What is Minkowski Space and what does it have to do with Landau-Lifshitz spacelike convention?
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Minkowski space: the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for representing a spacetime.  Formally, Minkowski space is a four-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form with signature (-,+,+,+).  This signature is often called the Landau-Lifshitz (spacelike) sign convention
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Basically, the space that you and I experiences daily is 3 dimensions: up/down, left/right, forward/backwards. But space doesn't have to be 3 dimensions. It can be any number of dimensions. Of course, it is very difficult for us 3-dimensional beings to imagine what these dimensions are, or where they are. The problems is that we can NEVER see the fourth spatial dimension, never mind the 5th or the 6th, etc. etc. Let me explain why.\n\nImagine for a moment that we are 2-d beings, not 3-d. And we live on the surface of a sphere (like a ballon). This sphere is our universe. Everything we see, touch and experience happens on the surface of this sphere. We can go left/right and forward/backwards. But we can never go up or down. When we travel in one direction, we will eventually circumnavigate the sphere and make a complete circle, which brings us back to the initial starting point. If we ask the 2-d beings to imagine 3-dimensional space, all we can tell them is that the 3rd dimension (up/down) is orthogonal (a fancy way of saying perpendicular) to the surface of the sphere.\n\nThis is exactly our situation when we talk about the 4th, 5th, 6th, whatever nth dimension. They are defined as the dimension that is orthogonal to the n-1 th space. So the 4th dimension is orthogonal to our 3-dimensional space, and the 5th dimension is orthogonal to the 4-dimensional space, etc. etc.\n\nNow about Super String Theory and these higher dimensions: what the theory predicts is that our universe does indeed have more than 3 spatial dimensions. Some version sof the theory say there are 10, some say there are 11. Whatever the number (it doesn't really matter in our discussion here), these higher spatial dimensions are currently curled up into very very tiny spaces. So what does that mean?\n\nWe now go back to our 2-d being living on a sphere example. For the time being, let's say we are now on a flat 2-dimensional surface (like a sheet of paper). Super String Theory says that the higher dimensions are curled up. So let's consider in our case the 3rd dimension as the curled up dimension. This means that if we were to magnify any point on the 2-d surface all the way to about 10^-35 meters (Planck Length), we will see that there is a tiny loop (like a loose thread on a knitted sweater).\n\nNow imagine if our 2-d being were (1) small enough to see and fit inside the loop, and (2) able now to travel in the 3rd dimension, then for it to go in the up/down direction, the 2-d being will now be just going around in circles inside the loop. Remember in the previous case, our 2-d being can travel in one direction on the surface of the sphere and then return to the same point. Well, here our 2-d being is traversing through the 3rd dimension, except that there is no up and down because the 3rd dimension is "curled up" into a loop. So the 2-d being just goes around the loop in circles.\n\nIf you understood the above, then now we can answer the question, what and where are these higher dimensions in our universe. They are culred up in every point in our 3-dimensional space, and if we were small enough and we could move into these higher dimensions, then we would be moving through these very convoluted curled up loops, but not really able to go anywhere.\n\nBecause these higher dimensions are curled up into such tiny tiny loops (remember 10^-35 meters), only the most elementary structures like "strings" are affected by them. The size of these strings are of course on the order of 10^-35 meters long, and can traverse through the curl up dimensions. When the strings vibrate, the vibrations in the 3 large dimensions don't have any effect, but in the curled up higher dimensions, the vibrations can create resonance like playing a musical note.\n\nThese resonances are the stuff of reality, meaning these vibrational resonances of the strings inside the curled up dimensions is what creates matter and energy and thus everything that&#
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Anew theory of gravity as well as a unified field theory (UFT) are being proposed, showing how the four force fields work together, implicating also a fifth field. A main feature of this new model is the proposal of the existence of diallel, gravitational-field lines that emanate from every entity. For the earth, for example, they emanate radially from its center. These diallel lines provide a conduit for combined particle and wave flow and exchange beyond the speed of light, and they have quantum states similar to the atom. On the web site, we present a general field equation showing the importance of energy density in this new UFT. We also present a new gravitational force equation, that has as a special case the commonly accepted gravitational equation.
