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14h
comment How could we travel to the nearest supermassive Black hole?
The total energy of the later states of the Universe would still be zero because the energy is conserved - but there would be both positive and negative contributions. It would be analogous to the pencil.When it's already falling, the potential energy is lower (negative relatively to the initial state) than zero, while the kinetic energy of the pencil is positive. In the Universe, the analogy of the falling pencil would be a cosmic cataclysm (cosmos filled with huge amounts of hot radiation etc.) that would begin immediately.
14h
comment How could we travel to the nearest supermassive Black hole?
The vacuum has to stay the vacuum, so it must be the state of the Universe with the lowest energy - often labeled by the number zero. If there existed states of negative energy (energy lower than the vacuum's), the set of configurations in which the Universe may be would resemble a pencil standing on its tip. Even if the forces were balanced, any tiny deviation would lead to the pencil's exponentially growing deviation from the unstable position, and the pencil would fall. Similarly, the Universe would get filled with lots of tachyon waves moving everywhere.
1d
comment What ds>dQ/T mean?
There is no objective identification which part of the system is the system and which part is the environment. It's the total entropy that grows more than the bound. This extra growth may take place both in the system and in the environment.If the environment has blue and red ink which gets mixed to a purple ink, the entropy of the environment goes up. If the same inks appear within the system, the entropy of the system goes up, even without any heat transfer.
1d
comment What experiment would disprove string theory?
Dear Philip, my answer makes it quite detailed that this isn't the only way to disprove the theory. However, if you could prove (or back by strong evidence) your statement, it would be a great news for the validity of string theory because that situation would mean that you have proven the theory, in any sensible sense of the word "proof". To think that it would be bad news means to misunderstand the basic rules of logic. The dream of physical theories is to survive (or survive for as long as possible), not to be killed.
2d
comment How do I calculate the differential cross section with respect to the transversal momentum?
Apologies, Shawn, you could force me (or someone else) to solve the full problem, but I think that if that happens, it could only be used to cheat. I am not really helping you anymore. You really shouldn't try to solve similar complex problems such as the parameterizations of differential cross section before you are sure enough about the relativistic dispersion relations, among other more basic things. In all these 2-to-2 problems, there are four momenta obeying $p^2=m^2$ with the obvious $m$, and 4-momentum conservation holds. One has to be sure about these matters before more complex stuff
2d
comment What is the difference between the Big Bang Model and the Λ-CDM Model?
These explanations appear in many reviews, textbooks, introductions. I was intrigued to mention the book "A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity" (buy at amazon.com or elsewhere) which has pages 315-320 about the FRW stuff etc. What I like about this book is that there is already a section in the updated book about the 2015 LIGO detection. ;-)
2d
comment What is the difference between the Big Bang Model and the Λ-CDM Model?
You may look at the Wikipedia entries for the radiation-dominated era en.wikipedia.org/wiki/Radiation-dominated_era and similar pages. The timing and average temperature and energy density and speed of expansion etc. are all calculable (obeying various power laws). Cosmologists first simplify Einstein's equations to the Friedmann-Lemaitre-Robertson-Walker equations for a uniform universe with uniform matter, and solve this much simpler ordinary differential equation that only depends on time. The different eras differ by different ratio pressure/energy_density.
2d
comment What is the difference between the Big Bang Model and the Λ-CDM Model?
Dear @SyedAli, we don't measure the value "50,000 years" directly in any way. I already answered this question in the previous comment. Nevertheless, we feel sure that the theory is right already from the first three minutes because it produces e.g. the right percentages of the light nuclei as we observe, and those were created in the first 3 minutes. This agreement (plus CMB curves etc.) is a source of a big confidence that the theory is correct, and that's why we also believe the equations of the big bang that the radiation era took 50,000 years, and many other things.
Apr
29
comment What is the difference between the Big Bang Model and the Λ-CDM Model?
Dear Syed, cosmologists aren't directly observing the radiation-dominated era. The matter dominated era ended relatively recently (now it's the cosmological-constant-dominated one, forever), and the transition was never sharp in any way. The standard evidence for TBBT is listed here: schoolsobservatory.org.uk/astro/cosmos/bb_evid
Apr
29
comment How do I calculate the differential cross section with respect to the transversal momentum?
The last equality follows tautologically because if you define the longitudinal parts of the 3-vectors as vectors at all, they are 1-dimensional vectors, so the inner product of the vectors is the same as the product of the "only components". No, the derivative of $t$ with respect to $p^T$ surely isn't zero.
