pogono

Ok, it has been published this night. I invite you to read something, what we are looking for last 100 years - General Relativity for waves of matter. The road to quantum gravity is now wide open. Maxwell-like picture of General Relativity and its Planck limit http://arxiv.org/abs/1301.2758 Have a good read ;-) It is an article draft, so if you have any questions or remarks - you know how to find me ;-)

Hi, I have prepared brief explanation of DaF framework ended with GR generalization for gravitational field. Introduction Thanks to Geroch's decomposition applied for Schwarzschild solution of the General Relativity one obtains metric in (3+1) decomposition (space + time). In this picture curved spacetime is equivalent to the flat space-like manifold minimally coupled to a scalar field [math]\Phi[/math], where [math]\Phi[/math] is equal to inversed gravitational time dilation factor. In obtained picture gravity is described as regular field with Maxwell-like equations on flat spacetime as follows. We define scalar fields as follows: [math] \Phi=\frac{1}{\gamma_r}=\frac{d\tau_r}{dt}= \sqrt{1-\frac{r_s}{r}}[/math] [math] \Theta= r \cdot \beta_r = r\sqrt{\frac{r_s}{r}} [/math] where: [math] t [/math] is the time coordinate (measured by a stationary clock located infinitely far from the massive body) [math] \tau_r [/math] is the proper time of stationary observer located in distance r to the massive body [math] r [/math] is the radial coordinate [math] r_s [/math] is the Schwarzschild radius As we know mass relation to Schwarzschild radius is: [math]M=\frac{c^2r_s}{2G}[/math] Therefore (assuming c=1) one may easy calculate that the gradient of the scalar field [math]\Phi[/math] is equal to proper gravitational acceleration "g" for Schwarzschild solution. [math] \nabla \Phi = \frac{r_s}{2r^2} \gamma_r = \frac{GM}{r^2} \gamma_r = g[/math] Above "g" acceleration is measured for the reference frame of the stationary observer located at distance "r" to the massive body. Using introduced above scalar fields one defines vector fields as below (we assume c=1 and [math]\hat{e}[/math] as directional versors). Vector field responsible for gravitational acceleration is denoted as “G”. [math] \vec{T} = \Phi \cdot \hat{e}_y [/math] [math] \vec{G} = -\nabla \Phi \times \hat{e}_y = -\nabla \times \vec{T} [/math] [math] \vec{V} = \nabla \Theta \times \hat{e}_x [/math] [math] \vec{B } = \nabla \times \vec{V} [/math] Utilizing relations between above fields, one obtains Maxwell-like equations for gravitation: [math] \nabla \cdot \vec{G} = 0 [/math] [math] \nabla \cdot \vec{B} = 0 [/math] [math] \nabla \times \vec{G} = \gamma_{r} \cdot \frac{\partial \vec{B }}{\partial t} [/math] [math] \nabla \times \vec{B} = - \gamma_{r} \cdot \frac{\partial \vec{G}}{\partial t} [/math] Therefore by analogy to electromagnetism we may introduce four-potential V in form of: [math]V^\mu=(\Phi,\vec{V}) [/math] and related gravitational field tensor: [math]F_{\mu\nu}=\partial_\mu V_\nu - \partial_\nu V_\mu [/math] After simple transformations one derives wave equation (d’Alembertian) as follows: [math] \gamma^2_r \cdot \frac{\partial ^{ 2}\vec{G} }{ \partial t^{2}} - \nabla^2 \vec{G} = 0 [/math] Above d’Alembertian describes the wave propagating in the flat spacetime with speed equal to: [math]v_{light}=c/\gamma_r=c \cdot \sqrt{1-\frac{r_s}{r}} [/math] In result, in considered case curved spacetime is physically equivalent to the flat spacetime with variable speed of light, where refracting index for light is equal to [math]\eta=\frac{c}{v_{light}}=\gamma_r=\frac{1}{\sqrt{1-\frac{r_s}{r}}} [/math]. It should not surprise us, that in above picture, spacetime around the mass behaves as gravitational lens. Lagrangian and Hamiltonian Analyzing Einstein-Hilbert action for considered case one may derive proper Lagrangian and Hamiltonian for gravity on flat spacetime with refracting index for light speed. For the stationary observer with rest mas 'm' that keeps his position against gravitation we obtain Lagrangian and Hamiltonian in the form of: [math] \mathcal{L}= mc^2 \frac{1}{\gamma_r}[/math] [math] \mathcal{H}= mc^2 \gamma_r [/math] Thanks to superposition principle for some test body with rest mass 'm'' and four-velocity [math]U^\mu=\gamma(c,\vec{v})[/math] we obtain proper Lagrangian