Boost Your Knowledge on Frame of Reference

The electric dipole moment for a charge distribution is:
$\stackrel{\to <!--\; \to \; -->}{P}={\int <!--\; \int \; -->}_V\mathrm{\rho <!--\; \rho \; -->}(\stackrel{\to <!--\; \to \; -->}{x})\stackrel{\to <!--\; \to \; -->}{x}{d}^{3}x$
If we imagine that $\mathrm{\rho <!--\; \rho \; -->}=const$ on a spherical volume then it's easy to see that $\stackrel{\to <!--\; \to \; -->}{P}$ depends by the origin of the frame of reference, it can be zero or something totally different. For this reason it's very difficult for me to give a physical meaning at this quantity (because it's strange that depends by the origin).

An object at the end of a rope is attached to the ceiling of an elevator. If the elevator is allowed to free-fall and hit the ground, at impact the tension of the rope is greater than if the elevator was standing still.
Why in the equation below, the acceleration is positive for $m\stackrel{\to <!--\; \to \; -->}{a}$
$T-<!--\; -\; -->mg=ma$
Is this because to the object at the end of the rope, the elevator appears to be accelerating upward? Why is the acceleration not negative, since the elevator was free-falling?

Do we assume that the rest mass of a fundamental particle is constant in all inertial reference frames? i.e. is the rest mass of an electron if it is travelling at constant velocity c/2 (relative to the distant stars) the same as the rest mass of the electron if it is travelling at velocity 0 relative to the distant stars?

If we take a source with a frequency of $230Hz$ moving at $30m{s}^{-<!--\; -\; -->1}$ right and the listener moving $25m{s}^{-<!--\; -\; -->1}$ right.Let the source be behind the listener at some distance so that the distance between them decreases.Take the speed of sound as $340m{s}^{-<!--\; -\; -->1}$.
With the speaker as the frame of reference, the listener is moving $5m{s}^{-<!--\; -\; -->1}$ towards the source.
$f_L=\left(\frac{c\pm <!--\; \pm \; -->v_L}{c\pm <!--\; \pm \; -->v_S}\right)f_S$
$=\left(\frac{340+0}{340-<!--\; -\; -->5}\right)230$
$=233.4328Hz$
With the Listener as the frame of reference, the source is moving $5m{s}^{-<!--\; -\; -->1}$ towards the listener.
$f_L=\left(\frac{340+5}{340+0}\right)230$
$=233.3824Hz$
With the system as the frame of reference, the source is moving at $30m{s}^{-<!--\; -\; -->1}$ right and the listener moving $25m{s}^{-<!--\; -\; -->1}$ right.
$f_L=\left(\frac{340-<!--\; -\; -->25}{340-<!--\; -\; -->30}\right)230$
$=233.7097Hz$
Why are the answers all different?

Consider the situation: a body slides down on a wedge with only gravitational force as an external force. The velocity of the body in the wedge's frame using work energy theorem. Now, use this velocity of the body to find the velocity of the wedge in ground frame using conservation of momentum. The point is that can I do this?

Unruh radiation normally applies to linearly accelerated frames
Is there an equivalent of the Unruh thermal radiation in a frame that is spinning?

A train is travelling towards a tunnel at relativistic velocity. The tunnel has two gates on either side that open/close to admit/reject the train. The train is the same length as the tunnel when it at rest in the tunnel's frame of reference. In the tunnel frame, the train is moving so it is contracted and can fully fit inside the tunnel. Thus, the gates are initially closed, open right before the train enters, close while the train is completely inside the tunnel, open right before the train leaves and closes right after the train has completely left.
How would you describe this sequence of events in the frame of reference of the train?

Does the mass of a moving object in empty space with a constant velocity change within its own frame of reference from its rest mass?

Nothing with mass can travel at the speed of light. Therefor one can not ask what happens when an observer in a car moves with the speed of light. But, is it possible for a frame of reference to move with the speed of light? Is there something preventing that?

In classical mechanics we have the notion of inertial frame of reference. If my understanding is correct, such frames are exactly those for which Newton's laws of motion have the usual form. I believe I have also heard, that they are the frames that are "moving at a constant speed" with respect to some distinguished point in space.
So we have to be careful whether our frame is accelerated or not. Which makes me wonder:
Does there exist a set of physical laws (of same descriptive power Newton's laws have, but perhaps expressing relations in some other set of physical quantities) such that the laws from this set look the same, no matter how the frame of reference moves?
Does such a set of physical laws exist if we restrict to polynomially moving frames of reference?