Apr
29
comment Question about the apparent loophole in principle of least action
For example, in the Euclidean path integral in QFT (using the action principle), one can find numerous solutions even with given boundary conditions, the instantons. That doesn't mean that something is wrong and we should discard these configurations or the theory. These instantons (even though they are just "local minima" of the action) are genuine and they give contributions to the probability amplitudes - and allow some otherwise forbidden processes.
Apr
29
comment Question about the apparent loophole in principle of least action
I don't think that your "safer viewpoint" is justified. This whole negative "spirit" in the interpretation of these things is largely irrational. When some laws of physics are formulated in terms of the action - and they often are in practice - and the solution to the least action principle says that there are no solutions or many solutions, then it is a right and important insight that means something and we should take it very seriously, regardless of the (naive!) prejudice that there always has to be a unique solution.
Apr
29
comment What is the difference between the Big Bang Model and the Λ-CDM Model?
So if the "consensus" came from a back-translation of the "concordance model", it isn't just an inaccurate translation - the very essence is completely wrong. After all, Steinhardt may have flaws but he was ahead in doing these things before the cosmological constant was discovered. And he is not among those who would ever worship a consensus, especially given his criticisms of inflation that pretty much is a matter of consensus. His criticisms of the inflation is something he cares about and he knows that any "consensus talk" would quickly backfire against this important thing of him.
Apr
29
comment What is the difference between the Big Bang Model and the Λ-CDM Model?
Dear @CuriousOne, I have actually encountered the term (cosmological) "concordance model" in the technical literature. I didn't know what it was at that time. But at some moment, I realized it goes back to a 1995 paper by Steinhardt and Ostriker, arxiv.org/abs/astro-ph/9505066 - but the concordance isn't an agreement between the people (majority of cosmologists). It's the interesection of the experimental strips in the "h" and "Omega_Lambda" plane - constraints by 3 main relevant types of experiments. Nothing to do with the society.
Apr
29
comment What is the difference between the Big Bang Model and the Λ-CDM Model?
But my point is that it's very likely that the "consensus model" was used for something that is actually rather controversial - because that's the very reason why people talk about "consensus". They use the word to exert some political pressure, pretend that those who disagree are a minority to be crushed, and they have to because they don't have any actual proof or even evidence. So it's up to the listener to be very careful when he is being "impressed" by the phrase "consensus model" - something wrong is probably going on in that case.
Apr
29
comment What is the difference between the Big Bang Model and the Λ-CDM Model?
Thanks, @CuriousOne. Dear Syed, if I were asked to guess what most cosmologists who would use the unfortunate phrase could have meant, it could have been both TBBT and LambdaCDM. But the cold dark matter could also be more specific, made of WIMPs, but in that case, the degree of people's belief that this is what we have in Nature could really be below 50% (and young researchers aren't too big WIMP fans, it may be seen), and our beliefs don't matter, anyway.
Apr
28
comment How many observations over an atom can be made?
So an exact measurement of the location unavoidably ionizes the atom. If you repeat the measurement of the location a moment later, the electron will be at a different place. You may also measure the momentum vector. You get some reasonable quantity agreeing with the expectation that the kinetic energy is comparable to the binding energy in the atom. After the measurement, the location becomes totally uncetain - in the whole space - so it's again almost certain that the atom got ionized. But there are also "limited accuracy" measurements, e.g. measure in which tenth of the atom the electron is
Apr
28
comment How many observations over an atom can be made?
Every measurement impacts the electron - or any physical system - in quantum mechanics. How long you may play - it depends what property of the electrons you measure. For example, if you measure the energy, you get one of the allowed energy eigenstates, and it's there again. Instead, if you decide to measure the exact location $\vec r$ of the electron, you may get it but the momentum $\vec p$ becomes infinitely uncertain after the measurement, by the uncertainty principle. This corresponds to almost 100% certainty that the momentum is large enough for the electron to escape the nucleus.
Apr
28
comment Fine structure constant and unit conversion
If you read e.g. articles about some X-rays relatively to the LHC, a TeV could get converted to a keV from the X-rays, and there would be $10^{6}$ there, too. But I understand your explanation why it happened, too.
Apr
28
comment Fine structure constant and unit conversion
It's very attractive to say that mega, $\text{M}$, is one thousand, because the lowercase $\text{m}$, or "milli", is one thousandth. ;-)