Mordred

Hrrm the Aichlburg-Sexl Ultraboost is rather tricky to find good articles on it. Though those I found do show the derivatives via the transformation rules. The log function I would like to get more clarity on. ( Though it makes sense to a degree) Do you have a good source ?

Thank you for the clarifications Markus I had looked at the Vaidya spacetime before. Though it has been awhile however I had never heard of the Aichlburg-Sexl Ultraboost. Rather if I have I can't recall it. As I previously stated my studies on BH metrics are lacking compared to my primary expertise in Cosmology studies. So many of these other coordinate systems I rarely look into. So I am glad for the clarifications +1

Good answers to you both however the correct answer gets more complex. @md65536 the Schwartzchild metric though it is still a horizon breaks down on different observers. The coordinate system cannot accurately describe the causal nature of events. @Endy0816 the radius of the null surface would change afiak however the mass determines the radius so it would not be an inverse relation. However the null surface isn't necessarily the event horizon..... (more on that later) ( I could very well be wrong on this) This is where we can finally get back to the Ops misunderstanding on Hawking radiation. There is another horizon called the Apparent horizon which represents the trapped surface. This trapped surface may or may not coincide with the event horizon described by R_s=2GM. Here is an article covering this. https://www.google.com/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/hep-th/9501071&ved=2ahUKEwixr5mr7IzqAhWHvZ4KHZhcA44QFjABegQICRAB&usg=AOvVaw2CyhOXTEPO_MkckNS8tQqE Now the Apparent horizon definitely changes according to the observer. I'm unclear if the Schwartzchild event horizon does due to its limitations of valid observers and being a coordinate singularity that doesn't truly describe causality for all observers. (Future past lightcones etc) Let's readdress this question now that we are looking at apparent horizons being involved in causal relations. @rjbeery Don't get discouraged the Penrose diagrams are extremely confusing with regards to causal connections even with lightcones. Hawking radiation is described in more detail in the last link but note which region it occurs. @everyone involved I am far more familiar with cosmology and particle physics than black hole dynamics. Lol unlike many I studied them to a certain extent but it's never been a focal point in my studies. 🤔 Anyways here is another article covering horizons in regards to BH's Black hole Boundaries. https://arxiv.org/abs/gr-qc/0508107

Unfortunately this is not true in the case of a coordinate singularity such as the EH. A coordinate singularity is not invariant under coordinate change. The r_s=2GM is an artifact of the Schwartzchild metric. I'm going to add a hypothetical question. Would a near c observer see the same radius for the event horizon as the at rest observer. You would see a different Blackbody temperature and as a result a different rate of Hawking radiation. Ie Unruh effect. (PS the answer cannot rely on the Schwartzchild metric ). One can argue the Schwartzchild metric is only suitable to a far away observer.

Here Proper acceleration (the acceleration 'felt' by the object being accelerated) is the rate of change of rapidity with respect to proper time (time as measured by the object undergoing acceleration itself). https://en.m.wikipedia.org/wiki/Rapidity However rapidity isn't strictly acceleration. You can have other relations that can apply rapidity. The link gives a few examples.

Be aware the dot notations can also mean the inner product of a vector. However in this case It is multiply

Agreed but let's ask a question. Where is the null surface located ? Under one coordinate choice its r_s= 2GM. However this isn't true for the Kruskal. Now ask yourself is it the spacelike or the time like that are invariant under the Lorentz transforms ? This question becomes important to understand the region's of the Penrose diagrams. Here is an examination of different causal connections in different coordinate systems the article is specifically dealing with Penrose diagrams. https://www.google.com/url?sa=t&source=web&rct=j&url=http://people.uncw.edu/hermanr/GRcosmo/penrose.pdf&ved=2ahUKEwixr5mr7IzqAhWHvZ4KHZhcA44QFjAAegQIARAB&usg=AOvVaw0E2LE-32F7TxpcK6vXujWN

Yes in that particular coordinate choice. However in the Schwartzchild metric itself you can get different results. The paper above described the differences. Though I am positive there are more current examinations. I would also not be surprised at different results within the same coordinate choice between authors. It's been a contented item for a number of years.

Sorry to disappoint you but the speed limit c applies to all forms of information exchange regardless of what you believe.

Ok this is bordering on getting off topic but consider two observers. One a rest watching an infalling object. That observer will never see the object cross the EH. Now in the infalling objects case that observer will cross the EH in a finite time period. Is the end no that observer will see future events that the observer at rest will not experience. Here is a relevant paper. Title is " Is it possible to see the infinite future of the universe when falling into a Black hole " http://arxiv.org/pdf/0906.1442v1.pdf Now note that this paper also describes a coordinate choice that can applied beyond the EH. Different coordinate systems can give different results. The EH is a coordinate singularity it is not a true singularity I already supplied a reference for that statement. Now given the last paper can you state the two observers experience identical causal connections ? Can you state an observer in the interior of the EH is causally disconnected from the universe outside the EH ? I will supply the coordinate system that describes this region. https://en.m.wikipedia.org/wiki/Kruskal–Szekeres_coordinates Just to give you better details on the Coordinate singularity and how to remove the singularity see https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.roma1.infn.it/teongrav/leonardo/bh/bhcap12.pdf&ved=2ahUKEwirjPaQ5YzqAhUJop4KHRgHCrQQFjALegQIAxAI&usg=AOvVaw3mu5haWWZXIJt3HKgb-vMS

Sigh your still not getting it. Here there is a clear example of the acceleration aspects and rapidity under the twin paradox https://www.google.com/url?sa=t&source=web&rct=j&url=https://arxiv.org/pdf/1701.02731&ved=2ahUKEwi9lqfzj4zqAhVCoFsKHRaXCHIQFjAFegQIAhAM&usg=AOvVaw3vAZGh65NWKJZiFukMfOZ0 Though a better example is under Introductory to GR by Lewis Ryder.

