Why is there a conserved quantity for every continuous global symmetry?

One of the most important theorems in physics is the theorem that states:

"To every differentiable global symmetry of a physical system there corresponds a conservation law"

This theorem is known as the first Noether's theorem, in honor of the great mathematician Emmy Noether, who proved it in 1915 in the context of classical mechanics (both relativistic and non-relativistic, but not quantum). By the way, Noether is part of the group of leading scientists and professors in her field who lost their jobs due to the intolerance of the Nazis when they came to power, as they immediately passed a law preventing Jews and Communists from working in universities and public institutions. This happened before the Holocaust and the Second World War. It is important to remember this so that it does not happen again.

This relationship between symmetries and conservation laws established by Noether is one of the most powerful ideas that human beings have ever had. Conservation laws are a very useful tool for finding out how the quantities of a physical system change over time. Knowing that there are physical quantities that do not change allows us to write equations where the unknowns are the quantities that do change. We can then use the quantities that do not change to find out how the other quantities change.

On the other hand, the symmetries of a physical system are related to its aesthetic aspect. For example, a sphere is beautiful because, no matter how you rotate it, it remains the same. Noether's theorem thus relates beauty to usefulness in physics in a certain way. Pragmatism and aesthetics go hand in hand.

However, for the physics student, it is not immediately evident that a continuous symmetry implies a conserved quantity. Apparently, they are two things that have nothing to do with each other. What is the reason for this relationship?

However, today we know that the world is not classical, but quantum, and that classical mechanics is nothing more than an approximation of the behavior of physical systems in a certain limit. Therefore, Noether's original proof does not serve us for the fundamental laws of nature. Does Noether's theorem still hold in quantum mechanics?

These are the two questions we are going to answer in this post.

Why don't protons in a nucleus repel each other?

The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom.

A model of the atomic nucleus showing it as a compact bundle of the two types of nucleons:

protons (red) and neutrons (blue).

By Marekich - Own work (vector version of PNG image), CC BY-SA 3.0, Link

The atomic number Z of a chemical element is the number of protons found in the nucleus of an atom. Since neutrons are neutral while protons are positively charged with a charge that is, in absolute value, equal to the electron charge, the atomic number is identical to the charge number of the nucleus. In an uncharged atom, the atomic number is also equal to the number of electrons, and that is why the atomic number uniquely identifies a chemical element. For instance, Z=1 is called hydrogen and Z=6 carbon.

The world nuclide is referred to a 'species of nucleus', characterized by its number of protons Z, its number of neutrons N, and its nuclear energy state. Identical nuclei belong to one nuclide, for example each nucleus of the carbon-13 nuclide is composed of 6 protons and 7 neutrons. On the other hand, the members of the group of all the nuclides of the same elements are called isotopes. That is, the nuclides with equal atomic number, i.e., of the same chemical element but different neutron numbers, are called isotopes of that element.

Stable nuclides are nuclides that are not radioactive and so (unlike radionuclides) do not spontaneously undergo radioactive decay. It has been found that there are 80 elements with one or more stable isotope. For instance, carbon has 15 known isotopes, from carbon-8 to carbon-22, of which carbon-12 and carbon-13 are stable.

But, because of the fact all the protons have the same charge and are very closed to one another in the nucleus, one expects that they repel each other by a strong electric force, a force that must be much stronger than the force acting between the nucleus and the surrounding electrons. This force should make the nucleus explode. Therefore, no nuclide with more than one proton should be stable, that is, the only stable element should be hydrogen!

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers. 

Why do astronauts float weightless in the International Space Station?

By NASA/Crew of STS-132 - http://spaceflight.nasa.gov/gallery/images/shuttle/sts-132/hires/s132e012208.jpg (http://spaceflight.nasa.gov/gallery/images/shuttle/sts-132/html/s132e012208.html), Public Domain, Link

We have seen on TV that astronauts float weightless in the International Space Station (ISS), and during spacewalks. In fact, the ISS serves as a microgravity research laboratory in which crew members conduct experiments in biology, physics and other fields. Microgravity is more or less a synonym of weightlessness and zero-g (zero gravitational field strength).