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I don't think that is correct.   I assume you mean fourth spatial dimension, rather than the Minkowski 3 space, 1 time geometry of relativity.  For that, time is actually a dimension, and the question pretty much answers itself.\n\nThe best insight into a spatial fourth dimension is to read the delightful classic little book, "Flatland". Almost any library or bookstore should have it. It works on the analogy of an individual who lives in a two dimensional world - his name is "A. Square" - who learns about three dimensions.\n\nAnother way to insight would be to look up "tesseract" on a search engine. Aside from hits on Madalyn L'engel's misuse of the term in her children's boooks (avoid her description), you should find many discussions on how to represent simple four dimensional objects in three dimensions, just as we represent three dimensional objects on a flat piece of paper with perspective.\n\nFour dimensions would have two directions that are perpendicular to our three dimensional world. They have been given the names "ana" for four dimensional "up" and "kata" for four dimensional "down" with respect to our space. Of course these are just the Greek words for up and down. \n\nSo borrowing a description from Flatland, if a four dimensional sphere were to start anawards of our world and move through it in the kata direction, what we would see, as it intersected our world woild be first a tiny three dimensional sphere (cross section of the four dimensional sphere near its kata pole) which grows in size to a maximum (the "equator" passing through), and then decreases gradually again to its mini size before vanishing, having passed commpletely through our world and out the other side.
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zero-point energy is one of the most widely misunderstood concepts of quantum mechanics.  all it says is that space has fluctuations of non-zero energy at a temperature of absolute zero.  the casimir force, which is a measurable effect, can be modeled as a consequence of zero-point energy, but it's essentially the same thing as the van der waals interaction - also a consequence of quantum fluctuations.  \n\nthese fluctuations do physically exist but are not 'harnessable' any more than the random thermal fluctuations of matter at non-zero temperature.  at least, the 2nd law of thermodynamics says it is so - any technology that did such a thing would overturn hundreds of years of evidence for the validity of the 2nd law.  i for one am not holding my breath.\n\ndon't let the crackpots confuse you with their nonsense.
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I would have to go with the Planck length on this one, which is a millionth of a billionth of a billionth of a billionth of a centimeter!  \n\nThe reason I say this: According to String Theory, strings are that long, and they are the fundamental building block of all matter.  Therefore, nothing exists that is smaller than that, so it is useless to have a smaller measurement.
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hyperbolic plane, you can check what the different models for hyperbolic geometry are.\n\nanotherone, would be Riemannian geometry, which is the basis for relativity theory.\n`
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Matter and Anti-Matter
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spacetime is finite and boundaryless.  if you wish to you may think of it as the 2d surface of a sphere, but remember, nothing inside or outside the sphere exists for the 2d surface.  in this model, nothing exists, not even space or time.
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When I think of "polarized", I interpret that to mean a system with a fixed, unchanging electric moment.  Electric  containment with static fields is generally not possible within any hollow space.  That's because those particular static fields can always be written as the gradient of a scalar field, the voltage field.  If the space is hollow, with no internal contributing sources, then the forces have zero divergence.  The Laplacian of the scalar field is zero.\n\nA fundamental result of the calculus of multiple variables is that if you have a volume completely enclosed by a surface, and the Laplacian of a scalar field is zero everywhere inside the volume, then the field takes its minimum and maximum values on the surface.\n\nThe energy state of a charged particle in such a field is the produce of the charge and the field value, which we've said is at its minimum and maximum on the surface.  So, either the field is constant everywhere within the container, in which case the container has no influence at all, so there is no containment, or the field is non-constant, and the minimum energy state is obtained when the particle collides with the containment somewhere.\n\nOK, so unchanging electric fields are not going to constrain a charged particle in a hollow space.  How about magnetic fields?  Well, apply a large magnetic field through the space, and inject charged particles into it such that they have no velocity in the direction parallel to the magnetic field, and those particles will orbit in the magnetic field.  If the radius of the orbit is smaller than the distance to the side walls, we can say that the particles have partial containment in the system.  A uniform magnetic field can hold moving charged particles in two dimensions, but provides no containment in the direction parallel to the magnetic field.\n\nIf we start modifying the electric and magnetic fields in more detailed ways, it becomes possible to contain a plasma.  That is, after all, what happens in a Tokamak.
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Mathematical method of quantum phase space is very useful in physical applications like quantum optics and non-relativistic quantum mechanics. However, attempts to generalize it for the relativistic case lead to some difficulties. One of problems is band structure of energy spectrum for a relativistic particle. This corresponds to an internal degree of freedom, so-called charge variable. In physical problems we often deal with such of dynamical variables that do not depend on this degree of freedom. These are position, momentum, and any combination of them. Restricting our consideration to this kind of observables we propose the relativistic Weyl--Wigner--Moyal formalism that contains some surprising differences from its non-relativistic counterpart
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According to the lorentz transformation the space time coordinate of two inertial reference frame moving at a relative speed v along x axis is\nx'=(x-v*t)/L\ny'=y \nz'=z\nt'=(t-v*x/c^2)\nwhere L is lorentz factor=sqrt(1/(1-v^2/c^2));\n\nThe tricky thhing is this if any object moves faster than light the L of above equation will become imaginary.\nSo for an observer the proper time t' and the relativistic mass m=Lm0 becomes imaginary...\n Thus it cannot be faster than light.\nHowever theoritically there exist tachyon particles which moves in superluminal(faster than light)  velocity with imaginary proper time.\n But those crazy physix!!!
