question it, and perhaps show that it is incorrect or incomplete, if evidence of frame dragging is not found.
Click to expand...
I find it unlikely that it will give a negative result for GR, but it would be nice, we are desperate for one.
 
OK, I gave a quick look into it on the other thread. I'm going to do a post now actually showing the equivalence of the two methods (algebraic and hyperbolic trigonometric Lorentz transform); I'll draw on a post I did a while back for material. There probably will be no surprised here for Yllanes, but others might find it interesting.

Some asked me, "Why must a vector, of how ever many components, be seen in terms of rotation? (not "how can it" but "why must it"!)"

I responded:

The best way to explain this is to start with the equations for converting from rectangular to polar notation. And of course, before I do that, I'll have to explain what those two notations are.

Rectangular notation is just the normal notation for a graph or chart: horizontal and vertical (and you just add depth for 3D, or color, or something like that). It's also called "Cartesian coordinates," because Rene Descartes invented it. You learned all about this in school, and I know you know how it works because you understood Feynman diagrams. So the horizontal axis (and here I will note that this terminology perpetuates your bias that a rotational axis is the same as a dimension, when they are not the same! The correct terminology would be "horizontal dimension") is "x," and the vertical is "y." The place where they cross is "x=0, y=0," and is called the "origin of the coordinate system." The x value increases to the right, and the y value increases upward; negative numbers are permitted. You can now specify any point on the plane by its x value and its y value. The x value is sometimes also called the "ordinate," and the y value the "abscissa," although I have to say that in many years' playing with math, I have only come across this terminology ever being actually used, as opposed to pointed out, once or twice.

Now, it turns out that there are all sorts of fancy uses for this; everything from population studies on the effects of various environmental conditions to calculations of the phase-shift in a radio oscillator tank circuit uses this type of math. It is worth stopping to note that instead of having two real number lines, if you're only dealing with two dimensions, you can make the ordinate be real numbers and the abscissa be imaginary numbers (which are just a real number times the square root of minus one). In this case, you can represent the number as what is known as a "complex number," which is simply the sum of the real and imaginary parts. The coordinate system is then known as the "complex plane." This is done a great deal in electronics, and it is very convenient when dealing with coils and capacitors in AC circuits. It turns out that there are all sorts of links between complex numbers and the equations that govern various shapes that are described on charts and graphs; and when you really get into it, calculus lets you do some really important things with data plotted on a graph. There are also some interesting things you can do with complex numbers in relativity, and we'll get there soon. In any case, that is what rectangular coordinates are all about.

There is another way of specifying something's position on a plane, and that is to start with an origin just like in rectangular coordinates, but instead of specifying x and y, you specify some particular direction (and traditionally it's horizontally extending to the right, because that makes the numbers come out nice) as "the angle zero," and then you just give the angle of the position you want to describe from "zero" and the distance from the origin. You assign some value to the angle, and traditionally the symbol for the angle is the Greek lower-case letter theta (θ), and some value to the distance, and there you have it. Various symbols are used for the distance; d, r, and others. We'll stick with r for now, for "radius," since without the angle, the distance from the origin specifies a circle.

It turns out that this is a much easier way to describe things when you deal with vectors. That's because you can then reduce the velocity vector to a length, and all you have to specify is that length and the angle, and you've got all the information you need to derive the position at any time you care to name. And what's even cooler than that is that you can add two velocities together with a very simple equation when you describe them this way. So this "polar" coordinate system, as it's called, is very much easier than fooling about with the position-based system Descartes came up with. But eventually the time comes when you want to deal with position, and do the neat stuff you can do with the mathematics based on the Cartesian coordinate system; and the polar coordinate system turns into a three-headed firebreathing monster; you don't even want to have nightmares about the math you have to do. So you can't use these nice convenient vector descriptions any more; you have to convert them into rectangular coordinates.

Some bright soul came up with a way to do this. As it turns out, from trig we know that we can describe a right triangle whose hypotenuse (the longest side) is the distance; and then we can just drop a vertical to the horizontal axis of the Cartesian coordinate system and do some nice trigonometry, and out the other side come the x and y values. And the equations to do this are:

System 1
x = r(cos θ)
y = r(sin θ)

The reason for this is because of the trigonometric relationships between the lengths of the sides and the sizes of the angles in a triangle.

