Mordred

This publication though extremely complex (field theory usually is) has probably one of the best coverage of fields in general. Chapter 1 alone will take considerable math skills and study. The article also covers twistors, spinors etc. "Feilds" https://arxiv.org/abs/hep-th/9912205# Over the years I've always kept this article handy. Its been a lifesaver when studying new metrics. Another handy tool provided you can buy textbooks is "One hundred Roads to Reality" by Sir Roger Penrose. For a non model specific coverage on various applications his 1000 page plus book is incredibly enjoyable. He drops a considerable amount of humour into his descriptives. In particular at his own models. ( try not to laugh too hard at his zig zag model) @Declan chapter one of fields will of particular use in clearing up some of your misunderstandings.

Thats true. As one that has tried it back when I first started studying Cosmology. I usually laugh my head off on such a claim.

For Declan. [latex]\frac{dx^\alpha}{dy^{\mu}}=\frac{dx^\beta}{dy^{\nu}}=\begin{pmatrix}\frac{dx^0}{dy^0}&\frac{dx^1}{dy^0}&\frac{dx^2}{dy^0}&\frac{dx^3}{dy^0}\\\frac{dx^0}{dy^1}&\frac{dx^1}{dy^1}&\frac{dx^2}{dy^1}&\frac{dx^3}{dy^1}\\\frac{dx^0}{dy^2}&\frac{dx^1}{dy^2}&\frac{dx^2}{dy^2}&\frac{dx^3}{dy^2}\\\frac{dx^0}{dy^3}&\frac{dx^1}{dy^3}&\frac{dx^2}{dy^3}&\frac{dx^3}{dy^3}\end{pmatrix}[/latex] This is just the Einstein metric tensor. Each 4×4 matrix has the number of degrees of freedom. You can't Not even a supercomputer can do so. Without tools such as tensors, twistors, guage groups etc. Even N body codes rely on those tools Lol a good example is the three body problem using Keplers laws. Now imagine an entire galaxy. Particle fields are far more complex with their added degrees of freedom.

Take this thought experiment. Remove every star and blackhole. Have a homogeneous and isotropic fluid. Gravity still works. I tried that once. I stopped after 11 pages of partial derivatives. (Most of them factored out) I never did complete all the partial derivitaves in the Einstein field equations. Most cases I work on is homogeneous and Isotropic fluids which the FLRW metric greatly simplifies. Twistors use of symmetry relations is a handy tool. One that does greatly simplify a lot of complexity. I ran into a professor who specialized in twister theory once. He ran me through some of the basics behind it. I honestly wish I had more time with him

Declan my advise is don't draw conclusions until you have time to study the material properly. You probably don't want this thread to get locked. Take some time from this thread to work out the mathematical details. If you have questions on GR don't hesitate to start a new thread in mainstream (as your questioning mainstream physics.) You can always come back to this thread when your better prepared and armed.

I seriously doubt we will find a black hole small enough to radiate Hawking radiation. The blackbody temperature of the BH must be higher than the blackbody temperature of its surroundings.

Excellent. The Schwartzchild metric is a static solution. The Kerr metric which includes frame dragging is not. Tensors are independent of coordinate choices. (Metric choice) Incorrect thats a conclusion drawn from not fully understanding of how the static solution of the Schwartzchild solution works to include particle movement. That is defined by the particles geodesic equations.

