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6

What happens if the twin in the spaceship doesn't return? Would he still be younger than his other twin? It's really a moot point, because you can't compare clocks. There is no absolute time! You can't say, "What's each twin's age at this instant?" because "this instant" depends on the observer. Is the symmetry broken simply by accelerating out of earth? ...


3

Stack all the $\phi^i$s into a column "vector" $\vec\phi$. The mass term $m^2\vec\phi\cdot\vec\phi$ is obviously invariant by $R^{-1}=R^T$. The same with the kinetic term $(\partial_\mu\vec\phi)\cdot(\partial^\mu\vec\phi)$ because $\partial_\mu R=0$. It is $SO(n)$ invariant because I take it $i$ runs over $n$ values. Thus your $r^i_j$ generates $SO(n)$. ...


3

Newton's third law states that if object A acts on object B with force $\mathbf{F}_{AB}$, then object B must act on object A with force: $$\mathbf{F}_{BA}=-\mathbf{F}_{AB}$$ When expressed in terms of A and B's momentum, the same equation can be written as: $$\frac{\mathrm{d} \mathbf{p}_A}{\mathrm{d}t} = -\frac{\mathrm{d} \mathbf{p}_B}{\mathrm{d} t}$$ ...


3

If there is no external force with explicit time dependence, then the harmonic oscillator contains no explicit time dependence. Then the system has time translation symmetry, i.e. the result can only depend on the difference $T =t_b-t_a$, not on $t_a$ and $t_b$ individually.


2

The laws of physics are discovered through a mixture of heuristics, modelling and inference. In case of momentum, the story goes like this: It is possible to 'transfer motion' from one body to another. However, experiment shows that it is not velocity that is conserved during such transfers, but another 'quantity of motion'. We give that quantity the name ...


1

You are correct in your interpretation that Weisner's method is geometric in nature: it is a method for finding generating functions for special functions using representation theory of Lie groups and Lie algebras. And as you know, Lie groups play an enormous role in modern geometry, on several different levels. Lie groups are smooth differentiable ...


1

The Lagrangian is what is integrated over spacetime in the action, i.e. has to be a 4-form. As such, it is necessarily a (pseudo-)scalar under Lorentz transformations. When wondering about Lorentz transformations and such, the Hamiltonian is, as a non-Lorentz-covariant object, not a good starting point, by the way. It is often better to start with the ...


1

The volume is defined as $V=|\vec{x}\cdot (\vec{y}\times \vec{z})|$, to avoid negative volumes. So under space inversion, volume would remain invariant, which is physically desirable since it should still take up the same amount of space. New: I guess you could define V to be a pseudoscalar, but that might have consequences. For example, mass is something ...


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It has always struck me as odd that Newton's 3rd law was never explained in any detail in my physics degree lectures. I have quoted it glibly for decades, but it was only when a 10 year old asked me WHY? that I thought about how to understand this law. In the macroscopic world, we apply a force and there is an "equal" reaction force. What's actually going ...



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