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The Theory of Relativity


Einstein's Theory of Relativity functions on these premises:

  1. The laws of nature are the same in different inertial frames of reference.
  2. The speed of light is constant in a vacuum. (Special Relativity)
  3. Gravitational mass is equivalent to inertial mass.(General Relativity)
The Theory of Relativity is a description of gravity at high speeds; it is a completion of Newton's Laws of Motion. Newton's Laws apply to bodies of matter at low speeds, while Einstein's model applies to matter high speeds. Although Einstein's model can account for and describe matter and gravity at low speeds, Newton's Laws fail at accurately describing the relationship between gravity and matter at high speeds.

EINSTEIN | SPECIAL RELATIVITY | GENERAL RELATIVITY

Intro

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 * 105 km/sec (or 186,000 mps). So, to someone stationary, the speed of light would be 3 * 105 km/s faster than them. Someone travelling in a space ship at 10,000 mph, the speed of light would be 3 * 105 km/sec faster than them. This led to a peculiar situation:

Einstein created a thought experiment: if someone were walking along and threw a pencil straight up in the air while walking, the pencil would appear to travel along the height of the triangle along with the observer moving along the base. However, the pencil actually travels along the hypotenuse while the observer is moving along the base. If the pencil were a beam of light, the speed of light would either have to speed up in order to keep time consistent, or time would have to speed up along the hypotenuse with respect to the base. Consequently, this speed of light (c) can be represented by c2 in the Pythagorean Theorem: a2 + b2 = c2. The photon must always travel at the speed of light (c). Even if the person were moving (at velocity aperson), the light would move at exactly the speed of light c. So, we have 3 variables: c, aperson, and the speed of the photon upward or downward along the height: bphoton. This leaves us with the equation c2 = a2person + b2photon. We can rearrange this to check the speed of the stationary photon: b2photon = c2 - a2person. This works out if one gets rid of the person (a2person = 0). This leaves b2photon = c2: The speed of the photon is equal to the speed of light. Because the speed of light is equal, either the time along the hypotenuse is shorter or the length of the hypotenuse shortens in relation to whoever is observing the photon. This leads us to:

Lorentz Transformations
from library.thinkquest.org

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 108 m/sec).


Jim is in a train car. He's looking at his light clock.


Bob can see Jim's light clock too. This is what Bob sees. Bob has his own light clock, which disagrees with Jim's.
Let's say:

  • 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.
Let's figure out the distance the light traveled, (according to Bob), using the Pythagorean Theorem (*x1/2 means square root x)

c is the hypotenuse. c2 = base2 + height2. The base is vt' / 2, the height is L. (c2)1/2 then equals:

2 * (L2 + (vt'/2)2)1/2 =

2 * (L2 + v2t'2/4)1/2 =

2 * (4L2/4 + v2t'2/4)1/2
multiply L2 by 4 in order to add the two together, which becomes:

2 * (42 + v2t'2/4)1/2 =

2 * (42 + v2t'2)1/2) / 41/2
the square root of 4 = 2, so the two 2's cancel out, which becomes:

(42 + v2t'2)1/2

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 (42 + v2t'2)1/2. Now how about the time? Let's work it out:

Square both sides: c2t'2 = 4L2 + v2t'2

Move term to the other side: c2t'2 - v2t'2 = 4L2

Factor out t'2: t'2(c2 - v2) = 4L2

Isolate t'2: t'2 = 4L2 / c2 - v2

Factor out a term on the bottom: t'2 = 4L2 / c2 (1 - v2/c2)

Get rid of squares (square root everything): t' = 2L / c (1 - v2/c2)1/2

Since we know that, on a time clock, 2L / c is the time for each tick, we can replace 2L / c for t: t' = t / (1 - v2/c2)1/2

For each tick, t, on Jim's clock, Bob observes t' ticks on his own clock. The equation t' = t / (1 - v2/c2)1/2 is called the Lorentz Factor.

E = mc2
from adamauton.net

Perhaps the most famous equation of all time is E = mc2. The equation is a direct result of the theory of special relativity, but what does it mean and how did Einstein find it? In short, the equation describes how energy and mass are related. Einstein used a brilliant thought experiment to arrive at this equation, which we will briefly review here.