There is an expression " matter tells geometry how to curve, while geometry tells matter how to move." The stress-momentum monentum tensor tells spacetime how to curve. The geometry tells matter how to move. In other words you cannot properly define the dynamics of the Einstein field equations with just the stress tensor. You need both sides of the equation. In a sense they are two sides of the same coin. Much like energy and mass. Change one you change the other. I'm glad to see your reading Sean Carrolls article. Keep in mind throughout this thread I've provided other resource aids. The lecture notes by Mathius Blau for example details nunerous misconceptions in GR due to artifacts of coordinates. The second part of the basic particle physics has excellent Relativity coverage as well as that section is all Relativity. Elements of Astrophysics has a huge collection of formulas with explanations used by any everyday astrophysicist. Including some detailed relations for galaxy rotation curves as well as GR. Granted it will take a great deal of time to properly absorb it all. See above. By the way +1 I'm glad to see your studying. Which puts you ahead of many of the crackpots we see in Speculations. A couple of hints to help. The EFE includes the equivalence principle as well as conservation of energy/momentum. (Even though its debatable if energy is conserved in GR) Pay close attention to the different geodesic equations for massive and massless particles. Also any details on the Levi Civita connection. https://en.m.wikipedia.org/wiki/Levi-Civita_connection I fixed the latex. If you quote this post you can see the corrections and how to latex symbols and fractions. This will help a bit. Coordinates in GR take the form (ct,x,y,z) this leads to a 4x4 matrix. For the moment we are ignoring everything but the exact specific real numbers the components of the metric take at a single point. Lets define a point as [latex]x^\alpha[/latex] and our new coordinate as [latex]y^{\mu}[/latex] these simple coordinates leads to [latex]g_{\mu\nu}=g_{\alpha\beta}=\frac{dx^{\alpha}}{dy^{\mu}}\frac{dx^{\beta}}{dy^{\nu}}[/latex] [latex]dx^2=(dx^0)^2+(dx^1)^2+(dx^3)^2[/latex] [latex]G_{\mu\nu}=\begin{pmatrix}g_{0,0}&g_{0,1}&g_{0,2}&g_{0,3}\\g_{1,0}&g_{1,1}&g_{1,2}&g_{1,3}\\g_{2,0}&g_{2,1}&g_{2,2}&g_{2,3}\\g_{3,0}&g_{3,1}&g_{3,2}&g_{3,3}\end{pmatrix}=\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}[/latex] Which corresponds to [latex]\frac{dx^\alpha}{dy^{\mu}}=\frac{dx^\beta}{dy^{\nu}}=\begin{pmatrix}\frac{dx^0}{dy^0}&\frac{dx^1}{dy^0}&\frac{dx^2}{dy^0}&\frac{dx^3}{dy^0}\\\frac{dx^0}{dy^1}&\frac{dx^1}{dy^1}&\frac{dx^2}{dy^1}&\frac{dx^3}{dy^1}\\\frac{dx^0}{dy^2}&\frac{dx^1}{dy^2}&\frac{dx^2}{dy^2}&\frac{dx^3}{dy^2}\\\frac{dx^0}{dy^3}&\frac{dx^1}{dy^3}&\frac{dx^2}{dy^3}&\frac{dx^3}{dy^3}\end{pmatrix}[/latex] The simplest transform is the Minkowskii metric, Euclidean space or flat space. This is denoted by [latex]\eta[[/latex] Flat space [latex]\mathbb{R}^4 [/latex] with Coordinates (t,x,y,z) or alternatively (ct,x,y,z) flat space is done in Cartesian coordinates. In this metric space time is defined as [latex] ds^2=-c^2dt^2+dx^2+dy^2+dz^2=\eta_{\mu\nu}dx^{\mu}dx^{\nu}[/latex] [latex]\eta=\begin{pmatrix}-c^2&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}[/latex] The sign convention can be confusing. In the above were using (-,+,+,+) so if you look at diagonal components of [latex]G_{\mu\nu}[/latex] (-1,1,1,1) were following the same sign convention. Some metrics use sign convention (+,+,+,-1). (Well hopefully that helps rather than confuse) Just in case here is a decent matrix algebra article. http://www.google.ca/url?sa=t&source=web&cd=1&ved=0CBsQFjAA&url=http%3A%2F%2Fwww.mheducation.ca%2Fcollege%2Folcsupport%2Fnicholson4%2Fnicholson4_sample_chap2.pdf&rct=j&q=matrix%20mathematics%20pdf&ei=WaBmVbjaCrDfsASK4YGwAQ&usg=AFQjCNFLoGWucTsDoKqVhBhrLWIaPeIHbw&sig2=P6W5USwrpu7eDNGAbRf4SQ

Show the equations as being unchanged then. Your papers have some command of latex within them. For this site its [latex]demo[/late$] replace $ with x. Its one thing to assume the equations will remain the same. Its quite another to prove they will do so. Ive worked enough with EFE and FLRW metric to seriously doubt those equations will remain unchanged by your model. In point of detail I'm confident they won't. Which is why I supplied the articles I did covering the Schwartchild Newtonian connections. As well as recommending you study the geodesic equations.

Both time dilation and length contraction are needed to cause gravitational lensing.

If your thinking his lecture notes on General Relativity. I would agree with you. Its an easier read than Wald and more specific than Scott Dodelson. "Lecture notes on General Relativity" Sean Carroll. https://arxiv.org/abs/gr-qc/9712019

I fully agree with that. I was looking into finding a simple enough coverage on geodesics for Declan. I was hoping that Master-Geodesics would help. Unfortunately like a lot of papers it skips a lot of pteliminary details that one would find in a textbook such as Walds "General Relativity". Scott Dodelsons "Moderm Cosmology" didn't particularly help either. One of the hassles when your not sure if the target audience understands Langrene density or the connection coeficient.