Look at the formulas do you require trigonometric functions for constant velocity ? There are six types of boosts and three rotations. https://en.m.wikipedia.org/wiki/Lorentz_transformation The example they give here is a rotation in spacetime which is a form of acceleration.

No rapidity is a term used to describe the hyperbolic rotations due to Lorentz boosts. It is not involved under constant velocity.

I will have to crunch some mathematics involving the Schwartzchild metric to demonstrate for the infalling as opposed to at rest observer. However you should recognize the Schwartzchild singularity is a coordinate singularity as per https://en.m.wikipedia.org/wiki/Eddington–Finkelstein_coordinates An infalling observer is undergoing rapidity (acceleration) also your dealing with curved spacetime so it will take me a bit to latex the examples. I should have time tonight for that.

Please provide the source of the diagram above. Some of the details above need clarification. Another point of detail is that on causal connections one must also take care on the observer. To an outside observer once the EH is crossed you have reached infinite redshift. However a coordinate change can somewhat differentiate where this will occur. An EH is an apparent horizon not a true horizon. So how one defines the light one also involves the coordinate choice. Also there is a significant difference from what an observer at rest will note and an infalling observer. They will have significantly different lightcones. To make matters worse a rotating BH can have four event horizons. The Penrose diagrams also show that a rotating electrically charged BH is different from both the static and rotating BH. An infalling observer for example will see a different location of the EH as opposed to an an observer at rest. (Black holes and causal connections are not easily described. There are non trivial factors to take into consideration) Hence needing further details on your diagram. Diagrams are no substitute for the mathematics ie the line element ds^s.

Correct at the moment prior to Singularity the temperature is Planck temperature [math]10^{32}[/math] correction to previous temp I mentioned at Planck time [math]10^{-43}[/math] in a Planck length. You will arrive at these values by extrapolating expansion backwards prior to those you have a singularity condition which we cannot mathematically describe. Note at this point the diameter is only 1 Planck length. What does this say for the argument particles have no room to move ? ( That argument stems from thinking particles are like billiard balls ) an elementary particle has no discernable volume. They can still vibrate as per the quantum harmonic oscillator. The other detail is we do not know if the universe is infinite nor finite. That Planck length is only our observable universe portion at 10^-43 seconds. The remainder of the universe at that time could very well be infinite or simply a much larger finite portion outside our shared causality described by the observable universe. With a BH we don't even know if the singularity described is even feasible. We can only speculate

Any measurement device would experience the same effects. Those effects do not require a conscious observer.

An atomic clock placed on Mt Everest recorded time dilation with no conscious observer present. Your theory is easily falsified.

Whoever wrote the article wouldn't know. However consider this the average surface temperature of a neutron star is roughly 600,000 K. The BB intial temperature is roughly [math]10^36[/math] K. Yet expansion causes cooling. So consider this how does a cold initial singularity lead to such an extreme high temperature due to the BB expansion ? Obviously the temperature would initially be hotter.

It would be more accurate to think of the drag being caused by the particles that reside in space.

Correct on both accounts. No microblackholes formed at the LHC other other particle accelerators. Also a BH must have an event horizon.

Your drawings really do not reflect the magnetic field. I'm positive you have used iron filings with two magnets. That test alone shows the curl aspects of a magnetic field.

I know what you mentioned. I still haven't seen anything of substance from you. Relativity is merely one example where higher dimensions is applied. The models that use higher dimensions is numerous however all physics theories treats the term dimension as per the definition I provided above.

Well the situation gets complicated by the commoving coordinate system however the Ricci tensor is still applicable. The steps to describe the FLRW metric for a homogeneous and isotopic universe with and without curvature terms is fairly lengthy. Here is a rather detailed article covering the main aspects. It also shows how the key FLRW formulas are arrived at via the GR formalism. https://www.google.com/url?sa=t&source=web&rct=j&url=http://icc.ub.edu/~liciaverde/Cosmology.pdf&ved=2ahUKEwj3k6qiv4DqAhW4IjQIHaI8CcEQFjACegQIARAB&usg=AOvVaw3mK36Mj5P8FmJLI4RPNXOX Mathius Blau also has a decent section on the FLRW metric in his lecture notes on GR. ( much later chapters) http://www.blau.itp.unibe.ch/newlecturesGR.pdf My main point is simply that all the key FLRW metric equations can be applied with the GR formalism. I'm point of detail in a sense the FLRW metric is a special solution of GR (Ie vacuum solution of a perfect fluid)

If you wish an intelligent response you will require greater detail. I am one of several physicists on this site. I do not need to prove myself to others. My response still stands in the correct meaning of a dimension under physics and regularly work with higher dimensions in numerous different physics theories. Higher dimensions do not require the time dimension however you need better clarity otherwise this thread goes nowhere. ( the use of higher dimensions is even applicable in engineering applications ) we also have several experts in that arena. However only 4 dimensions are required to describe gravity under spacetime as per GR. So the question Totally makes no sense. You cannot separate gravity from spacetime curvature. They are one and the same.