and Hamiltonian in form of: [math] \mathcal{L}= mc^2 \frac {1}{\gamma_r} - mc^2 \frac {1}{\gamma} [/math] [math] \mathcal{H}= mc^2 \gamma - mc^2 \gamma_r [/math] To comply with the Newtonian approximation: [math] \mathcal{H}=mc^2 (\gamma - 1) - V \left ( r \right ) [/math] where: [math]V \left ( r \right ) =mc^2(\gamma_r-1)[/math] Classic Newtonian approximation we obtain using Maclaurin expansion of above Hamiltonian, the same way that we do it for Kinetic energy approximation: [math]\mathcal{H}= mc^2 (\gamma-1) - mc^2 (\gamma_r-1) \approx mc^2 \frac{\beta^2}{2} - mc^2 \frac{\beta_r^2}{2} = \frac{mv^2}{2} - m \frac{c^2 r_s}{2r} =\frac{mv^2}{2} - G\frac{mM}{r} [/math] If we consider above field V in the Planck limits, we obtain proper quanta equal to rest energy value. For [math]r_s << l_{P}[/math] we calculate: [math] \lim_{m \to m_P; r \to l_P} V \left ( r \right ) = m_{P}c^2 \left( \frac{1}{\sqrt{1-\frac{r_{s} }{l_P} } } -1\right) \approx m_{P} \cdot \frac{c^2 r_{s} }{2l_{P} } =\frac{c^{ 4}r_{s} }{2G}=Mc^2 [/math] Obtained quanta may be treated as some rest energy (some rest mass M) related to given radius [math]r_s [/math] (Schwarzschild radius). Equations of motion and relation to Newton-Cartan theory The introduced Lagrangian locally satisfies the Euler-Lagrange condition: [math]\frac{d \frac{\partial \mathcal{L}}{\partial v}}{d\tau_r}= \frac{\partial \mathcal{L}}{\partial r}[/math] what yields to: [math]\frac{d (mv\gamma)}{d\tau_r}= mc^2 \cdot \frac{r_s}{2r^2} \gamma_r [/math] where LHS is relativistic force and RHS is just equivalent to "gravitational force" in Schwarzschild solution. Using introduced scalar field [math]\Phi[/math] we may rewrite it as: [math]RHS=mc^{2}\cdot \nabla \Phi[/math] Considering above we see, that equations of motion should fulfill transitional condition (assuming c=1 to facilitate): [math] \frac{d(v\gamma)}{d\tau_r}= \nabla \Phi [/math] Therefore equations of motion for considered case are in the form of: [math] \frac{d^2\vec{x}}{d\tau d\tau_r}= \nabla \Phi [/math] where: [math] \tau [/math] is the proper time of the test body [math] \tau_r [/math] is the proper time of stationary observer located in distance r to the massive body [math] \gamma=d\tau_r / d\tau [/math] Above equations may be explained as relativistic form of Newton-Cartan theory equations of motion. In DaF, the proper times were taken in place of coordinate time "t". The equations of motion in terms of four-velocity U and four-acceleration A may be rewritten for stationary observer reference frame as: [math] \frac{d U^\mu}{d\tau_r}= \partial_\mu \Phi [/math] and for the test body reference frame as just: [math] A^\mu = \partial_\mu \Phi [/math] General Relativity generalized for gravitational field In the Lorentz gauge equation of motion may be rewritten as: [math]A^\mu=\frac{\partial V^\mu}{\partial \tau}[/math] Above formula says, that gravitational acceleration is equal to derivative of the gravitational four-potential. Reversed it also says, that any move is the source of gravitational acceleration. It means, that any body with four velocity [math]U^\mu[/math] is at the same time the source of gravitational potential [math]V^\mu[/math]: [math] U^{\mu}= \left ( \gamma , \gamma \vec{v} \right ) \to V^{\mu}= \left ( \frac{1}{\gamma}, \vec{V} \right ) [/math] Four-acceleration A may be generalized to the tensor in the form of: [math]A^{\mu\nu}=\partial_\nu V^\mu [/math] showing it is equal to the first element of introduced gravitational field tensor: [math]F_{\mu\nu}=\partial_\nu V_\mu - \partial_\mu V_\nu[/math] where the second part is the acceleration for the other bodies. If we introduce gravitation four-current by analogy to electromagnetism: [math]J^\mu=\partial_\nu F^{\mu\nu}=\frac{2\pi r_s}{V}\cdot(\gamma_r,\gamma_r\vec{v_r})[/math] and generalize to the tensor multiplying by four-velocity U of the source of gravity: [math]J^{\mu\nu}=J^{\mu} \cdot U^{\nu} [/math] we obtain stress-energy tensor T with respect to the constant [math]J^{\mu\nu}=\frac{4 \pi G}{c^4} T^{\mu\nu}[/math] what drives to General Relativity main formula in the form of: [math]G_{\mu\nu}=2 \cdot J_{\mu\nu}[/math] This way we have created General Relativity main equation equivalence as wave-based formulation that might help us with explaining the wave nature of mater.

