timo

Not sure if I understand the idea, so the following may not apply here: It sounds tempting to think of vertical rotors in which the torque is extended on the "downwards side" and removed on the "upwards side", e.g. by extending and shortening arms of the rotor. However: In a cyclic process with no outside interference each tiny element in your construction, say each atom, follows a closed cyclic path. I.e. ignoring losses after one cycle the net energy gain/loss of each atom is the same as at the start of the cycle. Hence, the total energy is the same. (btw.: This is not a standard argument against such constructions - it's one I just came up with that sounds somewhat convincing to me, and which does not require to go through all the details of the mechanism that extends the weights on the "downwards side")

A few comments: - Be aware that the reasons you stated for orignally going into engineering (better job market) are still the same as before. - The job market for physics PhDs is generally good, just not as good as for engineers. - I am undecided about the question whether to go to a PhD in physics from an engineering undergrad degree or from a physics one. The physics one prepares you better and the engineering undergrad degree is unlikely to help much in job applications once you have a PhD in physics. On the other hand there are two reasons that may shift the favor towards the engineering undergrad: 1) you may have spent some time on it already and would have to start from scratch in physics, and 2) you already changed your mind about your future once and may do so again in the future: engineering gives you a much better opportunity to decide between going for the PhD route or the "real job" route. An undergrad in physics doesn't prepare you for much except a PhD program in physics.

Books/sources from different decades was supposed to be included in "some books use newer, more precise values" . If you meant something differnt, then I did not understand what you mean.

The (45) represents that the last two digits of the number, 15 in this case, are a best estimate but with an uncertainty. The (45) is the uncertainty of these digits. Usually, you could interpret it as there being a 67% probability that the real number (I am omitting units and exponent for simplicity of writing) is within an interval +- 0.00000045 around the given number 9.10938215. The exact meaning can differ, but it always reflects an estimate of how accurate the number is. You usually don't need the exact error estimate. What usually is important is the following: If you need the number of 0.1% of accurancy then you can consider the number as precise. If you need 15 digits of accurancy then you can completely forget about getting a result because you only have ~7 relevant digits. And if you need an accurancy of 7-8 digits then you have to use statistics. For your first question: I am surprised if different sources give really different values. There is two ideas that come to my mind: 1) Maybe they state the same quantities in different units which causes a different number. 2) Perhaps the differences are on the last digits. That may well be, because different groups may measure the same property (and will come to slightly different results), perhaps some books use newer, more precise results than others. It is very rare that you need this accurancy, so for most practical reasons all the different values should be identical. In the rare case that you need an exact number I recommend getting them from sources that specialize on measuring or collecting them, not from textbooks. My first source for the electron mass would be the Particle Data Group (PDG), for example

The idea to actively capture carbon is not completely new. I never thought it through so far. But two of the three obvious questions towards the implementation (the third one is the business, economic or political model driving this which is an arbitrarily broad discussion) are 1) What do you do with the carbon you capture? Dumping it underground, perhaps. Note that despite talking about gasses we are talking about huge masses, which probably still have huge volumes when liquified or solidified. 2) What is the technology you want to use? Something that need lots of electric energy is probably not smart, except if you cannot get enough of these renewables even after you already fullfilled all your other energy needs. A theoretically very simple and cost-effective technology would be trees, btw.

Few random comments, since I do not know if that actually is supposed to be the discussion thread and if there really was a question: - Electricity prices in Germany have dropped on the market level due to renewable generation. - End-user prices have increased, particularly due to technology subsidies for renewables being paid by the consumer. Subsidies for other technologies are paid by the state, i.e. they are contained in taxes and do not appear on the electricity bill. - More generally: Theoretically you can get to zero electricity costs, irrespective of technology. Just increase taxes and provide free electricity for everyone. Similarly, you can get to any high electricity cost irrespective of technology: Just put a huge tax on electricity consumption. The price of a unit of electric energy depends possibly more on the market design than on technology. And it may not be the relevant quantity, either - even for electricity consumers who usually happen to be tax payers at the same time. - In other words: You may be asking the wrong question, or at least ask in in a form that is too simple.