Now, this leads us to the conclusion that there must also be a means of converting from one coordinate system into another that is at an angle to the first, and there is; and of course, this conversion involves two polar-to-rectangular conversions, one for each of the two axes (again, remember, I'm using this terminology because it is traditional, not because it is right- the correct terminology would actually be "dimensions," not "axes"). And this pair of equations is, of course, the basis of the Galilei transform, that transforms one motionless frame of reference into another, provided they have the same origin. (To account for the origin, you just add the difference between the two origins after the equations below are done; we can safely ignore that for the purposes of this discussion.) And here are the equations to do this transform:

System 2
t' = t
x' = x(cos θ) + y(sin θ)
y' = y(cos θ) - x(sin θ)
z' = z

You can also add terms for z in there if you want to, and for two or three angles (one for each axis, and this time it is the right term), but the math gets a little complex, and what you see here is sufficient for my purpose, so I'm not going to take this particular example any further. I figure you can see how you would be able to derive the new x and y at the new angle.

Now, you also need to know how to derive the distance and the angle from the rectangular coordinates; it's simple enough. The distance is, of course, derived from Pythagoras' theorem, so

Equation 1
r² = x² + y²

and, since the tangent is defined as the opposite over the adjacent, and x is the adjacent and the opposite is the same length as the y coordinate, we also know that

Equation 2.1
tan(θ) = y/x

so the angle is simply

Equation 2.2
θ = tan⁻¹(y/x)

tan⁻¹ is also called the arctangent, abbreviated arctan.

So there you have it; equations to move from polar to rectangular, and rectangular to polar coordinates. Plus a system of equations to convert two frames of reference (did you catch that? My "coordinate systems" are now "frames of reference-" they got promoted) into one another.

You might think at this point that we're ready to move to four dimensions, but you'd be wrong. There's something else we have to discuss first, because it changes everything. That something else is the speed-of-light limit on all velocities, or more precisely, the fact that energy travelling at the speed of light (in other words, light) is always observed to be travelling the speed of light. This rather (from a mathematical standpoint) peculiar requirement imposes some very curious conditions on the way that things work in our four dimensional universe, compared to the way they might work if this were not true. And you've encountered those conditions, and seen the peculiarity of the math; that's what you're having trouble with.

So now we come to the Lorentz transform, which attempts to reconcile this observed constancy of the speed of light with how things seem to operate in our spacetime according to our measurements. And this turns out to be relatively complex, as we have seen in previous posts. There is a way to make it simpler, but it traverses some rather esoteric headspace. Stick with me here, and I promise you will understand relativity much better than you do now when I am done.

The esoteric part is a trigonometric function called the hyperbolic tangent, abbreviated tanh. This particular function has a very special property, which is that it can map its range of values between -1 and 1 to its domain of the real numbers, and do so uniquely in both directions. In other words, every tanh of any real number is between -1 and 1, and there is one and only one tanh per real number, and one and only one real number per tanh. A bidirectional mapping of this type is called a bijection.

Now, it turns out that defining the speed of light as 1 makes the Lorentz transform very simple. So this is really a good idea. And if we do this, then we can define a value called the rapidity (symbolized as "s") such that

Equation 3
v = tanh(s)

and because of the bijective mapping of tanh, we can know that we will get a unique value of s between 0 and ∞ for every value of v between 0 and 1, and that we will get a unique value of v for every value of s. (Remember, v=1 is the speed of light.)

OK, so what do we do with s? Well, it turns out that we can use s in a system of equations that are equivalent to the Lorentz transform, but that use "angles" (because this is not trig, it is hyperbolic trig, and the very meaning of the word "angle" is different) instead of coordinates; in other words, we can use polar coordinates! Here are the equations:

System 3
t' = t(cosh(s)) + x(sinh(s))
x' = x(sinh(s)) + t(cosh(s))
y' = y
z' = z

Now, remember, these are equivalent to the original Lorentz transform:

System 4
t' = γ(t-(vx/c²))
x' = γ(x-vt)
y' = y
z' = z
Where,
Equation 4
γ = 1/√(1-v²/c²)

And if you go back and compare the system of equations using the hyperbolic functions, System 3, with the prior system using the trig functions, System 2, you'll see that they're extremely similar. This is why people say that velocity is equivalent to a rotation; it is a rotation, but not the kind of rotation you're used to thinking of. Note that System 2 is the formula for transforming a coordinate system into a rotated coordinate system. And so is System 3; but the two coordinate systems for System 2 are comoving and have the same origin, so they differ only in the angle their axes (and there's that word again, and this time, it's the wrong one again- should be "dimensions") make with one another, whereas in the second case, the angles are the same, but they are non-comoving so there is a velocity difference, and this velocity difference is also a rotation, just like in System 2.