Ufff, generalization to General Relativity finished! In the way to arXiv. I will appreciate your remarks and comments. Title: Maxwell-like General Relativity formulation in the Planck limit File: http://www.dilationasfield.net/gaf.pdf Abstract: We show that Geroch decomposition leads us to Maxwell-like representation of gravity in (3+1) metrics decomposition. For such decomposition we derive four-potential [math]V^\mu[/math] and gravitational field tensor [math]F^{\mu\nu}[/math] that may be associated with gravitational interaction. Next we introduce valid Lagrangian and equations of motion. Then we show that gravitational four-current [math]J^\mu[/math] derived for introduced four-potential produce energy-stress tensor and generalize main General Relativity formula. At the end we introduce new approach to quantization of gravity that results in proper quantum values and is open to further generalization.

Hi, I have prepared a presentation for one of my seminar that explains the article: http://dilationasfield.net/Seminar.pdf Take a look if you are curious how is it possible to: - consider photon acceleration - explain light as disturbances of spacetime structure

Hi everybody, I have just put on arXiv my 4th version of the article (after 2 reviews) http://arxiv.org/abs/1301.2758 You my find there, that gravity and electromagnetism acts as two different consequences of one filed equation describing spacetime disturbances. If you would like to discuss the idea in real live - come to my seminars. I was invited to give open speech at Moscov Lemonosov Universtity (may 14th). I also wait for confirmation of my speech at GR20 conference in Warsaw (July). For those who are interested in my idea - I put news page at my webpage: http://www.dilationasfield.net/eng

Hi eytan_il, I understand you refer to my first article. Then I have to point, that 73 is not under Schwarzschild metric, but it is for plane Minkowski. So, the formula is ok. I recommend you to read my newest article: http://arxiv.org/abs/1301.2758 There you may find 3 vesions of the field equations, when 2 of them are covariant.

Hello Everybody I recommend you my newest article in the subject: http://arxiv.org/abs/1301.2758 Have a good lecture.

Every scientist working at quantum field theory knows, that present QFT framework in not final, but it is only approximation of some higher lever theory. (f.e. read it in preface to S. Weinberg, Quantum Theory of Fields, 1995) Take a look at my newest article. In subsections of section 4.2 you will see, that presently used formulas for: rest mass, elementary charge and photon energy are just approximations (for small energies) of the formulas I propose. My formulas works fine for high energies, and then, suddenly... - field quanta phenomena has simple explanation (formula's limits for Planck's scales) - inflation phase appears to be obvious - we may at least understand why elementary particles behaves as they are (not only count it's properties like in QM, but understand) - and so on... I do not know if you know Standard Model (SM) and 19 parameters standing behind it. We may now recalculate SM with my field idea. All we need is transforming my formulas to Lie Algebra and count angle dilations. A lot of work to do. A lot to discover.

exactly! Formulas that we have obtained following my reasoning are just General Relativity formulas. As I wrote few posts ago - it is very uncommon GR derivation. Starting with assumption what would hypothetically happen if we accelerate photons - we obtain GR in result.