Wrote a long post and then hit a wrong button. I'll try with short bullet points instead: - Think of matrices as a compact convention how to write down operations performed on vectors. - The notation is unique: Two operations are the same operation exactly if their matrix is the same. - Only operations with a certain property (linearity) can be expressed by matrices. That implies that if something is expressed as a matrix you immediately know that these linear properties exist in your problem, which may not be obvious otherwise. - Linear operations can be combined, which has a compact notation equivalent on matrix level: First applying operation A on a vector x, and then applying operation B on the result has the matrix notation BAx. - The whole matrix thing gets really interesting when realizing that the combination of two linear operations is a linear operation by itself (the operations form "an algebra", in this case "a ring"). Above combined operation could be written Cx, where C=BA. When B and A are represented by a matrix, there is a direct construction rule telling you how to calculate the matrix of C. In other words, you can do math with operations/matrices as if they were numbers (except that a few rules are different). That is extremely powerful. - Deliberately expressing something as matrices can be extremely helpful on an advanced level. In advanced math, your main problem often is to understand the mathematical behavior of your system. If parts of it can be expressed in matrix formalism you directly see that matrix math holds. In fact, just yesterday I solved an optimization problem by re-arranging my equations/objects such that they could be expressed as the extremely common equation Ax=b, with x being the sought-for parameters. Methods to solve this equation are well known, and in fact I stopped by work at this point (I was merely looking for a construction algorithm for a solution, so my work was done after getting to a point where well-known standard algorithms can take over). Bottom line: Main point of matrix notation and matrix math is that it appears very often, so knowing to work with matrices can help you in many problems. Be it on the calculation level (knowing how to efficiently perform calculations) or on the mathematical level (knowing abstract mathematical properties of your system because you are familar with the abstract mathematical properties of matrices).

Actually, the question literally was that if an alien lands in my backyard and told my that some of (or maybe even all of - it's not really specified) the scientific theories I believe in are wrong, "what would my reaction be?". And my statement was that I would be interested to know more, because I assume someone who builds interstellar spaceships will have something interesting to say. It's a bit like assuming a scientist may know something about science, and not asking for evidence of every statement in a discussion (even though it does not rule out this in key parts - since evidence is a process in which the theory is relevant it also helps the understanding process). The rather extreme interpretation of this as "trusting everything" or considering an alien all-knowing or infallible was not really what I meant. I admittedly did imply some mockery about "an alien with a spaceship does not impress me"-attitudes, though. Especially in the light of seldomly seeing physics professors whose students ask for evidence of their teachings.

I think our speculation posters could well compete with the average caveman when it comes to science (or writing, or oral hygiene, or knowing about the existence of such a thing as oral hygiene). Maybe not with the one who invented the use of fire (who may well be a senior expert of stone age oral hygiene). But surely with those who believe that the clan's hunters suppress the truth that mammoths and sabretooth tigers in reality are the same thing . EDIT: As a side-note: Those hunters probably know more about sabretooth tiger anatomy than all the biology experts on this forum. But that's no excuse to neglegt oral hygiene.

And still a caveman with an interest in science might be tempted to listen to what I have to say about our nuderstanding of chemistry and atoms.

Most of the discreteness that is often attributed to light comes from the discreteness of the process that created the light. Not from the nature of light, which is the nature of the electromagnetic field. The electromagnetic field has very few discreteness constraints, especially non that are as trivial as "light only comes in packets" or "you can only have certain energies" or "you can only have certain frequencies". The most famous/important constraint on light itself would be "if you enforce a given frequency (which usually would be an effective condition you have from having a certain creation process in mind), you can only have certain energies". And strictly speaking I think even this is only true when restricting yourself to a certain regime, the so-called free-field solutions (a very important regime for practical work, though). Separating between the discreteness of light and the discreteness of the creation processes may seem like hairsplitting. It is pointless to talk about light that is not created, after all. But the discreteness conditions of different creation processes are different, and I think making this destinction helps in understanding physics.

For me, being from a race that builds starships to travel vast distances through space (assuming "landing" to relate to something like that) and having the capability to speak alien (i.e. human) languages to a point where you can discuss about science would count as evidence that the person in question has relevant things to say. We should have a forum rule that having built a functional insterstellar space ship exempts you from having to supply further evidence to speculation posts.

I find the adjective "modern" much worse. Last statement about the meaning I got (years ago when we re-organized the forums) was "it means what is written in the sub-title". Begs the question why the sub-title isn't the title in the first place ("Nuclear, atomic and particle physics").

Since you did not really explain any of the notation (admittedly not very easy to do if the notation is what confuses you in the first place) replies you get will have to be guess-work. One guess would be the following: If h(x) denotes the value of the function h at a given position x, then [math]h^{-1}(x)[/math] often means the value of h's inverse function [math]h^{-1}[/math] at value x, whereas [math]h(x)^{-1}[/math] would mean the value of the function h at position x taken to the power of -1. You'll have to check if that makes sense in your context, though.