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L = 8pi G rho/c^4 is the definition of the cosmo constant L (1/m^2); where c = light speed and rho is the energy density of the known universe in a vacuum.\n\nF = GmM/r^2 is the equation for the attraction force F between two masses m and M, r distance apart.\n\nL was created by Einstein to make his equations come out right.  By that, he meant the universe should be spherical (not flat or saddle shaped).  His original results showed a flat universe, and he just "knew" that was wrong; so on came L to fix the shape problem.\n\nF was the result of experiments by Newton.  Through observation, he found that the attractive force between two bodies was proportional to the inverse square of the distance between them and to the product of the magnitudes of the two masses.  That is, F varies as (~) (mV/r^2).  \n\nWhenever there is a proportionality, as in F ~ mM/r^2, there has to be a constant of proportionality.  In this case Newton called it G; so that F ~ mM/r^2 becomes F = GmM/r^2 when the constant is inserted.\n\nThe G in both equations is the same constant.  But Sir Newton came upon it long before Einstein found it could be used in his equations and the calculation of L.  So there is no way that Newton (1642-1727) could know about Einstein's (1879-1955) use of it much later.
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Two distinct systems of hypercomplex numbers in n dimensions are introduced in this book, for which the multiplication is associative and commutative, and which are rich enough in properties such that exponential and trigonometric forms exist and the concepts of analytic n-complex function, contour integration and residue can be defined.\n\nThe first type of hypercomplex numbers, called polar hypercomplex numbers, is characterized by the presence in an even number of dimensions greater or equal to 4 of two polar axes, and by the presence in an odd number of dimensions of one polar axis. The other type of hypercomplex numbers exists as a distinct entity only when the number of dimensions n of the space is even, and since the position of a point is specified with the aid of n/2-1 planar angles, these numbers have been called planar hypercomplex numbers.\n\nThe development of the concept of analytic functions of hypercomplex variables was rendered possible by the existence of an exponential form of the n-complex numbers. Azimuthal angles, which are cyclic variables, appear in these forms at the exponent, and lead to the concept of n-dimensional hypercomplex residue. Expressions are given for the elementary functions of n-complex variable. In particular, the exponential function of an n-complex number is expanded in terms of functions called in this book n-dimensional cosexponential functions of the polar and respectively planar type, which are generalizations to n dimensions of the sine, cosine and exponential functions.\n\nIn the case of polar complex numbers, a polynomial can be written as a product of linear or quadratic factors, although it is interesting that several factorizations are in general possible. In the case of planar hypercomplex numbers, a polynomial can always be written as a product of linear factors, although, again, several factorizations are in general possible.\n\nThe book presents a detailed analysis of the hypercomplex numbers in 2, 3 and 4 dimensions, then presents the properties of hypercomplex numbers in 5 and 6 dimensions, and it continues with a detailed analysis of polar and planar hypercomplex numbers in n dimensions. The essence of this book is the interplay between the algebraic, the geometric and the analytic facets of the relations.\n\nAudience \nUniversity libraries. Libraries of research institutes for mathematics and physics. Departments of mathematics and physics of universities. Departments of mathematics and theoretical physics of research institutes.
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I fear i had even thought about the ∞ dimension once. Nearly anything could be described as rotations there. It has took a long time to get a foot on earth again.\n\nTime is not the real 4th dimension. Perhaps c*t. Look at this 'Minkowski space' thingy:\n\nhttp://en.wikipedia.org/wiki/Minkowski_space\n\nIf you measure across space AND time, there are negative distances. What shoud a negative distance mean in reality?\nIf you consider time as the 4th dimension, it is just a convenient way to write down long terms shortly or to visualize complex relations concisely. It isn't more than just mathematics.\n\n\n"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." \n\nAlbert Einstein (personally himself)
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the spacetime is finite yet has no edge and no center.  it may be something like the two dimensional surface of a sphere.  anything inside or outside the sphere exists for the surface.  nothing exists outside spacetime in this model.  the universe did not begin in empty spacetime.  spacetime began with the universe.\n\nhttp://map.gsfc.nasa.gov/m_uni.html\nhttp://universeadventure.org/\nhttp://www.sciam.com/print_version.cfm?articleID=0009F0CA-C523-1213-852383414B7F0147
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In the Standard Model, quarks, electrons and the like are considered to be pointlike. This gives problems at short distances where for instance the electromagnetic force between two of them becomes infinite.\n\nThe basic idea of stringtheory is that the fundamental constituents of matter are strings of a length of order of 10^35 m which vibrate at specific frequencies. However, the simplest string theories give more problems then they solve and consistent theories requires more then four dimensions. \n\nMembranes take the idea even further.  The particles are represented by vibrating surfaces and higher dimensional objects.