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## The Theory of RelativityEinstein's Theory of Relativity functions on these premises: - The laws of nature are the same in different inertial frames of reference.
- The speed of light is constant in a vacuum. (Special Relativity)
- Gravitational mass is
__equivalent__to inertial mass.(General Relativity)
EINSTEIN | SPECIAL RELATIVITY | GENERAL RELATIVITY
One of Einstein's assumptions that led to the development of Special Relativity was that the speed of light is constant, yet relative to all observers.
The speed of light in a vaccuum is 3 * 10
Einstein created a thought experiment: if someone were walking along and threw a pencil straight up in the air while walking, the pencil would
Before we derive any equations, we must understand the concept of a light clock. A light clock looks like this:
We can use it to measure time because we know that time = distance / rate. (If you went 10 miles at 20 miles per hour, it took you 0.5 hours, right?)
The distance the light travels is twice the length of the clock (this clock is on end, so it's the height here) and the rate at which it travels is the speed of
light. So with a light clock, time = 2Length / c (remember that c is always the speed of light, or 3 x 10
- L is the height of the light clocks
- c is the speed of light
- t is Jim's time, the time it takes for one tick of his light clock, or t = 2L/c.
- t' (read "t prime") is Bob's time, the time is takes for one tick of Bob's clock.
- v is Jim's velocity.
^{1/2} means square root x)
c is the hypotenuse. c
2 * (L
2 * (L
2 * (4L
2 * (4
2 * (4
(4
Since we can also say distance = rate x time (2 hours driving 60 miles per hour takes you 120 miles), we can say distance = ct', and ct' also equals (4
Square both sides: c
Move term to the other side: c
Factor out ^{2}(c^{2} - v^{2}) = 4L^{2}
Isolate ^{2} = 4L^{2} / c^{2} - v^{2}
Factor out a term on the bottom: t'
Get rid of squares (square root everything): t' = 2L / c (1 - v
Since we know that, on a time clock,
For each tick, t, on Jim's clock, Bob observes t' ticks on his own clock. The equation Lorentz Factor.
Perhaps the most famous equation of all time is E = mc First of all, let us consider a photon. One of the interesting properties of photons is that they have momentum and yet have no mass. This was established in the 1850s by James Clerk Maxwell. However, if we recall our basic physics, we know that momentum is made up of two components: mass and velocity. How can a photon have momentum and yet not have a mass? Einstein’s great insight was that the energy of a photon must be equivalent to a quantity of mass and hence could be related to the momentum. Einstein’s thought experiment runs as follows. First, imagine a stationary box floating in deep space. Inside the box, a photon is emitted and travels from the left towards the right. Since the momentum of the system must be conserved, the box must recoils to the left as the photon is emitted. At some later time, the photon collides with the other side of the box, transferring all of its momentum to the box. The total momentum of the system is conserved, so the impact causes the box to stop moving. Unfortunately, there is a problem. Since no external forces are acting on this system, the center of mass must stay in the same location. However, the box has moved. How can the movement of the box be reconciled with the center of mass of the system remaining fixed? Einstein resolved this apparent contradiction by proposing that there must be a ‘mass equivalent’ to the energy of the photon. In other words, the energy of the photon must be equivalent to a mass moving from left to right in the box. Furthermore, the mass must be large enough so that the system center of mass remains stationary.
Let us try and think about this experiment mathematically. For the momentum of our photon, we will use Maxwell’s expression for the momentum of an electromagnetic
wave
P The box, of mass M, will recoil slowly in the opposite direction to the photon with speed v. The momentum of the box is:
P The photon will take a short time, Δt, to reach the other side of the box. In this time, the box will have moved a small distance, Δx. The speed of the box is therefore given by: v = Δx / Δt By the conservation of momentum, we have: M(Δx / Δt) = E / c If the box is of length L, then the time it takes for the photon to reach the other side of the box is given by: Δt = L / c
Substituting into the
MΔx = EL / c
Now suppose for the time being that the photon has some mass, which we denote by m. In this case the center of mass of the whole system can be calculated. If the
box has position x
We require that the center of mass of the whole system does not change. Therefore, the center of mass at the start of the experiment must be the same as the end of the experiment. Mathematically:
The photon starts at the left of the box, i.e. x ML = MΔx
Substituting the
mL = EL / c Rearranging gives the final equation:
E = mc Using 1 kg of mass, we get the following amount of energy:
E = (1 kg) * (299,792,458 m/s) In practice, it is not possible to convert all of the mass into energy (For instance, the "Gadget"-style bomb used in the Trinity test and the bombing of Nagasaki had an explosive yield equivalent to 21 kt of TNT. About 1 kg of the approximately 6.15 kg of plutonium in each of these bombs fissioned into lighter elements totaling almost exactly one gram less, after cooling (the heat, light and radiation in this case carried the missing gram of mass). This occurs because nuclear binding energy is released whenever elements with more than 56 nucleons fission). However, this equation led directly to the development of nuclear energy and the nuclear bomb - probably the most tangible results of special relativity.
The term mass in special relativity can be used in different ways, occasionally leading to confusion. Historically, mass can refer to either the - The
**invariant mass**(also known as the rest mass, intrinsic mass or the proper mass) is an observer-independent quantity - m_{0} - The
**relativistic mass**(also known as the apparent mass) depends on one's frame of reference - m_{r}
Because of this, m
p = m
Does this equation look familiar? Apart from the variables p, m
L = L Let's say a road is observed as 20 meters by a stationary observer. If the observer then moving at a significant fraction of the speed of light, say .98c, then the length of the road will shorten drastically:
L = L
L = 20 [1 - (0.98) L = 3.98 meters
However, this only hold true for the traveller, since he is
Δt
Δt
Δt
Δt
Muons are particles created high up above the Earth's surface by cosmic rays hitting the atmosphere. They can be detected in large numbers at the Earth's surface.
They travel at around 0.998 times the speed of light, but they don't live for very long - typically, they decay after only 2 x 10
The Theory of Relativity, like all scientific theories, is an explanation and model of natural phenomena. The Theory of Relativity provides a highly predictive model of how
gravity works in many different contexts. However, what actually The stakes are high. If we can find out how gravity can be represented and function in a quantum environment, we might be able to build a model of what happened before the Big Bang. But, because of our macroscopic bias:
Quantum Physics prevents us from seeing "past" the singularity of the Big Bang. |