The relativity of simultaneity
Simultaneity refers to our idea that different things happen at the same time. Einstein’s relativity complicates this simple idea. At speeds approaching that of light, events that are simultaneous in one frame of reference may not be simultaneous in another frame of reference. This is known as relative simultaneity.
[Thought experiment 2]
-A train carriage has light-operated doors. The light is in the middle of the carriage. When the light is switched on, it travels equal distances to each door and opens them at the same time. At least, that’s how someone inside the train sees things happening. The train is at rest relative to this observer, and the two events occur simultaneously.
-However, a person outside the train sees the train moving. When the light is turned on it travels out towards each door, but at the same time, the train moves forwards. So the light reaches the back door sooner than it reaches the front door because the back of the train has moved closer to the light rays approaching it. Hence the back door opens first, and the two events do not occur simultaneously.
The equivalence between mass and energy
The energy-mass equivalence is where energy and mass are equivalent and are interconvertible. Their relationship is quantitatively governed by the equation:
If we apply a force to an object, then work is done on it. This energy would take the form of increased kinetic energy as the object speeds up. But since nothing can move faster than the speed of light, once an object already has sufficient kinetic energy it cannot speed up as we would normally expect. Instead it acquires extra mass.
Relativity results in a new definition of energy as follows:
E = Ek + mc2
Notice that when an object is stationary, it has no kinetic energy, and still has some energy due to its mass. This is called the rest energy.
Note that this equation challenges the law of conservation of energy and the law of conservation of mass, which state that their mass nor energy can be created nor destroyed. Einstein’s equation, which was derived by special relativity, caused a modification to create: The law of conservation of mass and energy: Matter and energy cannot be destroyed or created. They can only be transformed.
Length contraction
Length contraction is when the length of a moving object appears shorter compared to the length of the object when measured at rest.
Length contraction occurs according to the formula:
Length contraction is only observed in the direction of the motion. If observers on Earth watch a spaceship zoom past, they will see the spaceship as shorter than its real length. Similarly the observer inside the moving spaceship will observe the diameter of the Earth as smaller than it really is. The faster an object travels, the shorter it appears to an observer.
Time Dilation
Time in Einstein’s special relativity loses its absolute nature and becomes relative.
Time dilation can be summarized as ‘a moving clock appears to run slower’.
Time dilation is the slowing down of events as observed from a reference frame in relative motion. The time taken for an event to occur within its own rest frame is called the proper time to. Measurements of this time, tv, made from any other inertial reference frame in relative motion to the first, are always grater. The degree of time dilation varies with velocity.
[Thought Experiment 3]
A train is moving to the right at a constant velocity v that is close to the speed of light.
A light source is placed on the floor of the train while a mirror is fixed onto the ceiling. A light beam is emitted at the light source.
Observer A, who is inside the train, will see the light beam going straight up and down. Thus traveled 2d.
On the other hand, Observer B, who is outside the train, will see the light beam traveling forward-upward and then forward-downward. So the light has covered a distance 2d’, with d’ being the hypotenuse. Since d’ > d (from pythag), then the time Observer B measures is always greater than the time Observer A measures.
Time dilation formula:
Mass Dilation
Mass dilation is when the mass of a moving object appears greater compared to the object’s mass at rest.
The mass of an object within its own rest frame is called its rest mass mo. Measurements of this mass mv, made from any other inertial reference frame in relative motion to the first, are always greater. The degree of mass dilation varies with velocity. More simply state as: moving objects gain mass.
Einstein believed very strongly that momentum must be conserved in all inertial frames of reference. In order to solve this dilemma he suggested that the mass of an object must increase, or dilate, at relativistic speed by a factor that compensates for the effect of time dilation on speed measurement.
At speeds approaching light speed, mass increase is exponential in the same way as time and length, approaching infinity at the speed of light. As mass increases so does the force needed to accelerate it. As mass approaches infinity, the force needed to accelerate it further approaches infinity. The consequence of this is that speeds beyond the speed of light are impossible.
Mass dilation formula:
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