You may say so. To be very concrete I have shown what would happen if we try to accelerate photon. He will not accelerate. We will increase spacetime curvature instead, where photons are still moving with "c" speed.

O, no. Better not ask for it :-D :-D My way of thinking was so twisted and messed up, that It is kind of miracle that I have something so valuable at the end. :-D As you may see, above field equations looks rather strange and unusual. And believe me, for me, it was very hard to understand what exactly I am thinking about. :-D :-D

Field equations do not work as one way cause-effect relation. You can not say what is caused by what. It is all related with all in a circle differential relations. Below I put image with my fields definition. As you see, it is just self-reinforcing, universal mechanism. You may explain light, gravity (by substituting l_planck wit r_Schwarschild), but you may also try to use it to explain expansion of the universe as some strange kid of "wave" (just like the man who used my equations to try to explain inflation phase in cosmology).

Ok, just look at my "Like-Maxwell Equations" for space-time phenomena. You will see, there is some rotation with c/gamma velocity. If you consider such move there mus exist some radial acceleration - in this case: gravitational acceleration. I show, that it is all related: - by some rotation in additional axis light acts as it would be accelerated in radial distance - so, it has local and instantaneous additional velocity responsible for spacetime curvature - curved spacetime force bodies to act like they would be attracted The same Like-Maxwell equations explains, that electromagnetism IT IS spacetime. I show that if we consider time flow as some rotary field to spatial axis, there must be wave acting as local disturbance in spacetime isometry moving inside this spacetime. So, light is just moving, local spacetime anizometry. Above very good explains why in Minkowski metric we have imaginary axis. Rotary field may be presented as imaginary axis. It is enough to look at [math]e^{ix}[/math] - it is helix. So gravity is the same phenomena then electromagnetism, but on the higher level of abstraction. In gravity phenomena - light is rotating (radially accelerated from our perspective).

Hello michel123456. Thank you for your posts. I had to pay for my publication to get valuable review and publish my work. My problem is quite similar to one described here: http://www.ptep-onli...es/PP-04-10.PDF I had problems with publication (nobody treated me serious), because I am unknown and suddenly I propose new explanation for few well established beliefs. For example I show, that Newton potential that we use as limit while deriving Shwarzschild metric, it is only approximation! So, Schwarzschild radius may be derived without this approximation. Sounds obvious, but it breaks some stereotype. But it was worth. For now, I have invitations for two scientific conferences and I have many confirmations, that I may be right. F.e. - In some ironworks company they have obtained strange metal structures by irradiation on metal with high energetic beams. Some local physicist explained it using my formulas showing relation between strong light beam and time dilation effect, that delayed chemical reaction... - Some physicist had calculated shrinking and exploding universe using my rest energy and photon energy formulas. It seems that thanks to my formula behavior in high energies, he has found theoretical explanation for the reason of cosmological inflation (it is not well explained in present cosmological theories why we had inflation phase). - Some physicist emailed me with the proof of my idea. He did not believe I may be right, so he sited down to prove me, that I am wrong. But instead I have nice, three pages long, covariant confirmation for my idea. - etc. I hope it is not the end. Regards pogono

You may also take a look at my newest article: www.dilationasfield.net/gaiws.pdf I show there, that we may derive General Relativity form weird idea: what would happen if we try to accelerate photons with Rindler transformation. In section "3.3. Rindler's transformation" it is clearly explained: 1. We take regular Rindler transformation used to describe accelerating body by temporary co-moving bodies. Equations: (42), (43) 2. We put in place of velocity and acceleration: - free-falling velocity - gravitational acceleration Equation: (44) 3. Then we transform the formula according to regular math rules Equations: (45), (46) 4. Then, we write regular Minkowski for this temporary co-moving body and stationary observer Equation: (47) 5. Surprise! We got null geodesics formula in Schwarszchild metric for stationary observer (he is called Killing observer in GR) Equation: (48) So, we have to agree, that we were considering photon acceleration. However, we did not increase photon's velocity - instead of its acceleration spacetime get curved. Do it by yourself. Otherwise you will never believe it works.

