decraig

Here’s something rough, in the context of differential geometry. We have a smooth parametric curve AB, on some manifold of arbitrary dimension, n. We want to define a length along the curve from A to B, such that it is natural to define the ‘length’ from B to A to be the negative of the length AB. Use coordinate basis (to keep it simple). At each point on the curve attach a coordinate system. The coordinate system at point A is arranged so that a dual basis vector ds(0) points along the curve. We need another coordinate system displaced along the curve by ds(0). It also needs a dual basis vector. Call it ds(1). And so forth. We want keep propagating in one direction along the curve, instead of bouncing back and forth or changing to some other dual basis vector. To do this we need to demand that the coordinate transformation matrix is a continuous function of our parameter. Most of the machinery seems to be set-up. Now integrate ds. [math] L_{AB} = \int_A^{B} ds[/math] [math] L_{BA} = \int_B^{A} ds[/math] There’s something still missing (I’m doing this off the top of my head). We have a parametric curve, and the parameter is missing from the expressions. I suppose that’s where the transformation matrices come in. Finally, we would probable like something to relate this back to the tangent space along the curve.

Ok... An incipient back hole is no more speculative than a back hole. Less, it seems, not requiring the central singularity of a black hole. I don't know if you intended to include this under the adjective 'speculative' or not. The evidence in support of incipient back holes is the very evidence in support of a galactic centered black hole. To be sure, these are not black holes, but might become black holes. After some reflection, I think there may be a way (within the ground rules: relativity theory, particle tunneling, and thermal dynamic) for this to happen. I'm going to wait and see if anyone can spot it. But let me ask you something. If I hadn't given this class of objects a name, and pointed to an ArXiv article that also used that name, would you have dismissed me as blathering? This is what I was up against before someone came up with a name I could use.

I proffered an incipient black hole as such an object. http://arxiv.org/abs/gr-qc/0609024 You should know the distinction between a mathematical model, and physical realisability. I have made several points, one not necessarily supporting another. This is something you seem to have read into what I have written. It is not I who am making a prediction of a galactic center black hole. In the spirit of scientific fairness the burden of proof really falls upon those making extraordinary claims. I, as yet, remain unconvinced by the arguments put forward on this thread.

Thanks. On the surface, the integral appears too ugly be true. It's hardly surprising that is fails general covariance. According to Wikipeda, the stress energy tensor with upper indices is integrated. Take for instance, a space-space component. It's really an element of per-distance squared. Now integrate over the cube of distance. We are off by 5 dimensions!

How is this being "out"? Relativistic mass is Newtonian inertial mass. It is that which resists a change in velocity, m = F/a. It can also be measured with a scale, not the invariant mass. You are free to use E= Fc^2/a. Should I have to need to use both masses, there already exists standard notation, without using the funny 'm'. Would you also object to defining torque because it has the same units as energy? There is also the action 4-density and energy density. They share the same units. Each are useful and measure different things. This was a thread on density, and has gone far afield, nit-picking over definitions that are already well defined. We could simply recognize that Endcreeper could be answered in terms of energy density.

"The black hole at the center of the galaxy..." is an oft repeated phrase on the science channels. 1) The cited evidence for this is the motion of stars at the center of the galaxy. This evidence is insufficient. An object of highly collapsed matter would have the same effect. How would one distinguish between an incipient back hole and a back hole? In an incipient back hole the bulk of the matter lies outside the surface where the event horizon would form. Can highly collapsed matter exchange Hawking radiation with it's environment? 2) The Schwarzschild solution having an event horizon is non-physical. The solution stretches over transfinite time. I imagine the same is true of the Kerr solution and the rest. Correct? Should there be a black hole at the center of our galaxy, it would be a primordial black hole, having gained or lost very little mass, and only through Hawking radiation. 3) The astronomy Gugus of the science channels also like to talk about things crossing the event horizon. Nothing crosses the event horizon to a stationary observer outside a back hole with the exception of virtual particles. To an Earthly observer, things don't cross the event horizon. Any objections? 4) We could imagine the Earth, some day, falling through the event horizon, the coordinates singularity defined by an outside stationary observer. Then we would see things falling through with us. But this cannot happen. The Earth would reach the horizon in transfinite number of seconds as measured by an outside observer. This gives our black hole an infinite amount of time to radiate into an expanded, very cold universe and vanish before the Earth, or any would-be adventurer crossed the horizon. [Edit:] One more thing: What is the theoretical support for primordial black holes and what quality?

