Length Contraction Inherent in Maxwell Equations?

There are two senses in which length contraction is “inherent” in the Maxwell/Heaviside equations.

One sense is that in some cases length contraction follows from Maxwell’s equations. FitzGerald and Lorentz showed that if the structure of matter is determined by electromagnetic forces then the equilibrium spacing of stationary particles due to purely electrostatic forces would become contracted when they are in motion due to the presence of additional magnetic forces. [And one effect of this contraction, together with a related slowing and desynchronization of clocks, would be that moving observers (using their own contracted measuring rods and slowed clocks) would not notice any effect of their own motion on the apparent speed of light (or in fact on any electromagnetic phenomena).]

Another sense is that length contraction is necessary to preserve the form of Maxwell’s equations. Poincare had already noted that if the coordinates used by a moving observer were related to those of one who is stationary by a standard Galilean transformation (ie by just a progressive shift of position with no change of length and time scales) and if Maxwell’s equations applied to (say) the stationary one then they would have to be modified in order to make correct predictions for the other. And he showed that the only kinds of coordinate transformation that leave the form of Maxwell’s equations unchanged are those involving length contraction and time dilation.

[Einstein then noted that if all the laws of physics, written in terms of the natural coordinates of an observer, have a form that is independent of the state of motion of the observer, then there is in fact no way to tell which of two relatively moving observers is “truly” stationary and whose clocks are “truly” synchronized. He therefore sought to express all the laws of mechanics (eventually also including gravity) in a form which used only the actually measured coordinates of each observer rather than those of some assumed fixed “rest”(or “aether”) frame (which would have been more complicated to do if possible, but was not actually possible to do properly because no observers could actually tell whether or not they were actually in motion).]

Source: (1001) Alan Cooper’s answer to Is length contraction inherent in the Maxwell/Heaviside equations? They do have some asymmetry. I remember reading something to this effect but can’t find it and could be wrong. – Quora

Why is photon neutral?

If a gauge invariant quantum field theory (QFT) has a Lie group gauge invariance, the matter fields are defined in the fundamental representation of the group, and the gauge fields are defined in the adjoint representation of the group. If both representations are non trivial, the quantum fields in both representations carry the gauge charges. In the case of the U(1) group the adjoint representation is trivial, therefore the gauge field (photon) is also defined in the fundamental representation. Here only the matter field (electron) carries gauge charge (coulomb charge) and the gauge field does not carry gauge charge. In other words the photon is electrically neutral.

Source: (1001) Sanjay Sood’s answer to Why in quantum electrodynamics the photon is electrically neutral? – Quora

Why is photon electrically neutral?

 If a gauge invariant quantum field theory (QFT) has a Lie group gauge invariance, the matter fields are defined in the fundamental representation of the group, and the gauge fields are defined in the adjoint representation of the group. If both representations are non trivial, the quantum fields in both representations carry the gauge charges. In the case of the U(1) group the adjoint representation is trivial, therefore the gauge field (photon) is also defined in the fundamental representation. Here only the matter field (electron) carries gauge charge (coulomb charge) and the gauge field does not carry gauge charge. In other words the photon is electrically neutral.

Source: (1001) Sanjay Sood’s answer to Why in quantum electrodynamics the photon is electrically neutral? – Quora

Superposition Independent of Space and Time

The spin observables for an elementary particle can be studied without any reference to the particle’s position. They are then modeled as operators on a finite dimensional Hilbert space of spin states that is essentially independent of the infinite dimensional Hilbert space of “wave” functions on position space (which identify those aspects of the particle’s state that are related to its position observables). And in this context for example the eigenstates for any spin component of a spin 1/2 particle are superpositions of eigenstates for other components.

Now you might (correctly) think that this talk of spin components means that we must still be thinking about directions in physical position space. But in fact the study of any two-valued observable (with values “Yes” and “No” or “True” and “False”) forces us to also consider other observables whose eigenstates are superpositions of the Yes and No eigenstates and whose relationship to the original observable are mathematically equivalent to those between spin components in different directions despite not actually having any connection with directions in physical space. And in quantum computing, although spin directions for a single particle are often used as a conceptual model, the choice of Yes/No observable might in practice be something different.

Of course, to study the progress of a quantum calculation, while disregarding space we still need to consider the time evolution of the system; but some aspects of the relationships between possible outcomes can be studied without reference to time in a similar way to how the time-independent Schrodinger equation can be used to study stationary states and energy levels of an atom or molecule. And in these types of analysis the question of whether and how some states are superpositions of other states is still relevant even though there is no reference to either space or time involved.

Source: (1001) Alan Cooper’s answer to Would quantum superpositions function in the absence of time and space? – Quora

Black Hole Fantasy

A black hole has been spotted heading towards Earth, and we have 200 years before it arrives. Does our species have a chance of survival?

Perhaps. But not on Earth. (Assuming any kind of astronomically plausible black hole, there is no way of avoiding substantial perturbation of the Earth’s orbit followed by tidal distortion and probable destruction of the Sun.)