Nevertheless, the ISS maintains an orbit with an altitude h of between 330 and 435 km so, according to Newton's law of universal gravitation, the gravitational field strength g in the ISS is:

where G is Newton's constant, M mass of Earth and R is the radius of Earth.

Taking into account that the gravitational field strength at the surface of the Earth is g=9.8 m/s^2, we conclude that things and astronauts inside the ISS weigh only a 10% less than they do on Earth! The weight is almost the same! How is this possible?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Something happens with the light

If we split a collimated light beam by using a half-silvered mirror, then the two resulting beams (A and B) have exactly the same intensity. Since the light is made of photons, that means that half of the photons go through path A, and the other half through path B.

If we now reflect both beams by a mirror and the two beams then pass a second half-silvered mirror and enter two detectors as explained in the picture:

then we expect the A beam to be split into two beams. We will call them A1 and A2. A1 goes to dectector 1, while A2 goes to detector 2. Each one contains 50% of A-photons, that is, 25% of the photons of the original light beam:

On the other hand, we also expect the B beam to be split into two beams. We will call them B1 and B2. B1 goes to dectector 1, while B2 goes to detector 2. Each one contains 50% of B-photons, that is, 25% of the photons of the original light beam:

So the amount of photons that should arrive to detector 1 is 25% + 25% = 50%, and the same for detector 2:

Nevertheless, once we have carried out the experiment, what we found is that 100% of photons arrive to detector 2  and no photon arrives to detector 2!

Moreover, what is even more puzzling, if we obstruct channel A (or B, it does not matter), then we detect the same number of photons in detector 2 as the number detected in detector 1. Are you able to figure it out? Try it!

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Why does the death of a living being affect the decay of carbon-14?

Carbon-14 is a radioactive isotope of carbon with an atomic nucleus containing 6 protons and 8 neutrons. Carbon-14 decays into nitrogen-14 through beta decay:

By emitting a beta particle (an electron, e^-) and an electron antineutrino (ν_e), one of the neutrons in the carbon-14 nucleus changes to a proton and the carbon-14 nucleus becomes the stable (non-radioactive) isotope nitrogen-14.

The equation governing the decay of a radioactive isotope is

$$ N=N_0 e^{-\frac{t}{\tau}}$$

where N_o is the number of atoms of the isotope in the original sample (at time t = 0, when the organism from which the sample was taken died), and N is the number of atoms left after time t. On the other hand, the mean-life τ is the average or expected time a given atom will survive before undergoing radioactive decay.

Since the amount of carbon-14 inside a piece of wood or a fragment of bone decrease as the carbon-14 undergoes radioactive decay, measuring the amount of carbon-14 in a sample provides information that can be used to calculate when the animal or plant died. The mean-life of carbon-14 is 8267 years, so the equation above can be rewritten as:

Nevertheless, radioactive decay is a process that takes place inside the nucleus, so nor a change of temperature neither chemical reactions affect radioactive decay. Carbon-14 atoms inside a living being are decaying after and before the living being dies. So why is this method used efficiently to measure when the living being died? How do we know N_o, the amount of carbon-14 the living being had at the moment it died, if carbon-14 was also decaying when the plant or the animal was alive?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Does the expansion of space apply in the solar system?

The prevailing cosmological model for the universe accounts for the fact that the universe expanded from a very high density and high temperature state, and that nowadays the expansion is even accelerating. This is an expansion of space, that is, the increase of the distance between two distant parts of the universe with time. It is an intrinsic expansion whereby the scale of space itself changes. This is different from other examples of expansions and explosions in that, as far as observations can ascertain, it is a property of the entirety of the universe rather than a phenomenon that can be contained and observed from the outside.

By NASA/WMAP Science Team - Original version: NASA; modified by Ryan Kaldari, Public Domain, https://commons.wikimedia.org/w/index.php?curid=11885244

Since it is an intrinsic expansion, it is natural to think that the planets in our solar system are expanding with time, as universe is. Moreover, our measurement devices should be expanding too. But, taking into account that a measurement is the assignment of a number to a characteristic of an object by comparing with other objects, why were we able to measure the expansion of the universe if our devices are expanding too?

