Orodruin

I just want to underline some of what has already been said: What you do for exercise is important, but what you eat and in what quantities is more important. A common mistake is to go "on a diet", meaning you see it as something temporary. In order to get rid of excess weight and keeping a healthy weight later on is to implement a nutritional plan that works for you long term. If you see it as something temporary you might get rid of some of those extra kilos, but as soon as you start eating "as usual" again you are going to regain them and possibly more. You need to find healthy foods that you like in order to be able to stick to the long term plan. Also, do not change everything at once, that is just a start for failure. Start little by little to phase out unhealthy stuff that you eat for healthier alternatives and do not overeat. Do not eat between meals. Good food will keep you sustained and let you go longer without feeling hungry between meals. In the end, losing or gaining weight is just about caloric deficit/surplus and what you eat is key to helping you with that. Also, if you are being really serious about wanting to live healthier, you should probably see a professional who can help you find a meal plan and exercise routine that will work for you long term. It is an investment that will not only help you feel better and live longer, but also save you future medical expenses as well.

This is true classically. The underlying reason for this being the linearity of Maxwell's equations, which allow superposition of electromagnetic fields. Once electrodynamics is quantised along with the introduction of some charged fields, you get light-by-light scattering at the one-loop level - as observed last year by ATLAS. However, this effect is really tiny. Ultimately because the electromagnetic field does not carry electric charge and the source for the electromagnetic field is electric charge and current.

As I said earlier, I do not want to put the label "real" on any concept in physics. However I do not see how one could consider the electron real without considering the photon real. Both are just excitations of different quantum fields, what would make a fermion field more real than a boson one? Or a fermion field in a certain representation more real than a gauge field? If you cannot call a photon real because it is always absorbed in its interactions I would not call electrons real because of the same reason. That another lepton is created in its stead in order to conserve lepton number I would consider inconsequential. Yes, (simplistically) each particle is an excitation of the corresponding quantum field.

I would tend to agree with you if you are saying what I think you are saying. As a scientist, I am interested in how well a theory describes the data. If it does a good job at this I am going to be happy with it. Whether or not the underlying constructs of that theory are "real" (whatever meaning is given to that word) are of less importance.

We do feel it. It is the fact that the Earth has to keep pushing us up. This is how gravity is working in GR, there is no gravitational force and objects follow the closest thing you can get to straight lines (known as geodesics) in curved space-time unless accelerated. The curvature of space-time at Earth's surface is such that you would start approaching the center if not accelerated by the ground. The satellite on the other hand is not being acted upon by any force and is therefore not feeling any acceleration. It is orbiting Earth because space-time is curved in such a way that this geodesic becomes an orbit.

Did we just compare a rate with a velocity? With the figures he put in: 53 Mly is approximately 16 Mpc, which would mean an expansion velocity of H*d of order 10^3 km/s, i.e., larger than the quoted velocity. This is of course assuming that the quoted velocity is relative to the comoving frame. Of course, it is borderline and the Milky way is also moving wrt the comoving frame which may change things. On the other hand, the quoted velocity is not even in accordance with the stated timings if the Universe was not expanding - if it is 53 Mly away, we see the light now, and the projectile will arrive here in 17ly, then it would need to travel at roughly 0.76c >> 900 km/s. Conclusion: Do not trust things you read in magazines blindly, they have put numbers that do not add up without telling you how they arrived at them. We see it 53 Mly away, which means that we see it as it was 53 Myr ago and that its total travel time would be 70 Myr. 53/70 ~0.76 < 1.

The GPS satellite is not accelerating. It is in a state of free fall and follows a geodesic in space-time. If you were sitting on the satellite you would not feel any acceleration.

Well ... pure plutonium-239 would have a critical radius of just under 10 cm. It is a bigger than a billiard ball but not several orders of magnitude.

With the conclusion that GR is needed to get the correct result for GPS satellite clocks as they actually tick faster than clocks on Earth. On the other hand, someone standing on Earth is accelerating in the GR setting while the satellite is not ...

A photon in itself is a quantum field theoretical object which makes it difficult to picture in terms we are familiar with. Just as a quantum mechanical particle does not have a well defined position or is a classical wave, a photon is also struggling with this. A better (for many purposes sufficient, but still not fully accurate) description of a photon would be in terms of a wave packet for its wave function.

To add a bit on the why bother issue: when you have a lot of photons the effects on the macroscopic level are well described by classical field theory of electromagnetic waves. Quantum effects become averaged out. At that level it is simpler to treat the EM wave and it still gives a good description - much like newtonian gravity is fine for everyday life without the need to go to GR.