Hello, I have prepared brief explanation of my article for non-physicist. You may find it at: www.dilationasfield.net/eng Have a nice read

Thank you imatfaal. In my article I have just shown, that we may consider accelerated photons (!) and the result is the same that GR formulas (what is generalized by Kuroneko with Killing vector fields in second link I provide). It means we may consider accelerated wave => "accelerated wave function". Since De Broglie (and then Schrödinger and Dirac) we do not imagine gravity other way, then just another quantum interaction. Higgs describes it this way. Now, we may leave this way and consider accelerated photon. We do not need "the mass" anymore!... I appreciate if someone will develop it farther with Lie Algebra.

Hello all, my article with Unified Field description has been published: http://www.scirp.org/journal/PaperInformation.aspx?paperID=17700 Moreover..., soon will appear an article written by some physicist/mathematician, who make citation of mine, confirms my results and generalize my equations using Killing vector fields (and Gauss-Codazzi equation). You may find draft of this article here: http://tp-theory.net/eng/proof-theory.html He confirms f.e. what I have shown: - we may consider reference frame assigned to photon!! (if we use Killing observers) - we may derive GR equations using Rindler's transformation on flat Minkowski space-time what digs a tunnel between GR and QM Have a good reading pogono

Yup, and now even more ;-) Revised version updated at: http://tp-theory.net/tpt_eng.pdf How do you like it? You will be shocked, what comes out from my Rindler transformation (formulas 15-25) I appreciate any comment.

Hmmmm. Nice has few meanings... Please, say if you think that it may be useful or if it is interesting point of view. I am also looking for someone experienced as co-author. There is a lot of things to do to finish the article, as I suppose.

And...? How do you like it?

No, no. It is just plane Minkowski. I have only pointed, that gamma is similar to Schwarzschild gamma factor, what will be important farther. F.e. if we consider above field equations for central rotation with time dilation around, then we have 2 options: - assume gravity IS just time dilation what force us to define some additional rotation in time axis for every test body (explained in 4-dimentional Lagrangian and Hamiltonian definition introduced in my article, section 4.1), or - we can make space-time curved and obtain regular Schwarzschild

