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25

This is a deep question. There are (at least) two definitions of mass: gravitational mass is how something is influenced by gravity, which is the $m$ in $F = Gm_1m_2/r^2$, and is more-or-less 'how much stuff there is'; inertial mass is how resistant to acceleration something is, and it's the $m$ in $F = ma$. If we call these two versions of mass $m_G$ ...


7

$d\vec p$ is not actually a vector, but a differential volume element. It is a bit of sloppy notation because sometimes $d\vec p$ means the vector $(dp_x,dp_y,dp_z)^T$ and sometimes it is the volume element $dp_xdp_ydp_z$. You can also use $d^3p$ to denote a volume element but both are used often. A more proper way to derive $m^3$ is using the Jacobian. ...


5

Please can someone offer some help to correct my intuition? Drawing a free-body diagram is an important exercise that takes some practice to get right. I have found that the following systematic approach works well, I will use your book on a table scenario to demonstrate. First, list all objects involved in the scenario. Here it is the book, the table, and ...


4

OP is essentially asking if Newton's gravitational formula $F=Gm_1m_2/r^2$ could work in GR if we interpret the masses $m_1$ and $m_2$ as relativistic masses (rather than rest-masses), say for the deflection of a relativistic test-mass in a Schwarzschild geometry? Knowing that gravity couples to energy (rather than rest-mass) that's a natural question to ask....


4

Consider for a moment electric field of a charge $q$ moving with speed $V^\dagger$. If we denote the angle between direction of motion and position $\vec R$ of test charge as $\theta$, we'll have (in Gaussian units) $$\vec E=\frac{q\vec R}{R^3}\frac{1-\frac{V^2}{c^2}}{\left(1-\frac{V^2}{c^2}\sin^2\theta\right)^{^3/_2}}.$$ We can see that, if the test ...


4

Your mistake is in saying that $\overline{\psi_{L,R}} = \overline{\psi}P_{L,R}$. In fact, $$ \overline{\psi_{L,R}} = (P_{L,R}\psi)^† \gamma^0 = \overline{\psi}\gamma^0 P_{L,R}^† \gamma^0$$ $$ = \overline{\psi} P_{R,L}$$ To see why that last line is true we have to expand $P_{L,R} = (I\mp \gamma^5)/2$ and note that $(\gamma^5)^† = \gamma^5$, $(\gamma^0)^2 = ...


3

Unfortunately, you’re trying to learn from a 50-year-old book that presents the concepts in what we now consider to be a very confusing way. Now, we (generally) use “mass” m to denote the rest mass of a particle: the mass that’s characteristic of any electron, for example. Then in the modern convention: $$ (mc^2)^2+ (cp)^2 = E^2$$ French uses “mass” to ...


3

Which one's the correct definition of what mass is? In a way, both. Mass is a fundamental measure of the amount of matter in an object, or as you say, a measure of the amount of "stuff" in an object. At the same time it is a numerical measure of its inertia. Because mass is a fundamental property, definitions of mass can tend to be circular, as a ...


3

Mass is gravity's equivalent of electric charge, with two obvious differences: Charges can be positive or negative, but masses are positive. We can discuss "what about the mass-energy stored in gravitational fields?" or any number of gotchas, but you and me and planets, as familiar examples, definitely have positive mass. If the dissimilarity ended there, ...


2

So what is illustrated in this figure is the energy band structure, these represent the dispersion, i.e. the energy $E$ in function of the wave vector $k$. Now, you can find in standard textbooks on solid state physics that the effective mass is defined as $$(m^*)^{-1} =\frac{1}{\hbar^2}\frac{\partial^2 E}{\partial k^2}$$ So the curvature of these (...


2

In the past 200 years physicists have manage to model mathematically the data from innumerable experiments, and this is the present model. All the data show that the particles called electrons have a charge opposite to the charge of the protons. They also show the enormous mass difference. The models created on this observation are continually validated by ...


2

No. The mass is a scalar quantity, and is therefore the same for each direction. Not having a copy of the book myself, there are three possibilities that come to mind. The first is that Susskind is about to say that it is an experimentally observed fact that $m_x=m_y=m_z$, and so one can simply drop the subscript from the mass and treat it as a scalar. ...


2

Dale's answer gives a great methodology for making free body diagrams, which are important for understanding the system in question. I wanted to specifically address the below question in a more "conceptual physics" way: For example, if we have a book on a table, and are drawing two separate force diagrams for the forces acting on each object, then surely ...


1

There are seveal other very nice answers here, but I want to focus very closely on "why do we say weight acts on the book?" Let's conduct a thought experiment Slide the table (or other support) out from under the book leaving it unsupported. What happens to the book? It falls down right? So it goes from not moving to moving downward which means it ...


1

When we are drawing Free body diagram of a particular object we are only concerned with the the question that what forces are acting on the object and not with what force acting on the surrounding? Why? Because to get the full knowledge of the dynamics of the object in question you just need to know the force (for acceleration), initial velocity, and ...


1

The text makes it clear that the summation is over three neutrino species. In which case when you take the number density out of the summation, it is the average density per species that you use, which is one third of 339 cm$^{-3}$.


1

Is there any physical violation in writing this out or it is just because of general relativity which doesn't consider gravity as force but as consequence of curved space-time? Yes, there indeed is one right away: You started with a result from Special Relativity. But from that framework you already know that there's a finite speed of causality. Newtonian ...


1

According to Special Theory of Relativity, we know that any equation that contains mass gonna be corrected as ... This is incorrect. This correction allows some Newtonian formulae to work, but not all of them, and Newton's gravitational equation is one of the equations that don't work.


1

You are not specifying it, but you define matter as the constituents of atoms basically, electrons and quarks, that do have rest mass, and are elementary particles. In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume.1 All everyday objects that can be touched are ultimately composed of ...


1

In an asymptotically flat spacetime, we have two well-known measures of mass, the Bondi mass and the ADM mass. They differ because the Bondi mass doesn't include energy being radiated away to null infinity by gravitational waves, whereas the ADM mass does. The ADM mass is conserved. In your example, a distant observer will feel a mass of $M+m$ at all times, ...


1

The Higgs potential receives quantum corrections from the Yukawa couplings with fermion doublets. The magnitude of the correction is determined by the Yukawa coupling constant, which is itself proportional to the fermion mass. As a result, the heaviest fermions make the greatest contribution to the effective action. Because the top quark is by far the ...


1

The gravity here is emergent not only from the body they are falling towards but also each other. They will fall on different paths due to the relative masses. The force of gravitational attraction is the same so the speed of movement in the vacuum is the same but they will not fall towards the massive body at the same rate. Individual objects on tiny scale ...


1

French is showing that 1) mass is an intrinsic property of just the object, and 2) mass’s effect on kinematic is understood & reliable. would redefining our scale resolve the inconsistency? If (1) was true, but (2) was not, then some other way of scaling the mass work work. For example, gravity and a spring scale. But if (1) was not true, mass was ...


1

Neutrino oscillation experiments have so far provided the only evidence that neutrinos have mass. Since they measure the accumulated phase differences, they are only sensitive to the differences in masses, and thus never the absolute mass scale. A different measurement is necessary to probe the absolute mass scale. There are several experimental programs ...


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