Unfortunately, it's broken twice, but it does depend upon who is doing the accounting. Say you are in some inertial frame of reference. Simultaneously, the momentum disappears from system A, and appears in system B. But, I have some non-zero velocity with respect to your inertial frame, such that the events are no longer simultaneous. For me, the momentum either disappears for a while, or doubles-up. But, this is the Classical section of this forum, so maybe this answer is out of bounds. Hello again, ajb. Could you share these expressions. I'm hard pressed to find an expression for the momentum of a system of ponderable matter fields.

This is really a problem in relativity. To simplify, I reduced the problem to 2 dimensions. L is the action density, represented not as a pseudo scalar, but an oriented 3-form. The integral is a Lagrangian. S is the scalar action of a space time 4-volume. It is an invariant over a curved manifold. S_x would be energy density of a system. y is the temporal coordinate. S_y is a momentum flux density term. Does that help?

jajrussel, On electron orbitals: The modern idea of an electron orbital is more like the vibrations of a drum skin. You hit the center of the drum, and the whole skin vibrates up and down. If you hit the drum off-center, the half you hit goes down. Then it oscillates. Half the drum head goes down while the other half is going up. These are fairly good models of the lowest energy states of an electron orbital (yes, they are still called 'orbitals') corresponding to the S1 and a P orbital. More complicated arrangements of these standing wave vibrations are possible. An even better model of an orbital would be air vibrating inside a hollow sphere. ---------------------------------- The idea of photons as propagating particles doesn't work very well in quantum mechanics. 1) Things propagate as waves and 2) display a particle nature only on interacting or being measured. The one exception is an interpretation of quantum mechanics known as Bohmian mechanics. Even in this model, particles are guided by extended fields filling up all space. As for taking zigzag paths, swansont may have been hinting at a theory called quantum field theory. In this theory, in getting from point A to point B, a photon takes all possible paths. So our 'photon' is a field spread-out everywhere. Added up, and averaged out, it can look like a classical straight line trajectory. Despite the almost universally held belief, the speed constant, 'c' is not the speed of propagation of light in a vacuum, but an idealization. Take an ordinary beam of laser light so we get some simple and well behaved waves. We measure the wave velocity, called the phase velocity, by how long it takes for a peak or trough to pass from one point to another over an interval of time. We measure the "group velocity" (a group of oscillations), by measuring how long a group takes to go from one point to another. Only in one direction, along the axis of the beam, do both the group and phase velocities have a speed of c. In any other direction the phase velocity exceeds c and the group velocity is less than c. A wave guide is a good example of this variation and for the very same reason. Velocity is multivalued function of direction.

This is a very abstract way of looking at it. I would have said 'quantization', or "the uncertainty principle". I like your mathematical approach. Is it possible to the tie this in with quantization? Again, a very insightful comparison. I deal almost exclusively with anticommutative structures with real valued entries (for the most part). I would like to make some sort of contact with quantum mechanics. Do you have any idea that may bridge the gap between anticommutivity and noncommutivity?

Sorry, but nothing "moves in time." Try to define "moves in time". dt/dt is not a very interesting nor useful quantity. However..... The idea can be a useful prop in some cases, as long as it's not taken too seriously. Even Yakir Aharanov appeals to the notion in his 2 State-Vector Formalism of quantum mechanics where he imagines two state vectors, one propagating forward in time, and one in reverse. You might look into this similar idea, Perkinsjc. The 2 State-Vector Formalism is a fascinating idea, if as advertised, simply resolves the issue of spurious oscillations in weak measurements. Given a few changes, one might develop an alternative, rather than adjunct, set of axioms to quantum mechanics. This is a mouthful of words, but you can find simple reading in the lighter journals, online, addressing resent and supporting research out of Rochester University.

What holds human cells together? http://answers.yahoo.com/question/index?qid=20080109122114AAHUJzT

As much as I appreciate and admire Sean Carroll, I find this ongoing argument of invariant mass vs. relative mass empty. There is no capital "The mass", in my opinion. It's perfectly fine that two physical quantities may have the same units.

I've got one of those infernal bombs sitting on my patio table. It's a left-over from a customer design job. These guys--the customers--needed a power supply for their device. These so-called HOH generators produce explosive mixtures of oxygen and hydrogen gas. The fringe science people call it Brown's gas. Close enough I guess. Not only do they not produce more energy then they can provide, you have a good chance of being blinded or maimed in an explosion, even if you do know what you're doing.

s This is not such a bad idea, and deserves better consideration. The cross product of two vectors yields a signed area of the parallelogram spanned by the vectors. This extends to volumes and so on. Signed values are very important in physics. Does this generalize to one dimensional curves? Anyone?