As a theoretical physicist (well, high-energy phenomenologist, but people tend to put us in the TP box), I am not sure I am ready to sign this statement ... On the other hand, what you and I see as a formality may be different. If I remember correctly, Einstein's 1905 paper on electromagnetism (the special relativity one) contains the Lorentz transformations of the Maxwell equations in component form ... I would describe my relationship with tensors as an appreciation of a general framework and amazement over how useful it is in different branches of science.

Voyager put an upper bound on the vacuum energy density using measurements of the solar system based on how the planetary orbits behave. In order to do this, it was necessary to know the masses of the planets to great precision as well as the orbits. This limit was around 2 x 10^-17 g/cm^3. The value measured by cosmology today is around 6 x 10^-30 g/cm^3 so it should not come as a surprise that Voyager did not see a significant signal. EDIT: typo

Well, yes and no. We do not know what the dark energy is made of or its exact equation of state - it could be increasing in energy density and it could be decreasing as the Universe expands. We do know that the normal forms of matter and radiation (including all particles we have seen so far) dilute their energy densities as the Universe expands.

Normally what can be measured is differences in energy except for one case - Einstein's equation - which govern the geometry of space-time and therefore also the expansion of the Universe. This means that you cannot measure the vacuum energy density locally (this is a good thing, it would be quite unhealthy if you could) but instead it must be inferred from measurements at larger scales through the expansion history. Now, on the other hand, differences in the vacuum energy of two setups may be measured through the Casimir effect, which results in a net force.

Some questions to get you started: How long does it take for each of the moons to complete a full lap? Using this result, how many laps will each of the moons have completed after an arbitrary time t? What is the condition for the time when the inner moon passes the outer one? That being said, I think you are already on to a good start from your mention of Kepler's third law.

We are not forgetting anything. The statement is in itself a contradiction since you cannot break conservation of momentum without breaking the law of momentum conservation. In order to change the momentum of a system, you need external forces - which are missing from your setup. Furthermore, an external force would simply transfer the momentum to whatever object was exerting the force due to the action/reaction counterpart acting on that object. When talking about stresses in continuum mechanics, you will often be studying a small part of the full continuum. This part may very well see a change in momentum, but this change will be compensated by a change in the momentum of the surrounding continuum.

So to summarize and put it plainly: Maxwell's theory is not only a theory of how light behaves but of how electromagnetic fields behave and interact with matter. Since light is a form of electromagnetic radiation it is described by Maxwell's theory. However, it is not based on optics and any typical textbook on the subject is not going to start from there but probably from static configurations. Once dynamic fields have been reached, optics will follow from the theory. This is why the book does not start by discussing optics.

There is no smallest wavelength for photons. However, the wavelength is inversely proportional to the energy of the photon. While a photon with a wavelength of 600 nm has an energy of around 2 eV (the typical scale of binding energies of electrons to atoms). As has already been mentioned, a photon with a wavelength of a Bohr radius (ca 0.5 Å or 0.05 nm) would have an energy of ca 23 keV, large enough to ionize any atom. This question is essentially of the form "If I could violate the laws of physics, what would the laws of physics be like?" See 1, there is no smallest photon, but photons with shorter wavelengths would do things to your atoms you would not be particularly fond of.

I would say this depends on your definition of noticeable. For everyday experience it is clearly not noticeable, but it is an effect which can be and has been measured (see eg http://www.sciencemag.org/content/329/5999/1630). The effect would also be quite noticeable if GPS satellites did not correct for it.

The matrix is directly related to the transformation. I suggest you think along the following lines: I want to map A to B, what does this mean? Which points in A should be mapped to which points in B? How can I write down the conditions describing that a certain point in A maps to a certain point in B? How can I use this information to deduce the elements of the transformation matrix? A different way would be to think along the lines: How are the x and y values of the points and the transformed points related? How can I describe this with a matrix equation?

If someone came up to you and told you they had invented a device, no larger than a pen, which could be used to create black holes. Would you believe them? Would you undertake serious experimentation to find out whether or not they were correct?

We know that what you are suggesting is impossible in the very well tested theory of how light behaves. This theory has been tested and found to properly describe observations to an extremely high degree.

No, this is not how quantum mechanics work. The electron is not full of photons that it can emit, it is interacting in a well defined way with the electromagnetic field, thereby creating excitations in it. This is very much alike how a rock thrown into a pond can create ripples on the surface without having ripples inside of it. Also, the size of the ripples may very well be larger than the size of the rock.

Exactly how big do you think a photon is? A photon is not a little billiard ball and the best definition of its "size" is the size of the typical quantum mechanical wave packet - which may depend on both the emission and absorption processes. As I said in another thread, elementary particles are not waves or small billiard balls - they are elementary particles and have some properties we typically would associate with billiard balls and some we would associate with waves. But they do not care about how you model them, they are what they are and all we can do is to try to model them as good as possible.