So you're looking at this, going, "Well, so what? That's neat, but what does System 3 have to do with System 4? I know System 4 is right, but I've never even heard of a hyperbolic tangent before." Here's the deal: remember that speed-of-light limit? That's what turns it into hyperbolic geometry.

But there's another important way to look at the relationship between Systems 3 and 4. The only way to make equations that look like System 3, but give the same answers as System 4 (which is the one we know is right), is System 3. There just isn't any other set of functions and definitions of values that will give that- but more importantly, there is a system of equations that is obviously, patently, and without question a description of a rotation- and that system of equations "just happens" to give exactly the same results as another set we came up with by a completely different process? Hey, guess what? That means that they match- and for these two to match, they both have to be describing different features of the same reality. And you know what? They DO!

And as it turns out, that's how our universe is constructed; that is, to quote Einstein exactly as he has been quoted many a time before, "the geometry of spacetime."

That was the end of what I wrote. I have not reviewed it for correctness; I'll try to do so before edit time runs out. Have fun!
 
Schneibster said:
And as it turns out, that's how our universe is constructed; that is, to quote Einstein exactly as he has been quoted many a time before, "the geometry of spacetime."
Click to expand...

SR is equivalent to hyperbolic geometry. This is true. It is not, however, what we mean when we talk about curvature of spacetime. The concept of curvature does not mean time and space mix in different ways when changing coordinates.

In GR, or in riemannian geometry, we have some manifold endowed with a metric, g, which tells us the norm of vectors, how much geodesics deviate, etc. Imagine you an a friend are on a euclidean plane 10 km apart and start walking in parallel directions in a straight line (the geodesic of the plane). No matter how long you walk, you will always be 10 km apart. No imagine you are on the Equator 10 km apart and start walking in parallel directions. This means following one meridian each. The meridians are great circles, which constitute the geodesics of the sphere. So you start walking along initially parallel geodesics. But as you walk you are constantly getting closer. This is the concept of curvature that is used in GR. If you drop two tests masses from a great height, they start their movement parallel to each other but start getting closer (because both point to the centre of the Earth). This is explained by saying spacetime is curved.

In differential geometry, the concept of curvature is carried by the Riemann tensor: Ra bcd. This object is a combination of derivatives of the metric. In Minkowski spacetime, [latex]$g =\left(\begin{smallmatrix} -1 \\ & 1\\ & & 1\\ & & & 1\end{smallmatrix}\right)$[/latex], so the derivatives are all zero everywhere.

In 2 dimensions, it should have 2^4 different components, but the many symmetries it has reduce them to only 1 independent component, essentially the Gauss curvature of classical differential geometry. In 4 dimensions, of its 4^4=256 different components only 21 remain. We need 21 numbers to describe the curvature of spacetime at any given point. All of them are zero for Minkowski spacetime.

For simplicity, I assume we have a coordinate basis, so that g is formed by the products of its vectors. In differential geometry, vectors are differential operators (equivalent to the partial derivative in their direction), so the basis defines partial derivatives.

Explicit formulas:

Christoffel symbol: [latex]$\Gamma^\gamma_{\alpha \beta}=\tfrac12 g^{\gamma\delta}(\partial_\alpha g_{\delta\beta} + \partial_\beta g_{\delta\alpha} - \partial_\delta g_{\alpha\beta})$[/latex]

(with tensors g^a v_a means u^1 v_1 + u^2 v_2 + ... u^n v_n, so you see the expression is quite complicated).

