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In relativity, we must modify the way we calculate kinetic energy for reasons similar to those that justify changing the calculation of momentum. The relativistic kinetic energy of an object is given by:
The constant term mc² is called the rest energy of the object,
The total energy of an object is its kinetic energy plus rest energy:
This is the general form of the relativistic energy equation. You obtain the familiar E=mc² expression when the speed of the object is zero, and g=1.
Using E = mc² we can relate mass to energy. When discussing atoms, nuclei, and particles, it is common to use the rest energy for particles in eV (or keV or MeV) rather than their mass in kg. Determine the rest energies (rest mass) of the electron and proton in MeV?
The mass of the electron and proton are me = 9.11×10-31kg and mp = 1.67×10-27kg.
Using E=mc² will yield the energy in joules, which can be converted to electron-volts (eV) by the conversion factor (1 eV / 1.60×10-19J).
For the electron:
Ee = (9.11×10-31kg)(3.00×108m/s)2 (1 eV / 1.60×10-19J) = 5.11×105eV = 0.511MeV
and for the proton:
Ep = (1.67×10-27kg)(3.00×108m/s)2 (1 eV / 1.60×10-19J) = 9.38×108eV = 938MeV.
At the accelerator where I perform research, protons are accelerated to an energy of 920GeV. What is the speed of these protons?
These protons have a relativistic energy of
E = gmp c² = gEp
where Ep is the rest energy of the proton.
From this we find that these g for these protons is:
g = E / Ep = 920GeV/938MeV = 991.
Now, we can solve for v since g = 1/Sqrt{1 - v²/c²}:
g2 = 1 / (1 - v²/c²)
1 - v²/c² = 1/g2
v²/c² = 1 - 1/g2 = (g2 - 1)/g2
v/c = Sqrt(g2 - 1) / g = Sqrt(9912 - 1) / 991 = 0.9999995
The sun radiates approximately 4.0×1026J of energy into space each second. (a) How much mass is converted into energy of other forms each second? (b) If the mass of the Sun is 2.0×1030kg, how long can the Sun survive if the energy transformation continues at the present rate?
(a) The energy radiated by the sun is derived primarily from converting mass into other forms of energy.
The energy radiated is related to converted mass by E = mc².
Therfore:
m = E / c² = (4.0×1026J) / (3.0×108m/s)2 = 4.4×109kg
(b) The above amount of mass is "burned" every second, so the total time that this can continue is:
t = (2.0×1030kg) / (4.4×109kg/s) = 4.5×1020s = 1.4×1013yrs = 14 trillion years.
This section is optional; this discussion is made to connect topics of popular science presentations to material in this course.
Einstein felt that his "most pleasant thought" was the realization of the "equivalence principle". The equivalence principle is the basis of his theory of general relativity. It states that a gravitational field is equivalent to a local acceleration.
To understand what this means, consider the following thought experiment (sketch). A Physics 2140 student is in an elevator, and drops a ball, noting that the ball falls to the floor of the elevator according to y = y0 - ½gt². This will be exactly the result if the elevator is on Earth, and stationary.
What would happen if instead the elevator were on a space ship, in a gravity free region of space, and accelerating in the upward direction with acceleration a = g? In this case, if a ball is dropped, it continues moving with the same instantaneous velocity of the elevator at the moment it is released. Because it is no longer attached to the elevator, it stops accelerating. But the elevator continues to accelerate, so while the position of the ball can be described by yball = y0 + v0t, the position of the floor of the elevator is given by yfloor = v0t + ½at², with a = g. So the distance of the ball from the floor is given by:
In fact, no experiment that we do can differentiate between a local gravitational field, and a local acceleration. Thus the equivalence between acceleration and gravity.
Einstein took this idea and developed it into the general theory of relativity, with some remarkable predictions. The first is that, with gravity replaced by acceleration, what we call the gravitational force comes from local curvatures of space. We can visualize this in two dimensions with a stretched out sheet to represent two dimensions of space. The presence of mass (like a star) can be simulated by placing a weight on the sheet. The weight causes the sheet to curve in its vicinity, similar to the way that a star curves space in its vicinity. If we roll a marble across the sheet in the region where it is curved, the marble will deflect. In general relativity, the curvature of space causes the path of planets to curve, resulting in what we call gravity!
The next surprising result is that the same rules apply to light. That is, light will also curve when passing near a massive object. The curvature is not great, but in recent years it has been clearly seen by many telescopes. (Pictures of gravitational lensing.)
A third result is that clocks run more slowly where gravity is larger. This again has been seen, and recalling the earlier discussion of the global positioning satelltes, the correction of their clocks due to general relativity is even larger than the correction for special relativity. The special relativity effects cause the clocks to run slow by 7.11ms, and the general relativity effects cause them to run fast by 45.7ms per day!
This is also the cause of gravitational red shifting. The light from massive stars is shifted in frequency from blue to red as the light emerges from the gravitational field of the star.
A fourth prediction is that if the mass of a star becomes sufficiently great, it will curve space to such an extent that even light cannot escape from the star. The resulting object is called a black hole. The evidence for the existence of black holes is now quite strong. It is assumed that nearly every galaxy has a black hole at its center, and that objects known as quasars are very distant black holes. But as you might guess, since black holes don't radiate like stars, the evidence for their existence is indirect.
Finally, general relativity predicts the existence of gravity waves. Gravity waves cause local distortions in space as they propagate, and measurable waves are presumed to be produced in only the most cataclysmic events, such a black hole swallowing a star, or the collision of two black holes. Physicists have been trying to detect gravity waves for over 30 years, but they are very feeble, requiring extremely sensitive equipment. The LIGO experiment is the latest attempt; its state of the art interferometer may reach sufficient sensitivity to see the first signals of gravity waves in the next five years.
© Robert Harr 2000