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16

So what you wrote here isn't exactly what Einstein writes in the paper, and the difference there is what's causing your confusion (also he changes what he means by $\phi(v)$ halfway through the paper, which is the real problem). On page 7 of the pdf you linked, these equations appear: $$\xi = a \frac{c^2}{c^2 - v^2} x'$$ $$\eta = a \frac{c}{\sqrt{c^2 - ... 5 Recall that the Faraday tensor in this form is a linear mapping that maps a charged particle's contravariant four-velocity to the latter's rate of change, wrt proper time (modulo scaling by invariant rest mass m and invariant charge q):$$m\,\frac{\mathrm{d} v^\mu}{\mathrm{d}\tau} = q\, F^\mu{}_\nu\,v^\nu\tag{1} Now let's think of a particle's ...

4

I think this question is a little more low-level then the one that it's being marked as a duplicate of, so I'm going to answer it. The basic thing that you need to know is that an inner product on the 4-vector space need not have the form it has in Euclidean coordinates. That is, defining $w = ct$, it is not necessarily the case for all 4-D vector spaces ...

2

The time measured by the clock you carry with you on your journey is called the proper time, and the proper time is just the length of your world line (give or take a factor of $c$). So if we calculate the length of your world line as you accelerate away then back, that gives us your elapsed time. Better still, the proper time is an invariant i.e. all ...

2

In one of my vector mechanics lectures, the lecturer said the the space time interval was the dot product of the four position vector. But then he proceeded to show it was this: $s^2$ = $\Delta r^2$ - $c^2\Delta t^2$ Where did the minus sign come from? See Einstein's Simple Derivation of the Lorentz Transformation. It's really simple. The light moves a ...

2

There is an argument based on deriving an arbitrary coordinate transformation that allows time to change as well as space for different observers. This transformation will turn out to have an undetermined parameter $v$ which corresponds to a maximum speed, and in the limit $v \to \infty$ the transformation becomes the Galilean transformation of classical ...

1

Besides the philosophical problems of not having a maximum speed at which interactions propagates: Leibniz was horrified by Newton's theory of gravity, and even Newton himself knew that his theory could not be the complete story. The best shots that comes to my mind right now at guessing that the speed of light has to be finite (ignoring post 1850 evidences) ...

1

No. The charge enclosed by a surface is really just the number of protons inside minus the number of electrons inside all multiplied by the charge of the proton (which is a constant). If the electric field has a large density in one region then it has less in another so that total flux over a closed surface is still the net enclosed by the whole surface is ...

1

Recall that the gradient of a scalar field is a (0,1) tensor, a one-form. With a vector field $\vec A=\vec A(\vec r)$ you can think of having three scalar fields $P(\vec r),$ $Q(\vec r),$ and $R(\vec r)$ and then $\vec A(\vec r)=P(\vec r)\hat x+Q(\vec r)\hat y +R(\vec r)\hat z.$ We will do the same kind of thing in flat spacetime. Note that the vectors ...

1

Vector spaces with mixed signature metrics (both pluses and minuses in the metric signature) are known and understood mathematical structures, but it's not surprising you might not have heard of them before this point: from an educational standpoint, you need everything you've learned about metric spaces with positive-definite metrics and more to make sense ...

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