And the Riemann tensor: [latex]$R^\epsilon{}_{\alpha\delta\beta}=
\Gamma_{\alpha\beta}^\gamma\Gamma^\epsilon_{\gamma\delta}-
\Gamma_{\alpha\delta}^\gamma\Gamma^\epsilon_{\gamma\beta}
-\partial_\beta \Gamma^\epsilon_{\alpha\delta}+
\partial_\delta\Gamma^\epsilon_{\alpha\beta}
$[/latex]

So it is a threatening formula, but the concept of curvature is well defined. If we repeat indices we say we are contracting them, and the order or the expression changes: R^a_bad = R^0_b0d + R^1_b1d + R^2_b2d + R^3_b3d = R_ad. This new tensor is called Ricci's tensor and is the one that appears in the famous Einstein equation:

[latex]
$$
R_{\mu\nu}-\tfrac12 g_{\mu\nu} R = 8\pi T_{\mu\nu}
$$
[/latex]

Notice this equation does not include all of the Riemman tensor. There is another part, Weyl's tensor, which is responsible for tidal forces and, indeed, for all graviational effects outside the sources. Notice that the particular values of each component vary from one reference frame to another, but they vary in the same way as T does (T is the momentum and energy density). we

This is the curvature we talk about in GR. It implies that SR is flat.
 
Schneibster said:
That was the end of what I wrote. I have not reviewed it for correctness; I'll try to do so before edit time runs out. Have fun!
Click to expand...

great post! Very interesting stuff. In fact it's the first post i've ever printed out for posterity...
 
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I was going to start a thread on gravity, but this thread will do just fine.

In my paper last night was an article about some Racine, Wisconsin guy (that is where *I* came from!) has discovered gravity...or somehow has some sort of lock on its fundamentals. The article claims the guy sleeps and drinks gravity. That is all he thinks about. He lays awake at night pondering gravity. He is currently trying to confer with physics profs on his ideas. The photo shows him in front of a blackboard with two formulas. One says E = MC squared, and the one below it says G = E cubed.

Do scientists think or KNOW that gravity has gravity waves?

Is there any possibility that gravity could be a result of gyroscopic particles? (The reason I mention this is...do you remember me telling a story where 25 years ago, while I was blowing off water lines down in Texas where I was in charge of a small community water system), some guy pulls over on the side of the road and starts a friendly conversatuion with me? Do you remember that story? In no time flat, he quizes me on my religious leanings. Then when he thought I was some cocky atheist, he lays on me how I really don't know all that I think I know... and he goes into this lecture on how gravity works and how there are 3 or 4 components to it... and one that I remember was gyroscopic particles. The guy left, and I never saw him before, nor after. (I knew practically all the regulars in that community). It's like he was sent from heaven.
 
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There are no direct observations of gravity waves. There is at least one pair of (neutron?) stars whose mutual orbit is decaying in a way consistent with them radiating energy in the form of gravity waves.
 
I brought in the article with me, today. The guys name is Raffi Abagian. An Armenian from Racine, Wisconsin. He is an amateur physicist, who is "hot on trail of theory involving gravity".

"Raffi Abaganian has a big idea. The kind of idea that cold change the world.

But there is a problem. He doesn't know if he is right.

Abagian's idea isn't the next big thing in cell phones or toys or cars or any other gadget someone might need to make life more complete. His idea is really big. Universe big.

..,......

Physics is a hobby he said. 'Quantum physics, dynamics. I read a lo't, he said.

.......

He thought so much that he couldn't sleep.

'I started postulating over this, trying to figure out where I went wrong', he said. 'I wasn't able to.'

( He called up a college prof and he recommended Raffi get in touch with Stephen Hawking.)

.......

