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Why do we differentiate a 4 vector with respect to proper time to obtain 4-velocity?

The coordinates of an event in spacetime are given by the 4-vector $(ct, \mathbf{r})$, where $\mathbf{r}$ is the spacial coordinates of the event. This 4-vector can be seen as 4-displacement of a ...
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4answers
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Definition of the Spacetime Interval

The spacetime interval is defined as follows: $$\Delta s^2 = -(c\Delta t)^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$$ or in tensor notation: $$\Delta s^2 = \eta_{\mu\nu} \Delta x^\mu \Delta x^\nu$$ ...
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2answers
362 views

Defining invariant spacetime interval

So, my textbook goes about defining the invariant spacetime in the following way: Consider two frames of references, S and S', with a relative speed to each other, coinciding at t=t'=0. At t=0, a ...
4
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2answers
703 views

Invariance and forms of the Lagrangian

I have been reading the 1st chapter of Landau & Lifshitz Mechanics, and due to its concise style been facing a few problems. I hope you can help me out here somehow. Does the "homogeneity of ...