BusaDave: Agreed, A force on a bathroom scale due to gravity is the same as a force on a scale due to acceleration. Well done. You caught me in an error describing Einstein gravity in terms of forces. We are not in agreement. You won't learn a great deal of physics on a forum. I won't argue much further. You have to do the hard work and let the equations speak rather than the written word. So this is it for me. (The four forces you spoke of in one post have nothing to do with Newton's F=ma. 'Force' is an anachronism. It's just a name, having nothing to do with Newton's definition of force. The 4 forces are expressed in the standard model using what are called Lagrangians. Force does not appear in a Lagrangian. A Lagrangian formulates a physical theory in terms of kinetic and potential energy rather than force. Einstein's gravity is formulated using a Lagrangian. Try the Wikipedia entry on Hilbert Action to see for yourself. Everybody likes Feynman, right? See what Feynman says about force.)

But you are accelerating. From the perspective of an observer, who is freely falling, or freely floating, not acted upon by external forces, you are accelerating upward due to the force from the scale upon your feet. One standing upon the surface of the Earth is in an accelerating reference frame. The acceleration is the result off the force of the scale acting upward, not a "gravitational force". In the context of general relativity there are no "gravitational forces". This is a Newtonian concept. [Edited] I see swansont has already commented upon this. -------------------- What's going on with all these posts, comparing gravity and acceleration? Apparently the word 'gravity' is not understood here. Gravity and acceleration are not even in the same phenomenological category.

I suspect you mean that a Klein bottle cannot be embedded in manifold of less than 4 dimensions. Maybe we should call the Klien bottle a Klein manifold to have consistent language. But in any case, the 3D space we've come to know and love is (at least locally) oriented. A left glove cannot be locally turned into a right glove. But what if the 3-space is itself non-oriented? Can a Klein manifold be embedded in a 3 dimensional manifold that is itself non-oriented? It occurs to me that the non-oriented Klein manifold can be embedded in itself by identity. Adding one more dimension doesn't nullify the embedding. So it does seem possible to do so in specific cases.

Is this the correct folder in which to pose a question on differential topology? To help keep things simple, suppose you have a 2-form on a manifold of dimension 2, having metric type (+, +), rather than an arbitrary number of dimensions and other metrics. As I understand it, we can integrate a k-form over an n dimensional manifold where k=n. But I want to do an incomplete integration: Given, L an antisemmetric tensor, [math]L=L_{[ij]} dx dy [/math] define [math]S = \int L = \int L_{ij}dx dy[/math]. However, I would like to partially integrate L. [math]S_y dy = ( \int L_{ij}dx ) dy[/math] and [math]S_x dx = ( \int L_{ij}dy ) dx[/math] Is this even legal? I could be missing the obvious. I’ve left out the limits of integration because I don’t know to use LaTex well. Assume integration over some given intervals. I would like to obtain [math]S_x[/math] and [math]S_y[/math] so that they appear to behave under the rotation group as elements of a 1-form: [math]S = S_x dx + S_y dy[/math] Is this possible?

It's good to find agreement. Thank you.

That is something I've also considered, but the meaning is different. Something displaced at a distance can still be existent. Our language(s) in it's verb tense has to be used carefully. You can point at a Minkowski diagram and say "here is the present", "below this line is the past", and similar statements. It serves well as a model of space-time. Equally true, the verb tense used to talk about the Minkowski diagram is not the correct verb tense to use for space-time. Some insightful guy--I don't know who--once said "the map is not the territory". One book of Brian Greene goes a long way in trying to convince us that that space-time is an existent whole. It's certainly good to think four dimensionally, but at the same time, it's simply not true. The repercussions from this book seem to have adversely tainted the accepted language.

Are you asking whether or not the increase in relativistic mass is due to a change in volume? If so, the answer to this within the context of special relativity is 'no'. Within the general theory, I'm no so sure. In special relativity we can pretend that relativistic mass is an element of a vector where the other vector elements are three momentum components. The invariant quantity is the rest mass. With general relativity we have to begin with something a little more fundamental--the action with units of energy-time, or even the action 4-density.

Busa. You claim that there are objective laws of physics. Then where are these laws existent? Historically, physicists became so sure of the objective truth of Newtonian mechanics, they began to call his axioms Laws with a capital L. Applying the word law, to a bunch of axioms was simply a mistake. To be sure, Newtonian mechanics is wrong. This doesn't mean it is extremely useful, but still fails to have one to one correspondence with the results of experiment. I'm sure general relativity also fails. It is the mood, these days, not to have so much hubris as to claim one theory or another to be a law of nature, or a law of God.

There are no decidable propositions, so the question whether or not 'time' is an illusion is also undecidable. The usual assumption in the study of physics is that there exists some sort of objective reality, and go on from there. If we assume that 'time' is something that is measured with a clock using some particular procedure then it immediately follows that 'time' does not exist. The past existed. The future might exist. But if we say time exists then everything happens all at once.