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33

For simplicity, consider the case $u=v$. The "slow" formula is then $2u$ and the "fast" formula is $\frac{2u}{1+(u/c)^2}$. In the plot you can see these results in units of $c$. The "slow" formula (red/dashed) is always wrong for $u\ne0$, but it is good enough [close enough to the "fast" formula (blue/solid)] for small $u/c$. The cutoff you choose depends on ...


13

I'm usually a little more specific with my students. Let's consider a car traveling down the highway at $\rm 30\,m/s \approx 60\,mph$. Then the denominator in the relativistic formula is something like $$ 1 + \left(\frac vc\right)^2 = 1 + \left( \frac{30\rm\,m/s}{3\times10^8\rm\,m/s} \right)^2 = 1 + 10^{-14} $$ In your car, then, the difference between ...


10

This is an area rife with potential misunderstanding, so we need to be absolutely clear what we mean. Suppose I take a ruler and a clock and I use rulers to mark out $x, y, z$ axes in space and the clock to note the positions of events in time. Assuming spacetime is flat, I now have a universal coordinate frame that everyone who is stationary relative to me ...


8

Actually, regardless of the velocity of the objects in concern, the 'velocity addition' formula is always $\frac{u+v}{1+uv/c^2}$. There is no transition point where the formula changes from $u+v$ to the special relativity one. Its just that the difference you get in both the formulas at 'low velocities' is very very negligible. $c^2$, in $m^2/s^2$, is ...


4

What you're missing is that the speed of light is not constant. There's this modern-day myth that says "Einstein told us that the speed of light is constant". But search the Einstein digital papers on "speed of light" or "velocity of light" for examples like this: The speed of light is spatially variable. And that isn't some discarded idea from 1911, see ...


4

It helps to write the full action: $$S = \int \frac{-mc^2}{\gamma}dt - \int U dt $$ The first term can be put in a much better form by noting that $d\tau = \frac{dt}{\gamma}$ represents the proper time for the particle. The action is then: $$S = -mc^2\int d\tau - \int U dt$$ The first term is Lorentz invariant, being only the distance between two points ...


3

This effect is called relativity of simultaneity. It means that two observers need not agree on the simultaneity of two events, or on their temporal order. This effect depends critically on whether the events are spacelike separated (i.e. $\Delta s^2=-c^2\Delta t^2+\Delta x^2>0$) or timelike separated (i.e. $\Delta s^2=-c^2\Delta t^2+\Delta x^2<0$). ...


3

In a rotating reference frame, the coordinate velocity of an object can exceed $c$. However, this doesn't mean that they're moving "faster than light". If we were to look at the light-cones at these distant locations, we would see that the four-velocities of these objects are still confined within the light-cones at those locations. To put this another ...


2

$p^2 = m^2$ is the definition (up to a minus sign) of the mass of a momentum eigenstate. He derived that the same quantity (the expectation value of $[P^2,D]$ w.r.t. $\ket{p}$) equals $0$ and $2\mathrm{i}m^2$, so $m^2 = 0$. The $s$ is the scale parameter of the scale transformation induced by $D$, and it is any real number, so, starting from a given state ...


1

Just write out the product $$u_\alpha u^\alpha = g_{\alpha\beta}u^\alpha u^\beta= g_{00}u^0 u^0\ ,$$ since $u^i=0\ ,\ i=\{1,2,3\}$. Then imposing $$u_\alpha u^\alpha = -1 \quad \Rightarrow \quad g_{00}=-1 \ .$$ I think the rest you can do on your own :)


1

In general relativity, the Minkowski metric plays a privileged role because it is the unique asymptotically flat solution to the vacuum Einstein equations that has zero ADM energy. The positive energy theorem in general relativity says all asymptotically flat spacetimes satisfying the dominant energy condition have non-negative ADM energy. Thus, one can ...


1

The cameras are at the same place, because we ignore the small distance between the cameras, right? When the cameras are at the same place, then the same information arrives at the same time at both cameras. The person in the train picture will look the same age as the person in the ground picture.


1

The "fast" formula is always the correct one. A "fast" speed is one that is comparable to the speed of light. However, when both the speeds involved are much smaller than the speed of light, the "slow" formula is a very good approximation.


1

I drew the spacetime diagrams for you. On the l.h.s. you may see two simultaneous events in the unprimed (x,t) frame. The axes of a frame going in the positive direction (the primed frame) should be drawn into the unprimed as I have done it. You can find the new space coordinates by drawing a straight line parallel to the new time axis through the event. ...



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