As I discussed Ricci Scalar in my last blog, I’ll discuss about a famous paradox related to Relativity. I will discuss the other parts of the Einstein field equations in a simplified way in later blogs in the series. Let’s come to the topic now. Twin paradox as it is called, is one of the important thing you will ever come across when you are making your way through Special Relativity. There are lot of explanations which deals with general relativity, but this problem can be solved with Special Relativity with some modifications which doesn’t change the originality of the problem. The Twin paradox is as follow:
“The thought experiment involves two identical twins, one of whom makes a journey into space in a rocket at velocity comparable to speed of light ”c” and returns home to find that the twin who remained on Earth has aged more. This is considered as a paradox because each twin sees the other twin as moving, and so, according to an wrong application of relativity principle, each should paradoxically find the other to have aged less”
*So let’s see the solution for the paradox with Special Relativity first and I’ll explain it using General relativity. Assume the twin in earth to be called a “strange” and the twin going to outer space to be “Tony”. While leaving to space Tony synchronizes his clock with Strange and set it to “t=o”. Now in Strange’s reference frame Tony leaves the earth and travels at velocity comparable to “c” and turns around and returns. Therefore Strange would think that time went slowly(time dilation) for Tony and hence he is older than Tony meanwhile Tony would think that he was at rest and Strange was the one who moved in the opposite direction and came back. If you had noticed , Strange was in a inertial reference frame throughout the journey ,while Tony had to accelerate to turn around, meaning he was not in an inertial frame anymore which causes the asymmetry and it isn’t really a paradox but a little confusion . But wait, this question was posted before General Relativity was even postulated and we are dealing with accelerations here , so how can we answer it with Special Relativity which deals with only inertial frames(moving with constant velocity and no acceleration’s) and That’s where we have to modify the question a little without losing generality .We can actually describe it with special relativity itself which requires integration of proper time for the world line the traveller follows in space time diagram(FIG 2.1). Now assume that there is a third character named “Thanos” and Thanos is from a nearby planet called Titan which is 20lightyears(just to simplify) away from earth(will be a nearby distance in future J). Now Tony starts his journey from earth with his clock synchronized with Strange (t=o) and travels to Titan. Tony reaches Titan in the same time Thanos leaves to Earth from Titan while doing so they synchronize their clocks . Yes , you are correct if you had noticed that now there won’t be any accelerations taken into account and Special Relativity can be applied with ease. Assume that all this journey was in x- direction. We will get the following Space-Time diagram.
Now if you have seen the relation for time dilation and length contraction respectively ( I’ll discuss it in next blog).
Assume that Tony leaves earth at a velocity of 0.6c which gives Lorentz Gamma value of 1.25 where gamma( Lorentz Factor) is
From Strange’s view Tony will take 33.33 years to reach Titan(20light years/0.6c =33.33 years) and now Thanos synchronizes his clock with Tony so that now Thanos clock measure’s 33.33 years now. Thanos now travels towards earth with same velocity of 0.6c and when Thanos arrives with same calculation above it will take another 33.33 years for Thanos to arrive at Earth according to Strange. Now keeping this aside , let’s do the same process in spaceship’s frame. At start of journey from Earth, Tony will measure the distance between Earth and Titan to be less than 20light years as there is a length contraction which when calculation is done using the formula gives us the distance between planets to be 16 light years. So Tony’s clock will measure 26.66 years (t=16light years/0.6c) and after synchronization Thanos clock will measure the same and the same calculation follows which shows that the clock will forward by another 26.66 years . so when the clock reaches earth it would read 53.32 year while the clock which Strange had will read 66.66 years which resolves the paradox and proves that Strange will be older indirectly. All biological process will be slowed when moving at high velocities.
*Another explanation for this is using Space time diagram shown above(fig1.1) instead of the above calculation’s . The twin in earth travels only along the time and remains stationary in space while other twin travels and turns back and return’s (assume it in x-direction only). The planes of simultaneity represent the simultaneous events for two twins at different times for each other. The Space time diagram for the problem is shown above from which you can see that the simultaneity planes skip over a large part of time of the twin in earth when the outbound twin accelerates (shown as discrepancy in the diagram). So you can easily say that the twin in space has aged less than the Earth twin as his acceleration has caused the time gap in which a whole lot of time passed(enough to compensate) for earth twin, but how does this gap arise ,will answer it with General relativity later in the blog.
With the two explanations above , I would introduce a term called Proper time. Simply proper time is the time measured by a clock moving. The proper time depends on the world line(comes from space time diagram). For an observer stationary his coordinate time and proper time are same. Proper time will be derived from space time intervals which are a crucial part of relativity(derivation in next blog). The pictorial depiction for proper time is as follows:
where t- time measured by stationary observer. Tau- proper time for moving observer.
Proper time is given by:
just the final term is required which is the proper time (which can be thought as the arc length of the world line).
Ill follow (t,x,y,z) convention.
Let’s solve the paradox with the proper time( time dilation can be derived from proper time). With same values as we used in thanos problem, lets solve it in strange’s frame.Stranges’s world line will be a straight line along time direction with intial coordinates (0,0,0,0) and final coordinates(66.66 years,0,0,0). While for the clock carried by tony initial coordinates are (0,0,0,0) and coordinates while synchronizing clock with thanos is (33.33 years, 20lightyears,0,0) (33.33light year is from strange’s frame) . And after return journey the final coordinates will be (66.66 years,0,0,0) . With these values now, the calculation is shown below,
we find that the time elapsed by the clock is 53.32 years which is same as calculated before with the other calculation. Proper time is the best way to calculate the time elapsed by clock moving knowing it’s world line in Space Time diagram.
Having said this through all this calculation we have assumed that the spaceship accelerates to 0.6c from earth in a negligible time so as to neglect those effects and we neglected gravitational fields by the planet’s . But in real situations the space ship will take a lot of time to accelerate. So in those cases the Space time diagram will look like the below one
The world line curved as the space ship accelerates. If you notice there is no instant change of simultaneity plane but a nearly constant change which happens in reality , but still the outbound traveller will skip a little parts of time of earth twinwhich accounts for time gap and hence the less age of outbound traveller. For calculating proper time in these cases, we need to integrate proper time in different intervals, which will give you the total time elapsed by the clock in the space ship which went to space, took a turn around and returned(no need of third character). Having said these I mentioned that I’ll explain how we tackle it using General relativity, which we will see below.
*As most of you know that acceleration is equivalent to gravity, so when the twin who travelled to outer space turns he is an accelerated frame, the travelling twin will think that he is stationary and the earth twin is falling in a gravitational field which he thinks is filled throughout the space. He will see that the earth twin is at a higher potential as spaceship is fired towards earth. The linearized Einstein field equations are an approximation to Einstein’s field equations that is valid for a weak gravitational field and is used to simplify many problems in general relativity .The approximation can also be used to derive Newtonian gravity as the weak-field approximation of Einsteinian gravity. One of the interesting results of linearized Einstein field equation is the Gravitational waves(will discuss briefly in other blog). With the weak field approximation we find a formula for time dilation approximately t’=t/(1+φ/c^2)whereφ- gravitational potential. In our case φ=gd, g-acceleration of the travelling twin, d- distance between the twins. As d is high the earth twin’s clock will speed up relatively that it compensates for the difference time experienced and this also compensates for the discrepancy during simultaneity plane shift and so the travelled twin will be younger. Thus we have explained the Twin paradox successfully. Hope you enjoyed the blog. See you with some interesting paradox and questions in next blog.
Source of some image’s : google images.