Explaining Pythagoras' Theorem & Its Impact on Acceleration

D.A.Peel
Messages
5
Reaction score
0
Can anyone explain to me why Pythagoras' Theorem governs the rate of change, of mass, length and time within accelertated bodies?
It's a simple theorem learned by most children by the age of eleven, so one would expect the answer to this question to be quite simple as well.
 
Physics news on Phys.org
It's actually not the well-known theorem of Euclidean space, but a different version belonging to Minkowski space, where the length of a vector (e.g. a time interval) is calculated from its components using
s² = t² - x² - y² - z². (or -t² + x² + y² + z² as a matter of convention)
In one spatial dimension, this becomes
s² ("true" elapsed time) = t² - x² = t²*(1-v²) (less than elapsed coordinate time).
The same logic gives relativistic mass: it is the "time component" of a vector (Energy-Momentum vector) which has a length equal to the rest mass. Modern usage is to call the time component energy, not relativistic mass.
It's a different situation for length contraction: what we define as "length" is actually not a component of a vector, but a one-dimensional slice of a two-dimensional entity, the measuring rod, which extends both in space and in time. Therefore the different result.
 
OK, so this has bugged me for a while about the equivalence principle and the black hole information paradox. If black holes "evaporate" via Hawking radiation, then they cannot exist forever. So, from my external perspective, watching the person fall in, they slow down, freeze, and redshift to "nothing," but never cross the event horizon. Does the equivalence principle say my perspective is valid? If it does, is it possible that that person really never crossed the event horizon? The...
From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by ##g_{\alpha\beta}## to get $$\delta g_{\beta\nu} = -g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \qquad(*)$$ (This is Dirac's eq. (26.9) in "GTR".) On the other hand, the variation ##\delta g^{\alpha\mu} = \bar{g}^{\alpha\mu} - g^{\alpha\mu}## should be a tensor...
ASSUMPTIONS 1. Two identical clocks A and B in the same inertial frame are stationary relative to each other a fixed distance L apart. Time passes at the same rate for both. 2. Both clocks are able to send/receive light signals and to write/read the send/receive times into signals. 3. The speed of light is anisotropic. METHOD 1. At time t[A1] and time t[B1], clock A sends a light signal to clock B. The clock B time is unknown to A. 2. Clock B receives the signal from A at time t[B2] and...
Back
Top