Abagian's idea is this: 'Gravity is independent and absolute. It's seperate from Einstein's theory of relativity. Gravity, in fact, is the thing that keeps the whole universe in place', he said. 'When the big bang expanded, gravity kept it in place, even then', Abagian said."

~~~~~~~~~~~~~~~~~~~~~~~~~~

Jasnine Anderson, from the Racine Journal Times, did this story.
 
Thanks!, Unnamed! THAT could have saved me a lot of typing. And what is neat about the site is all those comments below it.

Edit: And you'll find MY comment at the bottom of the heap!! LOL. I could not resist in commenting with the others.
 
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Yllanes said:
This is the curvature we talk about in GR. It implies that SR is flat.
Click to expand...
I think that what we have in mind when we say "flat" is somewhat different from what most people mean when they say "flat," and in fact that if they actually visualized it (or even messed with it mathematically and considered the implications of what the math they were using meant) they would say that it is "curved."

In other words, given your definition of "flat," I agree with you- it is the definition that I am questioning. I don't make this as an argument that you are wrong; merely that the interpretation you (and I don't question that most physicists agree with this interpretation) are using is not the only one available, or even the most intuitive one from most peoples' point of view.

Let me put this another way: the only spatial location at which what you see is simultaneous with what is happening is the origin of the frame of reference you are currently using- in all frames in which that origin is different, you can only observe what was happening in that frame at the time that the light you see now left it, which is a time that is necessarily in the past. This is also implicit in the fact that the geometry of spacetime is hyperbolic, and in fact it indicates the degree of observed curvature; that curvature is just sufficient to ensure that you observe all points' "now" to be related to your "now" by a temporal difference that increases with spatial difference between them. "Now" is in fact an extremely slippery concept, except at the origin point of the frame of reference you are currently using; and the meaning of "now" must be translated (i.e. transformed) when you change that origin, supposing that you do so instantaneously rather than via continuous physically permitted translation and rotation.

Do you see what I mean?
 
Schneibster,

When I first began reading your explanation you shed sufficient light on the matter for me to see what you were talking about but (funny as it might seem) the moment I finished reading it I was in the dark.

Gene
 
Schneibster said:
Do you see what I mean?
Click to expand...
Yes. Yours is a concept of curvature, but not the one used in GR. You say that because of the fact that the signature of the metric is Lorentzian (-+++) a concept of cruvature arises. There is nothing really wrong with it, but the other point of view is more convenient, because it is the one that appears in Einstein's equations and governs the behaviour of gravitation.

For example, in cosmology we talk about three possible geometries: positive curvature, negative curvature and flat. This refers only to the spatial part of the metric. If the universe were 2D, those geometries would be a sphere, a saddle (better, a pseudosphere) and a plane. And the idea that a plane is flat, while a sphere or saddle aren't is, I think, intuitive. The curvature of a surface is described in classical differential geometry by several mathematical objects (Wingarten's operator, etc.) summarised by Gauss's curvature K. We can arrive at K by considering geodesics on the surface or the radius of curvature of curves in different directions. If we want to jump to higher dimensions, we see that a single number is not sufficient: we need a tensor (Riemann's tensor). But the concept of what is curved and what isn't is the same as in 2D (in fact, Riemann's tensor in 2D reduces to K).

I agree that your concept may be more intuitive in the context of SR, but a disclaimer saying that it is not the curvature used in GR could be a good idea. Otherwise, people reading your very nice explanations about SR and hyperbolic geometry would get the impression that that is the curvature used in cosmology, black holes, etc.
 
Agreed, Yllanes. I think both concepts are useful, and separate. I will do my best to bear this in mind in the future. Thanks for your advice and review.
 
Here's another thing to think about.

If gravity is curvature of space, and the equivalence principle is correct, then isn't acceleration productive of curved space?

And by the way, if the equivalence principle is correct, then how can there be a difference between inertial and gravitational mass?
 
Gravity Probe B results are available, but unfortunately they have encountered a glitch. Newly discovered torque and sensor effects must be accurately modeled and removed from the results before they can be evaluated for agreement within the accuracy of the experiment against the predictions of frame dragging in GRT. They have, however, confirmed the geodetic effect to within 1% or so.

Details here.
 
Yllanes said:
A related interesting problem is that Einstein's equations admit several mathematically equivalent formulations. These formulations, when discretised and plugged into a computer, give different results... It is very important to understand why this happens because it is very likely some physics is behind it.
Click to expand...

Could you expand on this please. If the continuum limit is the same, why is it likely there is physics behind it?
 
As far as the whole curvature thing goes, I don't think I understand what Schneibster means by Minkowski space being curved. The most intuitive definition I can think of for curved space is that parallel lines don't remain at a fixed distances from each other. In special relativity, parallel lines (space-like, timelike, or even null) do remain at a fixed distance from each other.
 
Schneibster said:
And by the way, if the equivalence principle is correct, then how can there be a difference between inertial and gravitational mass?
Click to expand...

Isn't that the entire content of the equivalence principle: that inertial and gravitational mass are the same? Isn't this one of the postulates of GR?
 
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