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Negative energy is a concept used in physics to explain the nature of certain fields, including the gravitational field and various quantum field effects. Gravitational energy Gravitational energy, or gravitational potential energy, is the potential energy a massive object has because it is within a gravitational field. In classical mechanics, two or more masses always have a gravitational potential. Conservation of energy requires that this gravitational field energy is always negative, so that it is zero when the objects are infinitely far apart. As two object move apart and the distance between them approaches infinity, the gravitational force between them approaches zero from the positive side of the real number line and the gravitational potential approaches zero from the negative side. Conversely, as two massive objects move towards each other, the motion accelerates under gravity causing an increase in the (positive) kinetic energy of the system and, in order to conserve the total sum of energy, the increase of the same amount in the gravitational potential energy of the object is treated as negative. A universe in which positive energy dominates will eventually collapse in a "Big Crunch", while an "open" universe in which negative energy dominates will either expand indefinitely or eventually disintegrate in a "big rip". In the zero-energy universe model ("flat" or "Euclidean"), the total amount of energy in the universe is exactly zero: its amount of positive energy in the form of matter is exactly cancelled out by its negative energy in the form of gravity. It is unclear which, if any, of these models accurately describes the real universe. Ergosphere For a classically rotating black hole, the rotation creates an ergosphere outside the event horizon, in which spacetime itself begins to rotate, in a phenomenon known as frame-dragging. Since the ergosphere is outside the event horizon, particles can escape from it. Within the ergosphere, a particle's energy may become negative (via the relativistic rotation of its Killing vector).
[2]Paolo Prandoni, [3]LCAV, EPFL
In this notebook we will look at the practical issues associated to
plotting the DFT and in particular the DFT of real-world signals. We
will examine how to map the DFT coefficients to real-world frequencies
and we will investigate the frequency resolution of the DFT and the
effects of zero padding.
In [1]:
[2]Paolo Prandoni, [3]LCAV, EPFL
The Karplus-Strong algorithm is a simple method to generate synthetic
sounds with arbitrary pitch. In this notebook we will explore a few
variations of the method to illustrate some basic notions of
discrete-time signal processing.
In [1]:
%matplotlib inline
import matplotlib
import matplotlib.pyplot as plt
import numpy as np
import IPython
In [2]:
plt.rcParams["figure.figsize"]