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Theory Of Relativity

Before set foot into this page,one must be known the basics of special theory of relativity so that you can understand this article very clearly.If you are not well known about special theory of relativity,check out the link given below  to learn more about it in detail and in a way that can be easily understood.We have learnt that time is relative.That is,it changes for different observers.This was termed as Time Dilation.In this article we are going to learn about some features of special theory of relativity.



Time Dilation:

      We went over the postulates of special relativity,and we saw that in order for the second one to be true,we have to rethink how time works.If the speed of light is the same in all inertial reference frames,then someone standing on earth and someone moving in a very fast spaceship must be experiencing the passage of time at different rates.This concept is called time dilation,and it tells us that the faster you move relative to the speed of light,the slower time will pass for you.This concept is best demonstrated through a thought experiment.Let's say that there is an observer on earth and another in a spaceship moving with constant velocity relative to the earth.On the spaceship there is a device that emits a light pulse,which hits a mirror and bounces back to a detector that sits right by the device.Each observer has a clock,and we pretend that the observer on earth can see what's going on in the spaceship.For the astronaut it's easy to tell how long it takes for the light to make the trip.If 
                                                     
               speed = distance       then,  time = distance 
                               time                                   speed

So it's just twice the distance between the device and the mirror to get there and back divided by the speed of light,or  t = 2D .Let's call this time interval delta t0.
                           c

                                  Image of distance calculated by an astronaut inside the spaceship

For the person on earth it's a little different,because the spaceship is moving relative to earth and therefore the light pulse has a horizontal component of motion in addition to the vertical one.The light follows the diagonal path up to the mirror and back,which is much longer than the other path.If this horizontal distance is L,then we use the Pythagoras theorem to get s as shown in the image below.And we get root D^2 + L^2,so twice this term will be the distance traveled by the light pulse according to the observer on earth.


But the speed of light is always the same ,so if the light travels a greater distance at the same speed,the observer on earth must register a greater time interval,delta t,for this event to occur than the person on the spaceship.This is why this phenomenon is called time dilation,because to dilate is to expand and the time interval delta t has expanded relative to delta t zero.This means that the clock on the spaceship is running more slowly than the one on earth,as this is the only way for both observers to get the same result for the speed of light.We can calculate exactly how much slower in fact.After doing a bunch of algebra expressions we attain a mathematical relation as shown in the image below.

Time dilation mathematical formula
Remember that although the spaceship is clearly in motion,it is the observer on earth that is in motion relative to the events in question,as the light is inside the spaceship,This equation allow us to plug in some velocity for the ship and calculate the precise time dilation that will result,or calculate the proper time given the dilated if we rearrange slightly.In the time dilation equation,as v gets very small relative to the speed of light.The mathematical relation is given as 


For objects moving nowhere near to the speed of light the dilated time interval is essentially identical to the proper time interval.That's why we don't notice time dilation on earth as we walk and run around,but for something like space travel,it becomes significant.The best example of this is twin paradox.

Twin Paradox:

            Let's say there is a set of twins and one goes on a journey through space while the other remains on earth.The traveler gets in a super fast spaceship and travels for 10 years,then turns around and comes back.Time dilation doesn't only affect mechanical clocks but biological clock as well,so upon returning on earth,they are no longer the same age.The supposed paradox arises when we consider inertial reference frames .To the twin on earth,the other one sped away and was moving very fast for 20 years,so the traveler should have aged less.But to the other one in the ship,it is the earth that sped away and came back 20 years later,so the twin on earth should have aged less.But when they reunite,one must be younger than the other ,so which one is it?The paradox is resolved when we realize that the ship was the one doing all the accelerating and decelerating,making it a non-inertial reference frame,where relativity doesn't apply.So we must treat the earth as the inertial frame.This means that the twin who left on the ship will indeed return home younger than the other one who stayed on earth.By precisely how much will depend on how fast the ship was moving.As strange as this all sounds,time dilation is only one of the many ramifications of special theory of relativity.