Does the expansion of space apply to the objects inside our solar system?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

The Moon is getting further away from Earth. Where does this extra energy come from?

All bounded orbits where the gravity of a central body dominates are elliptical in nature. In the case of the Moon orbiting the Earth, the eccentricity of the ellipse is so small (0.055) that it is almost a circle:

Therefore, the gravitational force F_g that the Earth exerts on the Moon is perpendicular to Moon's velocity v, so it is a centripetal force F_c, making the trajectory of the Moon bend:
$$ F_{g}=F_{c} \\
\frac{GMm}{r^2}=\frac{mv^2}{r} $$ where G is Newton's constant, M is Earth's mass, m is Moon's mass and r is the radius of the orbit.

This implies that the kinetic energy of the Moon is
$$
K=\frac{1}{2}mv^2=\frac{GMm}{2r}
$$ which is smaller than the absolute value of the potential energy
$$
U=-\frac{GMm}{r}
$$ So the mechanical energy of the Moon is
$$
E=-\frac{GMm}{2r}
$$

We know that at the time of its formation, the Moon sat much closer to the Earth, a mere 22,500 km away, compared with the 402,336 km between the Earth and the Moon today. So the Moon is getting further away from Earth, now at the rate of 3.78 cm per year. Nevertheless, according to the last equation, a larger r means that the Moon has more energy every year. Is its energy non conserved? Who is giving energy to the Moon?

Can an object exceed the speed of light if we push it for enough time?

The second Newton's law of motion establishes that, in an inertial reference frame, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration vector a of the object. That means that if we apply a constant force on a body without friction, then the object will move with constant acceleration, increasing its speed by the same amount every second.

The acceleration is proportional to the force, so if the net force is 100 Newtons and the mass is 2 kilograms, the acceleration will be 50 meters per second every second. But if the force is 2 N, the body will increase its speed by 1 m/s every second. Notice that this is not a huge acceleration. Nevertheless, if we keep pushing and wait for 300000000 seconds (9.5 years) the object will move faster that light.

But we know that nothing can exceed the speed of light. This is a well-established law of nature whose confirmation has become routine in current particle accelerators.

Try to find the solution to this contradiction!

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Is expansion of gases a violation of inertia law?

The first Newton's law (also called inertia law) says that, when viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a net force.

Let us suppose that we have a box with a wall splitting it in two halves. The left half contains a rest gas (no wind inside), while the right half is empty. Suddenly, we remove the wall. Because of the fact that gases expand to fill their containers, we know that the gas of the left half will move to the right side in order to fill the whole box:

But there is no net force acting on the gas, so why has it moved to the right half of the box? Do gases violate the inertia law?

Are you able to give an explanation? Accept the challenge!

If nuclear fusion is the reverse of fission, why is energy released in both processes?

Nuclear power is the use of nuclear reactions that release nuclear energy to generate heat. There are basically two ways to release energy from nuclei:

• nuclear fission, which is either a nuclear reaction or a radioactive decay process in which the nucleus of an atom splits into smaller parts (lighter nuclei).  This released energy is the one that is frequently used in steam turbines to produce electricity in nuclear power plants.

Public Domain, https://commons.wikimedia.org/w/index.php?curid=486924

• nuclear fusion, which is a reaction in which two atomic nuclei fuse to form a heavier nucleus. Nuclear fusion reactors are not yet economically viable, but this technology is currently under research and it could become viable in a few decades.

By Wykis - Own work, based on w:File:D-t-fusion.png, Public Domain, https://commons.wikimedia.org/w/index.php?curid=2069575

If splitting a nucleus into two smaller nuclei releases energy, it seems that combining two smaller nuclei into one larger nucleus would require energy, not release it, because it is the inverse process. So, why can we obtain energy from both processes?

 Are you able to resolve this cognitive conflict?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.