Lol provided he can see outside the box. Which is why I specified a closed box. Grandpa the belief that antimatter had anti gravity effects was previous to our understanding of antimatter. Some very old literature still describes antimatter as opposing gravity. Today we know this is false as we can create antimatter at the LHC. We also measured the Earth being bombarded with antimatter.

Lol sure I know your aware of the principle of equivalence. With the accelerometer if the elevator is moving at constant velocity. The inside observer would not be able to distinquish between gravity or inertia. This is one aspect not particularly definable by medium properties.

Yes but he needs some form of reference. Even if it is an accelerometer. Otherwise he would have no means of knowing if he is moving at a constant velocity or at rest.

You didn't have to. Its one of the basic thought experiments of Relativity

Does he? What about the equivalence principle and inertia.? Assume the observer is in a closed box. Does he know the difference unless there is an acceleration change

Do you understand the difference between a change in wavelength as opposed to velocity? In a medium the speed of light is less than c. This isn't an observer effect. You as the observer watch light slow down in a supercooled medium. You have precisely the same reference frame as that of the supercooled medium. It has insufficient mass for time dilation. This isn't the case in spacetime. The speed of light isnt slowed down according to any observer. Due to length contraction its wavelength changes. Not its propogation speed.

Here is gravitational redshift [latex]\frac{\lambda}{\lambda_o}=\frac{1}{\sqrt{(1 - \frac{2GM}{r c^2})}}[/latex] It is the wavelength that changes between observers not the invariant c.

No light still measures c to the distant observer. The wavelength of light varies not its speed

If you design a metric you look for what is common between all observers. Then develop your metric for what changes for each observer. In Relativity the only factor that is common between all observers is c. Both distance measures and time vary between observers. (Makes for a poor baseline geometry). This is one of the fundamental reasons why SR isnt as robust as GR. As SR trends towards the rest frame as the pteferred frame of reference. GR fixes the preferred frame in a non cooordinate dependant manner with tensors

So what about the outside observer? The quantity that is constant or invariant is c.. Look at the mathematics. I posted a basic textbook answer. One that you can find in any Relativity textbook. What did you think the term "invariant" means?

Sure but there is no preferred frame. The laws of physics are the same for all observers. This includes the distant observer. C is the only constant between observers. By the way you also need length contraction.

Look at the postulates of GR postulate 2. ALL OBSERVERS will measure c as the same speed. The metrics I posted show the details. This is the standard GR definement. If you like here is a simplified coverage. https://briankoberlein.com/2015/02/14/burden-proof/ Of course you could just read any GR textbook. They all teach the same postulates. I will link you to a few free ones including a reprint of Einsteins paper http://www.gutenberg.org/files/30155/30155-pdf.pdf: "Relativity: The Special and General Theory" by Albert Einstein http://www.blau.itp.unibe.ch/newlecturesGR.pdf "Lecture Notes on General Relativity" Matthias Blau

Light doesn't slow down. Its invariant all observers will measure it as c. Antimatter follows the same rules as regular matter. The only difference between antimatter and its matter counter part is charge. This is why I wanted you to look at the null geodesic equations in greater detail. While looking at those null geodesic equations. Also look at the interactions for a photon. Then consider that the universe is considered electromagnetically neutral. Consider this. A thousand light years of lead would certainly count as a medium. Yet a neutrino can pass through it without a single interaction. Its path follows the normal space-time geodesics. (spacetime as opposed to time-time (null)) So how can you account for that? Next how can you account for gravitational lensing where there is no nearby galaxy or black holes? Galaxy rotation curves isn't the only piece of evidence of dark matter. Quite frankly we may well be on our way to solving both dark matter and dark energy. Further research and tests are still underway. DARK MATTER AS STERILE NEUTRINOS http://arxiv.org/abs/1402.4119 http://arxiv.org/abs/1402.2301 http://arxiv.org/abs/1306.4954 Higg's inflation possible dark energy http://arxiv.org/abs/1402.3738 http://arxiv.org/abs/0710.3755 http://arxiv.org/abs/1006.2801 This is my cuurent field of study specifically the Pati Salam contributions to the SO(10) MSM model. More particularly on the GUT symmetry breaking aspects. I trend towards the single seesaw mechanism over other seesaw MSSM models. But thats a personal feeling