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Relativity

 

The Einstein's theory of relativity is based on the principle that the physical laws are the same to every freely moving observer, whatever his velocity is. That means that we all shall measure the same light speed, no matter the direction to which we move and how fast we do it. According to this theory, nothing can be faster than light (299 792,5 km/second), as it is demonstrated next.

 

Einstein proved that:

______ ______E =

__Mc2__

(1-s2/c2)1/2

E         = energy

M       = mass

c          = speed of light

s          = speed of the moving object

 

When s is close to 0 (as it is for the velocities reached by human artifacts), then we get:

E = Mc2

 

On the other hand, P can symbolize the linear momentum, or the total quantity of movement, and now we get:

____________ P =

__Ms___

(1-s2/c2)1/2

 

At minimal velocities we obtain:

P = Ms

The first equation of P tells us that when s becomes close to c (the speed of light), then P approaches the infinite. So, s can never overtake c.

 

The Rhythm of Time

We have already seen that the speed of light is the same for an observer who is static regarding to a reference vector and to an observer who is moving in regard to that reference vector. If we suppose that the static observer watches the light moving 10 metres and that the moving observer advances in the direction of the photon (light particle) at 1/5 of its speed, then the latter will advance 2 metres and will see the photon moving away only 10 - 2 = 8 metres. So, the light will have traveled different distances for each one of them. As the speed of light is constant and as Velocity (speed) = Distance / Time, then:

Time(divergent for the 2 observers) = Distance(divergent) / Velocity(equal)

We conclude that the time flows at a different rate for a moving observer and for a stationary observer. The equation that expresses this idea is:

___________ Tm =

__Tst___

(1-s2/c2)1/2

Tm      = Time between the ticks of the pointers of a moving clock

Tst      = Time between the ticks of the pointers of a stationary clock

 

Length and Mass

Just like time, the length and the mass of the object are also affected by its velocity:

Lm    = Lst * (1-s2/c2)1/2

Lm      = Length of the object moving at the speed s

Lst      = Length of the same object, when it is stationary

___________ Mm =

__Mst___

(1-s2/c2)1/2

Mm    = Mass of an object moving at the speed s

Mst    = Mass of the same object, when it is stationary

 

In other words, the object is compressed and gets heavier when its speed nears the speed of light.

 

Adding Velocities

The finite nature of the speed of light also implies that the addition of high speeds doesn't work in the same way as the addition of velocities that are common to us, because otherwise we would obtain superluminar velocities (higher than the speed of light), that are forbidden by relativity. Being so:

____________ S =

__s1 + s2__

1+(s1* v2)/c2

S         = final speed

s1        = 1st added speed

s2        = 2nd added speed

 

If s1 and s2 are 10% of c, then:

____________ S =

__0,1+0,1__

1+(0,1*0,1)/1

In other words, S = 0,2/1,01 = 0,198 (approximately 0,2 = 0,1 + 0,1)

 

If s1 and s2 are 90% of c, then:

____________ S =

__0,9+0,9___

1+(0,9*0,9)/1

In other words, S = 1,8/1,81 = 0,994 (much lower than 1,8 = 0,9 + 0,9)

 

Gravity and Inertia

Relativity also tells us that the gravitational mass is equal to the inertial mass (force that opposes the changes provoked in a massive body). Exemplifying, a person that descends inside a lift falling under the influence of a gravitational field feels like there isn't any gravity force acting on him. If that person weighs himself on a balance located inside the lift, this will show that he's weighing 0 kg, because the descending acceleration of the balance is equal to the person's. It's like he weighed himself in the interplanetary space, in the interior of a non-accelerated vehicle (where he would feel an inertial force = 0).

In the same way, the effect felt by someone under the influence of a gravitational field is equal to the one that is felt when he travels in a vehicle far from any gravitational field but moving at an increasingly high velocity - the person is pushed towards the inverse direction of the vehicle's movement, as a car driver that suddenly steps on the brakes and feels pushed against the front window. This is generally called the "principle of equivalence", according to which the effects of a gravitational field can't be distinguished from the effects of a uniform acceleration.

 

Consequences of the Principle of Equivalence

The immediate consequences of this principle are:

    1. The light beams are curbed under the effect of a gravitational field, as in a lift accelerating in the space (because the light follows a straight line when it's seen from the outside but, given the acceleration of the lift, it follows a curved line when it's seen from its interior). If in the presence of a gravitational field a light beam is curbed, then the space-time (whose geometry is defined by the course the light beams) is no longer flat, or in other words, the pathway between 2 points is no longer a straight line. Given this, gravitation determines the form of the space-time. Far away from the source of the gravitational field, space-time becomes flat again. If the mass of the object that creates the field is very small, then the space-time around it also tends to be flat;
    2. An object, when it gets close to the source of a gravitational field (for instance, a ball falling on the floor), gains velocity as it descends and, this way, to its inertial energy (E = Mc2) must be added the kinetic energy (speed). If that energy is converted into a single photon (particle of light), then this one will be more energetic when it reaches the floor, so its frequency will be higher and, consequently, its wavelength will be smaller (deviation to the blue). The opposite happens when a ball is thrown to the air (or when a photon is sent to the opposite direction of the source of the gravitational field). There is a deviation to the red. Since a light wave with 1 mm of length is elongated, for instance, to 5 mm when it reaches a given height, then to the observer standing at the top, the events occurring on the floor will look to be happening at a slower pace because the time lapse between 2 light wave peaks (or troughs) is higher than when this is observed from the floor. As the accelerated reference vector is equivalent to a reference vector under the effect of a gravitational, then the effect of the time dilation shall be felt in a vehicle acceleratedly moving. So, the period between the ticks of a clock placed inside this vehicle shall increase when that clock is observed from the outside.