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[1] having a given energy. If the energy of the photon is E and the speed of light is c, then the momentum of the photon is given by:

Pphoton = E / c

The box, of mass M, will recoil slowly in the opposite direction to the photon with speed v. The momentum of the box is:

Pbox = Mv

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 conservation of momentum equation and rearranging:

MΔx = EL / c2

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 x1 and the photon has position x2, then the center of mass for the whole system is:

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. x2 = 0. So, by rearranging and simplifying the above equation, we get:

ML = MΔx

Substituting the conservation of momentum equation into the above yields:

mL = EL / c2

Rearranging gives the final equation:

E = mc2

Using 1 kg of mass, we get the following amount of energy:

E = (1 kg) * (299,792,458 m/s)2 => (approx. 90 * 1015 Joules)

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.

Mass, Length, and Time In 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 invariant mass or the relativistic mass.

  • The invariant mass (also known as the rest mass, intrinsic mass or the proper mass) is an observer-independent quantity - m0
  • The relativistic mass (also known as the apparent mass) depends on one's frame of reference - mr

Because of this, m0 is also treated as a constant and mr is equivalent to E in E=mc2. Thus, the actual equation between energy and mass becomes E2 = m2c4 + p2c2, where p is the momentum. An object at rest has 0 momentum, so it simplifies to E=mc2. If the object is massless, then the equation becomes E = pc, which is the case for a photon. The equation for p is given as:

p = m0*v *1 / (1 - (v/c)2)1/2

Does this equation look familiar? Apart from the variables p, m0, and v, it's the Lorentz Factor from above. Notice that in the square root part of the equation, if the object's velocity (v) is at or greater than the speed of light, some wackiness ensues. A velocity equal to c means that we divide by zero resulting in an infinity -- sqrt(1 / 1 - 1) -- and if the velocity is greater than c, then we're dealing with an irrational number -- sqrt(1 / -n); neither of which exist in reality.

Length Contraction
In the equation below, L0 is the length of the object at rest in relation to a motionless observer - in other words, the normal length you see your computer at in day to day situations. However, because the speed of light has to be constant for all observers, length contraction occurs to objects moving at a significant fraction of the speed of light. Utilizing the Lorentz Factor, this new length comes out to be:

L = L0 * (1 - (v / c)2)1/2

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 = L0 * (1 - (v / c)2)1/2

L = 20 [1 - (0.98)2)]1/2

L = 3.98 meters

Time Dilation
Time Dilation also occurs on objects travelling at a significant fraction of the speed of light - which is also a result of the Lorentz Factor. Let's say that a space traveller wants to travel to Alpha Centauri, which is 4.3 lightyears away. If they could travel at a significant fraction of c - say .95c - it would take them 4.5 years to reach the star... Δt0. Δt0 is the change in time, which is derived simply by finding the quotient of dividing length by speed. 4.3 lightyears / 0.95c = 4.5 years (Δt0).

However, this only hold true for the traveller, since he is NOT moving relative to his speed of light. A person on Earth would measure the time spent by the traveller going to Alpha Centauri as 1.4 years! As in, the traveller would only age by 1.4 years in the 4.5 years of travel! This is because we see the space traveller moving at such a high fraction of c, that we witness the dilated time - Δtdilated - on Earth. This leads to what is popularly called the Twin Paradox. The explanation is as follows:

Δtdilated = Δt0 * (1 - v2/c2)1/2

Δtdilated = 4.5 * (1 - (.95c)2/c2)1/2

Δtdilated = 4.5 * (1 - 0.9025)1/2

Δtdilated = 1.4

Muons

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 10-6 = 0.000002 seconds. Multiplying their speed by their lifetime, they should only travel 600m before dying - but they are produced in the atmosphere (6,000m above the Earth) and can be detected at the surface! Due to their speed, their frame of reference says it's only 300m from the atmosphere to the surface.

So What IS Gravity?

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 causes gravity is still left unexplained. There are many hypotheses as to what causes gravity. The other three fundamental forces (electromagnetism, strong and weak nuclear force) can be explained by their quantum physical counterpart (photons, nucleons/gluons, and vector bosons respectively). No such quantum counterpart exists for gravity. The "graviton" has been proposed, but no evidence has been presented to demonstrate its existence. However, Loop Quantum Gravity might be a promising hypothesis. Along with String Theory and Casual Dynamic Triangulation, these make up the most promising prospects for a theory of quantum gravity.

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.