Length Contraction:

        It's not just time that's relative,it's length as well,and as we will find out the math tells us that when you approach the speed of light,your measurement of space changes just like time.This is because if an observer on earth and an observer in a fast spaceship register different spans of time for an event they must also be recording different distances since both observers agree on the same relative velocity between them.Specifically,the faster you go,the smaller objects seem to be,and the shorter the distance you perceive yourself to be traveling.This is a a phenomenon called length contraction.
Here we can see two versions of a spaceship traveling very fast to a faraway celestial objects.In one we see the things from the perspective of an observer on earth,and in the other we see the things from the perspective of someone on board the spaceship.The earthbound observer sees the spaceship moving at some velocity v,notices time moving at the familiar rate,and can measure some distance for the journey which we label as L0 ,and refer to as the proper length,as this is the length that is measured by an observer that is at rest with respect to the objects demarcating the distance.But on the spaceship the only thing that is the same is relative velocity v,though in this case it represents the velocity of Earth and the destination as they move relative to the ship.And because this velocity must be the same as for the earthbound observer,everything else must be different.Because of time dilation,the time interval will be different.,delta t0 rather than delta t,and we label the contracted length of the journey as L.The relationship between L and L is given by the equation as shown in image below,which can be derived from the time dilation equation.
    
Not only do the two observers arrive at different values for the length of the journey,they even arrive at different values for the length of the spaceship,as this dimension is parallel to the direction of travel.The earthbound observer would see the ship as being much shorter than the astronaut does.In fact this discrepancy in length neatly explains how the two observers could perceive different rates for the passage of time.It also explains how fast moving particles can defy certain expectations,because at such speeds,time slows down and the distance traveled contracts.

Mass - Energy Equivalence :

     Time dilation and length contraction are two of the bizarre ramifications of special theory of relativity,but we have one more to get through,and it comes in the form of his most famous equation,which is arguably the most famous equation in all of science.E = mc^2 .Almost everyone can recite this equation from memory,but far fewer know precisely what it means.It outlines the concept of mass-energy equivalence,stating that energy is equal to mass times speed of light squared.One may wonder what the speed of light has to do with energy and mass,especially since light has no mass.Nevertheless,we will find that the two are inextricably intertwined.We know that momentum is equal to mass times velocity,so as velocity increases momentum increases,but as velocity approaches the speed of light relativistic effect take hold,and we have to look at the relativistic momentum.That will involve this additional term here(shown in image given below),which should be familiar from the time dilation and length contradiction equations.

Mathematical formula of relativistic momentum
As velocity increases,it makes v^2 increases 
                                                     c^2                          
and it makes relativistic term decreases and momentum will increase overall.It is the square of this relativistic momentum,known as a Lorentz invariant,that is conserved in all inertial reference frames.At speeds much less than the speed of light,there is only negligible difference between the relativistic and non relativistic momentum,which is one of the reasons that special relativity is imperceptible in our everyday lives.But as an object approaches the speed of light ,it gains momentum exponentially,and the momentum approaches infinity as the velocity approaches the speed of light.This gives us possibly the best explanation for why the speed of light  truly is the universal speed limit,because it would require literally infinite energy to accelerate anything with mass to the speed of light.We can find similar logic in the other equations as well.With time dilation,we can see that time slows down as you go faster,with the limit of something moving the speed of light experiencing zero time.So if you could have the perspective of a photon,time wouldn't exist.To move faster than this would mean experiencing imaginary time,because you would then be taking the root of a negative number,and this can have no correlation with physical reality.
Perspective photon in time

The same goes for length.In the limit of the speed of light,distances contract to zero,so a photon experiences zero space as well.To go faster than this would mean measuring imaginary length,which again is absurd.
Perspective of photon in length

Beyond this,faster than light travel implies problem with simultaneity whereby an effect could happen before its own cause,which violates causality,so we must accept that for all these reasons,the speed of light absolutely can never be attained or surpassed by any massive object,for reasons that are fundamental not just to physics,but to mathematics and logic as well.E = mc^2 has many implications given the incredible conclusion that energy and mass are equivalent.Mass is energy,and incredibly dense energy at that,since to get the energy contained in some amount of matter,you multiply its mass by the speed of light squared,which is a huge number.Mass-Energy elucidates the incredible source of energy that has been exploited during the atrocities of nuclear warfare in World War 2.The conversion of matter to energy via a nuclear process is a mighty force indeed,one that must be met with great maturity.While the atomic nucleus could hold the key to energy needs of future civilization,its misuse can also bring about our destruction.
                                                         
                                             Image of violating causality.
Hope this article helped you to understand about theory of relativity clearly.



  



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