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The Webfooted Astronomer - December 2000

 

Aspects of Special Relativity

By Dave Irizarry

TODAY, I'd like to take a few moments to talk about some aspects of Einstein's Theory of Special Relativity, which depicts some of the "strange" properties of the universe in which we exist. The theory is concerned with time and distance measurements occurring across differing reference frames. Let us consider four formulas that describe the properties of an object moving at uniform relativistic speeds (high speeds approaching the speed of light). The formulas apply only to frames of reference in uniform motion to one another (no acceleration).

Time Dilation

There is no absolute time. Time is relative to the frame of reference in which a time interval is observed. It has been verified that clocks in a moving object run slower than a reference clock that is at rest. This effect is called time dilation. The formula below derives the time interval between events in a moving frame of reference as measured by a stationary observer, given the relative speed of the moving reference frame and the same time interval as measured from within the moving reference frame.

Let T be the time interval measured from a stationary frame of reference.
Let T0 be the time interval measured within a moving frame of reference.
Let v be the speed of the object.
Let c represent the speed of light.
Note: sqrt means take the square root of the quantity enclosed in parenthesis:
T = T0 / sqrt(1- v2/c2)

So, for example, if an event occurred in a fast moving spaceship, which moved along at 0.4 times the speed of light, and that event took 10 seconds for an observer aboard the craft, a stationary observer would measure the same time interval to be 10 / sqrt(1 0.16) = 11.9 seconds. In other words, a time interval of 10 seconds in the space craft appears as a time interval of 11.9 seconds to a stationary observer. Strange... isn't it?

Length Contraction

An object moving at a given speed measures shorter along the direction of motion than when it is at rest. This phenomenon is known as length contraction. The formula used to calculate the amount of contraction reads as follows:
Let L0 be the length of the object at rest.
let L be the length of the object at speed v when measured by a stationary observer.
Let v be the speed of the object.
Let c represent the speed of light.
L = L0 * sqrt(1 v2/c2)

So for example, a projectile with length 10 meters moving at 60% the speed of light would be measured by a stationary observer as having a length of 8 meters..

Relativistic Mass Increase

An object moving at a given speed will measure more massive than when it is at rest. The Mass of the object at rest is called the object's rest mass. The object's measured mass when it is in motion is called the relativistic mass. The relationship between an object's rest mass, relativistic mass, and speed is given by the formula below:
Let m0 be the rest mass of the object.
Let m be the calculated relativistic mass of the particle.
Let v be the speed of the object.
Let c represent the speed of light.
m = m0 / sqrt(1 v2/c2)

So, for example if our hypothetical space ship from equation No. 2 had a mass of 1,000Kg while at rest, it's measured mass while moving at 60% the speed of light would be 1,250 Kg. At 98% the speed of light, the ship's mass would be measured as 5,025.2 Kg. At 99.5, its measured mass would be 100,250.63 Kg!

Relativistic Kinetic Energy

To calculate the kinetic energy of an object moving at relativistic speeds, simply calculate m from equation No. 3 and subtract from it the rest mass m0.
Multiply the difference by c2 to derive the kinetic energy.
Let KE be the kinetic energy of a moving object.
Let m0 be the rest mass of the object.
Let m be the calculated relativistic mass of the particle.
Let c represent the speed of light.
KE = (m m0) * c2

So, for example, the ship in equation 3 traveling at 60% the speed of light and having gained 250 Kg in mass would possess a kinetic energy of 2.25 x 1019 Joules.

I hope the foregoing mathematical relationships between mass, time, speed, and energy have enriched your understanding of our physical universe. Although in our everyday experiences, we encounter speeds far less than the relativistic speeds at which the above phenomena become apparent, the foregoing relationships help us see that under the everyday Newtonian principles lurk Einstein's laws of space and time. Furthermore, equation No. 4 presents us with the stark truth that attaining relativistic speeds requires an enormous amount of energy, and will present a formidable engineering challenge for the developers of future interstellar spacecraft.

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