It is not clear what an increase of gravity is supposed to mean. Unlike "mass", "energy" and "momentum", "gravity" is not a physical quantity. You wouldn't say that an object's electromagnetism or its color increase, would you? That may sound pedantic - because it is . The reason for being pedantic at this point is that in the context of your question, which implicitly begs for being treated in the framework of a relativistic theory of gravity, there are two possible statements to be made about an increase of gravity caused by an increase of energy. And they both conflict with each other (caused by two different interpreations of how to quantify gravity). And the reason for the conflict is not that the relativistic theory of gravity had flaws. The reason is that strictly speaking an "increase in gravity" is not a properly defined physical statement. Energy kind of is a variable in the mainstream formulation of relativistic gravity. The famous equation for describing relativistic gravity are the Einstein equations (it is only one equation but for some reason the plural is commonly used ). This equation relates an origin of gravity, described via the energy-momentum-stress tensor, to the effects of gravity, described via the curvature of space-time. One variable in the standard representation of the energy-momentum-stress tensor is energy (as you might have guessed from its name).

Depending on the context there may be different interpretations. The simplest one would be the following. You take a random variable x and a random variable y. The probabilities for each combination are given in your table. Question: What is the probability that x and y do not have the same value? Sidenote: Strictly speaking is it impossible to answer questions about mathematical expressions when the meaning of the letters that appear is not given. Effectively, their meaning can often be assumed - e.g. in the context of a physics problem with only one body one often assumes that "m" means the mass of the body. But for X and Y (capital letters) combined with a table that lists x and y (non-capital letters) assuming things about the meaning from only the letters is highly speculative.

Strictly speaking, but possibly not what you meant, that is not even possible but fairly common. In the sense that people have a diploma in field A and during their initial career drift into another field B. Generally, "fields of research" are not very sharply defined in practice (e.g. yesterday I attended a talk by a climatologist who was very proud of their ice-simulation model - I'd bet not everyone working on that model started out in climatology).

Formally, it is one line. I don't think that accounts for much. Syntactically, it is two equation signs. That is, you made two statements (note that "=" signs appearing in calculations also indicate statements, even if they are as simple as "I can re-arrange this term to look like that"). That accounts for a bit more. One may be tempted to arrange the three expressions in a triangle and write "=" signs between them to better show the symmetry. Accounts for not so much, because that is just my taste of aesthetics. Information-wise, it seems equivalent to two equations - the third equation gained by closing the equation circle does not contain any new information. That sounds like the most solid statement to me. I wouldn't disagree with any number in {1,2,3}. If I was forced to pick one I'd go for 2.

Looks like an excellent example of a unit of measurement that is very impractical. My limit is the Maß (https://en.wikipedia.org/wiki/Ma%C3%9F), and even that is a bit of a Bavarian freak idea.

@Greg: I don't quite see how that equality holds. And also not what "resolves to one" means. You probably meant something that is correct but used an incorrect mathematical symbolism to represent it. On topic, and from a more abstract point of view: If you have a property X and a known invertible function Y=f(X) over the domain of interest then Y holds the same information as X. Usefulness and ease of understanding then are the main reasons to chose one over the other - and the fact that 0 m/s is a velocity that is relevant in lots of applications may well be the reason for choosing this over seconds per meter (both are arguably similarly easy to understand). In everyday science, particularly in spoken conversations, this equivalence of information is very well understood and implicitly used all the time. It may be a bit less common in internet forums where nitpicking is considered a way to demonstrate knowledge about a subject.

Slightly off-topic, but whatever: If you are using a different computer then you are probably going to have different browsers in a sense (that may share name and version, but are not the same unit of installation). The German language has two words for that, one that says "same model" and one for "same object". Would the English language offer such a distinction, like "equal browsers" (same model) and "same browser" (same piece of installed software)?

Everything you ever wanted to know about the term "theory" and more: https://en.wikipedia.org/wiki/Theory .

Naively, the Big Bang scenario gives you a dynamics for the distances between points: Distances between fixed points do increase over time, which is called the expansion of space. What is very interesting is that if you follow this dynamics into the past, then there is a point in time where all distances are zero. I would chose this Big Bang time as t=0. There may be a problem with this if some physics close to the so-expected t=0 causes the expected point to not exist (i.e. if the dynamics of distances is different close to t=0 then extrapolating today's dynamics through this region may cause nonsense results). Sidenote: What you call "Planck Time" is probably what I would call "Planck Epoch". Planck time usually refers to the length of a time interval, not a time-stamp. At least in my understanding.

It is indeed more likely that you keep the physical meaning of mid-day being in the middle of the day and drop the artificial meaning that mid-day had to be when the clock reads 12:00 (not sure how mid-day is understood at the poles today, btw). As a side-note: The modern term of GMT is universal coordinated time UTC. We use that in computer programs, for example. Reason is that internally you then have a clear definition, and you only have to care about time zones when displaying dates to an end-user. Internally using time zones is a recipe for disaster.

In case this more basic point was the issue: The book does not say 16 bit were equal to several thousands of bytes (they are equal to two bytes, of course). What it says is that if memory locations are encoded with 16 bit then there are 2^16 different possible memory locations that can be encoded: 2^15 in each direction with one left for the nitpickers or those who count zero as having a direction.