 

Curbed Space-Time (MoonRunner Design UK)

 

 

Clock at the base: 10:10 a.m., Clock at the top: 10 a.m. (MoonRunner Design UK)

 

Immediate Consequences of the Theory of Relativity

 

The Twins Paradox

The most evident consequence of the theory of relativity is the so-called "twins paradox", where is posed the hypothesis of a twin (30 years old) that stays on the Earth while the other (also 30 years old) departs to an interstellar trip at a velocity close to the light speed. At the end of the trip, the one that stayed on the Earth (40 years old) should acknowledge that his travelling twin would be younger (only 35 years old), because his watch and his life functions would have been delayed by the movement. However, the opposite should also succeed, since the twin brother that stayed in the Earth moved in relation to the traveler twin (as movement is relative and not absolute).

What actually happens is that the traveler twin comes back really younger:

    1. The reference vector of the brother that stayed on Earth is the inertial, since it wasn't subject to significant velocity changes, besides the relatively modest movements over the Earth's surface;
    2. The reference vector of the traveler twin is not inertial, since the departure and the return imply that there had to be at least one place where he changed the speed (between the time when he was still going away and the time when he started to come back again).

So, the age gap results from the use of 2 different reference vectors, by each of the twin brothers. The accelerated reference vector is, as it was mentioned, equivalent to a reference vector under the effect of a gravitational field, which provokes the dilation of time. Therefore, when the accelerated twin brother comes back, he shall be actually younger than his brother, just as a twin brother that would have lived under the influence of a strong gravitational field.

 

Black Holes

Other interesting consequences of relativity are the phenomena occurred around a black hole. These objects are bodies where the mass is so concentrated that, at its surface, the gravitational field is strong enough to prevent the escape of the light itself, which moves at the maximum speed allowed by the laws of physics. This hindrance is due to the fact that, under such conditions, the course of the light is subject to an extreme bending, so high that it can never be deviated from the centre of the gravitational field. As light isn't able to escape from it, then nothing can be. The space-time (as the light) is disfigured in such a way that a tunnel, called "wormhole", is formed. That tunnel will link the black hole to a white hole (which expels matter, instead of swallowing it) placed in a different universe or in a distant region of the same universe.

 

Black hole that is formed and gives origin to a "wormhole" (MoonRunner Design UK)

 

In a rotating black hole, the tunnel takes the shape of a ring and not a singularity. Opposing to what happens in a Schwarzchild black hole (without rotation), it's possible to cross a Kerr black hole (with a rotation) at a speed lower than that of the light. Nevertheless, the black holes with stellar origin exert so strong tidal forces that they destroy everything that dares to enter in them (as it happens to a massive body in the interior of the Roche limit). A super-massive black hole, as the one that possibly exists in the heart of our galaxy, doesn't pose so many problems concerning to this issue, because its surface (the events horizon) is too far away from the centre of the gravitational field. Obviously this kind of black holes is very seldom, since they occur only in very special locations (it is thought that only one exists in the Milky Way). Whoever may cross it will have, though, the chance to reach another universe or a distant place in our universe.

 

Tidal forces on the surface of a stellar black hole (MoonRunner Design UK)

 

Backward Time Travel

The existence of wormholes suggests that it is theoretically possible for someone to effectuate backward travels in time. The wormholes are comparable to shortcuts in the geometry of space-time, or in other words, they allow the connection between 2 different points of one or two universes, with local times that can be distinct.

Let's suppose that one wormhole is linking Earth to Alpha Centauri (the closest star apart from the Sun). If one of the mouths of the wormhole is kept stationary while the other is accelerated and braked further, then this one, as the travelling twin, will be younger that the mouth that didn't move.

Let's imagine that the mouth close to the Earth is now 6 years younger than the mouth in Alpha Centauri. If we travel from the Earth to out neighbouring star in 5 years, get into the tunnel through the mouth that is located there and get out of the tunnel through the mouth located at the Earth of 6 years ago, then we will have arrived to the starting point one year before we left.

 

The Paradoxes of Time Travels

This kind of travels in time generates a paradox: what happens if I begin a trip until a time when my father was not born yet and, when I arrive there, I decide to kill my paternal grandfather? Without my grandfather I could have never be born, so I wouldn't be able to make the trip. This way, my grandfather would be kept alive and I, after all, would have been born.

There are 2 proposed solutions to this paradox:

    1. The approach of the coherent stories supports that even if the trip to the past is possible, whatever happens in the space-time shall agree with the laws of the physics. According to this theory, when I would begin the time travel I would already have the memory of all the consequences of that trip and wouldn't be able to change, in any way, the course of the events. That implies that the traveler won't have neither freedom of choice nor free will;
    2. The approach of the alternative stories supports that when the traveler makes the trip he enters in an alternative story, different from the memorized story. He would then be free to act with freedom, not constrained. This approach seems to agree with the Feynman's quantum theory of the sum of the stories, which advocates that there is an infinity of possible stories, each one with a determined probability.

 

 

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