Relativistic mass = $m_0\mathrm{\gamma <!--\; \gamma \; -->}$ where $\mathrm{\gamma <!--\; \gamma \; -->}$ is the lorentz factor.
So, if a mass that is $.5$ at rest then it is safe to say that the relativistic mass will be $1$ if it goes at $\frac{\sqrt{3}}{2}c$.
What happens if that $.5$ is actually a radioactive isotope and is decaying while speeding up? Then at what speed will it approximately equal $1$?

According to the equivalence principle, gravity and inertial forces are similar. And according to general Relativity, If there’s a large object in space-time, it warps the space time’s geometry and causes the light to bend ( or to be seen as if it was coming from a different source). Now, if there’s an observer in an accelerating frame of reference and there’s no mass, will the observer still see the light to be coming from a different source considering the frame’s acceleration to have the same effect as that of a mass according to the principle of equivalence?

Massless particle (such as a photon) cannot have a frame of reference. But the electromagnetic field does have mass; does it have a frame of reference? If so, if one takes a charge as "setting up" its own EMF, then the field's frame of reference is the charges, so the distinction doesn't matter. But if we take the EMF to pervade all of space, and charges merely to alter the value of the field at any given location, then it makes sense to ask about the frame of reference of the field per se. However the EMF varies from point to point; how could a volume of the EMF, which is not homogenous, have a single frame of reference?

How does translational kinetic energy of a system depend on frame of reference where as rotational kinetic energy does not depend on reference frame?

In Schwarzschild coordinates ${\displaystyle (t,r,\mathrm{\theta <!--\; \theta \; -->},\mathrm{\varphi <!--\; \varphi \; -->})}$ the Schwarzschild metric (or equivalently, the line element for proper time) has the form
${\displaystyle g=-<!--\; -\; -->c^2{d\mathrm{\tau <!--\; \tau \; -->}}^2=-<!--\; -\; -->(1-<!--\; -\; -->\frac{r_{\mathrm{s}}}{r})c^{2}d{t}^2+{(1-<!--\; -\; -->\frac{r_{\mathrm{s}}}{r})}^{-<!--\; -\; -->1}d{r}^2+r^2{g}_{\mathrm{\Omega <!--\; \Omega \; -->}},}$
where $g_{\Omega }$ is the metric on the two sphere, i.e. ${\displaystyle g_{\mathrm{\Omega <!--\; \Omega \; -->}}=(d{\mathrm{\theta <!--\; \theta \; -->}}^2+{\sin }^2<!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}d{\mathrm{\phi <!--\; \phi \; -->}}^2)}.$
Isn't $(t,r,\mathrm{\theta <!--\; \theta \; -->},\mathrm{\varphi <!--\; \varphi \; -->})$ representing the spherical coordinates of an observer, looking at the black-hole from infinity?

"A high-speed train is traveling at a constant $150$ m/s (about $300$ mph) on a straight horizontal track across the south pole. Find the angle between a plumb line suspended from the ceiling inside the train and another inside a hut on the ground. In what direction is the plumb line on the train deflected?"
we're looking for is a relationship $\tan <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})=\frac{a_n}{a_t}$, because for the plumb line in the hut $a_t=1$ but not for the plumb line in the train due to the Coriolis force.
However,
1. What is the frame of reference here? Is it rotating, fixed so that the train is standing still? Is it rotating with the earth?
2. Since the train is on the south pole, isn't that the same to say that the earth's rotation doesn't affect it?
3. Can you show me how to get the right answer? The right answer is supposed to be $0.13$ degrees. In which direction?

If an object in free fall near earth, then an inertial frame of reference and I see the earth accelerating towards me with no force acting upon it. What causes that acceleration?

Cosmic microwave background temperature is about $2.7K$. But what temperature we will measure in space using a simple Kelvin thermometer in the shadow? Can it be lower than $2.7K$?

Why do the principles of physics get more complex when the frame of reference undergoes acceleration?

If I float motionless in outer space, I will measure spacetime to be approximately flat. By the equivalence principle, I will get flat spacetime if I am free falling near earth. Is this right? otherwise I can distinguish between the two scenarios.
Let say I declare myself (while free falling) to be at rest in an inertial frame of reference under GR. I see the earth accelerating towards me. What causes that acceleration? It's not the curvature of spacetime, right?