However, it may be possible to colonize some asteroids and by small manipulation of their orbits ensure that they get slingshotted so as to end up at sufficient distance from the accretion disc (into which the sun and planets will be converted by tidal forces) so that the body of the asteroid is sufficient to shield its residents from the radiation.

Since the colonies will of necessity be small, most humans will be stuck on Earth and not survive; but if sufficient genetic diversity is brought along in the form of germ cells and/or frozen zygotes or blastocysts then perhaps the species itself might survive and adapt to the new low gravity environment.

Source: (1001) Alan Cooper’s answer to A black hole has been spotted heading towards Earth, and we have 200 years before it arrives. Does our species have a chance of survival? – Quora

Less Time for Same Distance? 

For the traveller it’s not the same distance; due to length contraction it’s a smaller distance, and so takes less time. For the observer who sees the distance as a full light year it appears to take more than a year, but the traveller’s clocks appear slowed down and so will advance by less than that. (The time experienced by the traveller will still be more than a year unless the travel speed  exceeds , ie just over 70% of the speed of light.)

Source: (1001) Alan Cooper’s answer to It takes light a year to travel a light year, but why would it take a person less time to travel the same distance due to time dilation? – Quora

Timelike vs Spacelike

If two distinct events are such that there is any inertial frame in which they have zero spatial distance between them, then there is no frame in which they are simultaneous and so they are said to be “timelike separated”. This is because the frame in which they have zero spatial separation corresponds to an observer who sees them both happening at the same place one after the other; and for any other inertial observer, the time between them is also nonzero (since for any v<c the Lorentz contraction factor is never zero).

On the other hand, any two events which are seen as simultaneous by some inertial observer (which is different from being seen simultaneously by that observer!) are said to be “spacelike separated”. But the appearance of simultaneity is relative to the observer and only happens in one particular frame. Other inertial observers won’t see the events as simultaneous but all will agree that it would take faster than light travel to see them both at the same place – eg to actually be present at both of them.

Source: (1001) Alan Cooper’s answer to Is there relativity in simultaneity for events that have distance between them from the prespective of one frame but don’t have distance between them from the prespective of another? – Quora

GR and Twin Paradox

General relativity theory does not “solve the twin paradox of special relativity”.

Despite being “paradoxical” in the sense of contradicting our intuition that the time ordering of separated events should be absolute, there is no “paradox” in the sense of internal contradiction in special relativity. Nor is it impossible to analyse the experience of an accelerated observer in special relativity; and in the case where one twin is turned back (eg by a rocket) this leads to the conclusion that both agree on the difference between their ages when reunited.

The only case in which general relativity is needed is when the acceleration is due to gravity (eg by slingshotting about a massive star) – and so does not lead to the feeling of applied force by the freely falling traveller. But as soon as gravity comes into the picture we are no longer talking about special relativity.

Source: (1001) Alan Cooper’s answer to How does the general relativity theory solve the twin paradox of special relativity? – Quora

Lagrangian Described Simply?

For particles moving independently and freely subject only to specified forces, the motion of each particle obeys Newton’s law F=ma which can be written in terms of the Cartesian coordinates in the form x”=(1/m)dV/dx where V is the potential energy (and the form is the same for all components).
For particles constrained to be parts of a rigid body we might prefer to use more natural variables like the angular orientation of the body, but then the equations become more complicated (eg with centripetal and coriolis “forces” coming in, so the equation for the r component is not just
r”=(1/m)dV/dr ). And it is often not easy to work out the translation of Newton’s eq into the new coordinates.
The Lagrangian approach rewrites the equations of motion in an equivalent form which has the same structure in all coordinate systems – which turns out to be given for every component q by d/dt(dL/dq’)-dL/dq=0 with L=T-V where V is again the potential energy and T the kinetic energy (which in Cartesian coords would be (m(x’)^2)/2 ). This provides a systematic way of getting the appropriate equations rather than having to work them out in terms of the desired coordinates by a complicated conversion process from the Newtonian form.
It turns out that the Lagrangian equations are also equivalent to a variational principle (as mentioned by other responders) – namely that the actual trajectory be a stationary point (eg min or max) of the path integral of the Lagrangian function L.

Source: (1001) Alan Cooper’s answer to In layman’s terms, what is a Lagrangian? – Quora

Feynman Diagrams – Quora

Feynman diagrams are labels for a way of breaking a complex integral into parts, each of which can be evaluated by applying fairly simple rules. As such they make it much easier to evaluate expressions which originally looked very difficult, and so make it possible for theoretical predictions to be made (and checked) much more quickly (and by people with lower technical skill levels). This led to a rush of progress in the theory and application of Quantum Electrodynamics, and paved the way for much more rapid progress in applying and testing other Quantum Field Theories (such as those used to model the strong and weak interactions of elementary particles).

Source: (1001) Alan Cooper’s answer to Why did Feynman diagrams revolutionize particle physics? – Quora