Yup. Here we go: Let us prepare to vector field description, describing at first body with Planck's mass m_p, with line velocity V_rot, on the circle with radius R. We will define velocity using parameter [math] \beta [/math] as some function of R [math] \beta=\sqrt{\frac{R_{const}}{R}} [/math] where R_const is some defined constant. [math] v_{rot}=c\cdot \beta [/math] [math] \gamma= \frac{dt}{d\tau}=\frac{1}{\sqrt{1-\beta^2}} [/math] (please, notice that it is based on gamma in Schwarzschild metric. For E-M waves we will use Planck's length instead R_const). Angular velocity for rotating body we will denote as: [math] \omega=\frac{c\beta}{R} [/math] Non-relativistic angular momentum we may denote as: [math] \vec{L}=\vec{R}\times m_P\cdot \vec{v}_{rot} [/math] [math] L=m_P\cdot Rc\beta [/math] Radial acceleration and its relation to relativistic force is: [math] a_{\vdash}=-\vec{R}\omega^2=R\frac{d\vec{\omega} }{dt}[/math] [math] \gamma a_{\vdash}=\frac{F_{\vdash}}{m_P} [/math] Now, we will construct some vector fields to describe whole class of above rotations defined for any place in space. Rest mass we will treat as parameter. Let us define at first three versors n_R, n_x, n_y. For any conductive vector R: [math] \vec{n}_R=\frac{\vec{R}}{R} [/math] [math] \vec{n}_R \times \vec{n}_x = \vec{n}_y [/math] Let us define scalar field [math] \frac{c}{\gamma } [/math] and two related vector fields: [math] \vec{A}=-\nabla\frac{c}{\gamma}\times \vec{n_y} =\frac{\gamma}{2c} \omega^2R \cdot \vec{n_x} [/math] [math] \vec{T}=\frac{c}{\gamma} \cdot \vec{n_y} [/math] As we can show: [math] \nabla \times \vec{T} = -\vec{A} [/math] [math] \nabla \times \left ( \frac{c}{\gamma}\cdot \vec{n_y} \right )= \nabla\frac{c}{\gamma}\times \vec{n_y} [/math] Let us define auxiliary scalar field equal [math] Rc\beta[/math] (related to angular momentum) and two auxiliary vector fields U and [math] \Omega [/math]. [math]\vec{U}= \nabla Rc\beta \times \vec{n_x}= \frac{c\beta}{2} \cdot \vec{n_y} [/math] [math] \vec{\Omega}=\nabla \times \vec{U}[/math] Let us notice, that: [math] R\cdot\left ( \nabla \times \vec{U} \right )=R \cdot \vec{\Omega}=\vec{v}_{rot} [/math] [math] R\cdot\left ( \nabla \times \vec{A} \right )=\vec{R}\cdot \frac{\gamma}{c}\cdot \omega^2=R\cdot \frac{\gamma}{c}\cdot \frac{d\vec{\Omega} }{dt}[/math] From above we derive: [math] \nabla \times \vec{A} = \frac{\gamma}{c}\cdot \frac{d\left (\nabla \times \vec{U} \right )}{dt}=\frac{\gamma}{c}\frac{d \vec{\Omega}}{dt} [/math] [math] \nabla \times \vec{A} = \frac{1}{c}\frac{d \vec{\Omega}}{d\tau} [/math] Let us also show, that: [math] \frac{1}{c}\frac{d \vec{T}}{d\tau}=\frac{\gamma}{c}\frac{d \left ( \frac{c}{\gamma} \cdot \vec{n_y} \right )}{dt}= \frac{\gamma}{c} \cdot \frac{c}{\gamma} \cdot \frac{d \left ( \vec{n_y} \right )}{dt}= \frac{d \left ( \vec{n_y} \right )}{dt}=\vec{\Omega} [/math] Using above we obtain: [math] \nabla \times \vec{\Omega}=-\frac{1}{c}\frac{d\vec{A}}{d\tau} [/math] From above we derive two d'Alambertians: [math] \frac{\gamma^2}{c^2}\frac{d^2\vec{\Omega}}{dt^2}-\nabla^2\vec{\Omega}=0 [/math] [math] \frac{\gamma^2}{c^2}\frac{d^2\vec{A}}{dt^2}-\nabla^2\vec{A}=0 [/math] Above d'Alambertians describe wave with line velocity [math] \frac{c}{\gamma } [/math] or – as you wish - time dilation around rotation center. But it also means, that in local time it is "c" speed. [math] \frac{1}{c^2}\frac{d^2\vec{\Omega}}{d\tau^2}-\nabla^2\vec{\Omega}=0 [/math] [math] \frac{1}{c^2}\frac{d^2\vec{A}}{d\tau^2}-\nabla^2\vec{A}=0 [/math] This way we have just described local time flow dilation traveling through time-space. P.S. From above you may easy derive Gravitational Potential using R_schw in place of R_const. If you do it - vector "A" appears to be gravitational acceleration (with an accuracy of "c") You may also derive Rest mass formula and photon Energy formula (using Planck's length in place of R_const) and so on...

By local time-space structure's disturbance I mean local time flow dilation. As we know photons caries energy. As we know from SR - increasing energy means slowing down time flow. Photon's energy may be then understood as dislocating disturbance in time flow. You may find exact field equations in 3.3 chapter of my article draft. I can also post it here if you wish, but i do not know how to put here Latex equations ([tex] [/tex] tags seem they do not work).

Hello Klaynos. Thanks for your voice in discussion. Yes, the same time-space we live in. We know already, there is no aether and E-M waves propagates in time-space. So, time space is the medium for E-M waves. So, we should be able then to describe E-M wave as local disturbance in time-space structure... It is exactly the description I make in my article draft.