# The Twin Paradox – Once and for All

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Every science buff has heard of the Twin Paradox. Movies have been made, books have been written, plots and sub-plots all exploring  the same theme, have been aplenty; including the recent hit Interstellar (in the movie, Cooper, the main character in the story, ages much slower than his daughter on earth, Murph).

I have spent so many hours debating this paradox with so many of my geeky friends, that I felt I had to put set some thing to rest. There are a host of misconceptions around the paradox – and Hollywood, IMHO, only adds to the confusion…

### The paradox  – Why is the traveling twin the one that ages less?

Two twins grow up together on earth. At age 20, twin B decides to fly in a spaceship to Andromeda and return to earth immediately thereafter.  If she travels at 0.867 the speed of light (at about 260,000 m/s), she will reach Andromeda in 20 years. By the time she returns (another 20 years), she will have reached the age of 60.  However, her twin on earth, would have aged twice as fast – for each 20 year trip, the earth twin ages 40 years. This means, by the time the spaceship returns, the earth twin is a 100 years old!  While the space twin is still at 60.

If all motion is relative, why does the earth twin not make the same argument (i.e. that SHE, on earth, is the one moving  – and the spaceship, of course is stationary from the spaceship’s reference frame)? It is a perfectly valid argument – and the theory of relativity does say that ‘all motion is relative’, but there are a few subtleties to the theory of relativity. The lack of understanding these subtleties leads to several misconceptions about the asymmetric aging of the twins.

This post will try to elucidate some of the more common misunderstandings around both – the special theory – and the so-called ‘twin paradox’.

### First Things First – Does NOTHING go faster than light?

There are several  – and I mean SEVERAL things that can (and do) move faster than light.  However, none of these can be ‘material’ points (i.e. have any inertial mass).  Think about  the blades of a pair of scissors. If you think about the imaginary point that is the intersection of the two blades – as the blades close in (say at speeds close to ‘c’), this imaginary point moves FASTER than ‘c’. Yes – it moves faster than the speed of light – as do a lot of similar geometrical constructs (take the beam of a large searchlight pointed to space – the beam sweeps out areas at speeds greater than ‘c’). Again, this does not mean any material object is moving / can move faster than light.

Geometric points and relative speeds are always allowed to exceed the speed of light – and to reach any conceivable speed!  It is only material bodies that are subject to the speed limit.

### Passing Space Ships

Before we get to the twin paradox (one earth twin and one traveling twin), let us look at the same twins – but this time, both are traveling . They travel in two separate spaceships – each moving towards each other – at 3/4 th light speed.  Each ship cannot, of course, go faster than ‘c’. However, as viewed from earth, the relative speed of the two ships is 3/4 th + 3/4 th = 1.5 c ! This is not incorrect  – this is the ‘relative’ speed as measured by an earth bound observer. For ship astronauts, the relative speed they measure will always be less than ‘c’ –  however, not so for an earth bound observer. Remember, relative speeds CAN exceed light speed – as long as none of the material objects involved are individually exceeding  light speed.

For two spaceships passing each other:

• Astronaut A (Twin A) sees time move slower in ship B, as well as the length of B contracted!  According to A, B is aging slower!
• Astronaut B (Twin B) sees the SAME thing in A – time is moving excruciatingly slow, A’s length is contracted! According to B, A is aging slower!

They BOTH see the effects of relativity. Neither one can claim that THEY are truly stationary – while the other one is moving. Such a statement makes no sense. They are both moving – and they are both subject to the same relativistic effects. Each twin sees the other aging slower than her own self.

These twins in moving spaceships will serve as a prelude –  to the real ‘twin paradox’.

### Which Twin is really moving?

Back to the original twin paradox (one twin on earth, and one on a spaceship).

The twin on earth claims that the earth is MOVING away from a stationary spaceship – and so – she is the MOVING twin. The spaceship twin argues that the earth is stationary and the ship is the one that is moving. Again, according to relativity, they are both correct. So –  why then, does one twin actually age slower than the other (the one on the spaceship)?

Before we get to the answer, realize that, if the traveling twin actually KEPT moving away from earth for her entire life – then, everything discussed in the spaceship situation above is true. Neither twin would age faster or slower – they would both age the same amount. They would each see each other’s clocks run slower (by the same amount)!  And they would both claim that the other is aging slower. And they would both be right  – because this question ‘Who ages slower/faster’ only makes sense if you COMPARE the two clocks at the end – i.e. if the twins actually re-unite. If they keep moving in their own worlds – and never meet again – this question (who ages more , who ages less) is meaningless.

So – to summarize, if they are BOTH moving, there isn’t really a discrepancy in their aging (at least not one that can be sensibly discussed or measured).  So, what is so special about the fact that the spaceship turns around and returns to earth – versus – it just keeps going forever?

### The Spaceship Acceleration and the Spaceship Turnaround

Most people argue that the astronaut twin ‘accelerates’ – both on the start of the journey and on the start of the return trip – in order to reach ‘cruising speed’. This acceleration shifts the spaceship into a non-inertial frame – which is substantially different from the inertial frame of the earth twin. This ‘shifting’ is what causes their clocks to get out of sync – and run at different rates.

This argument is only partly correct. This is because we can avoid the whole ‘acceleration’ business and still end up with the same paradox.

Imagine that the ship (with the traveling twin on-board) started out elsewhere – and reached ‘cruising speed’ – as it passed earth. Now, the twins compare clocks as the traveling twin passes earth. So – no acceleration from the time that the two twins first synchronized their clocks.  Now, imagine the same thing on the return –  when the traveling twin reaches her destination, she JUMPS aboard another ship traveling towards the earth – again at the same speed. This means – both on the forward journey – and the return journey – the traveling twin’s ship experiences no acceleration. And yet, on return, she sees the same aging difference (earth twin is substantially older).

So – acceleration at start and end of the journey  – is part of the answer – though not quite the full story!

### Who is really moving (relative to the Universe)? – The Solution to the paradox…

The correct question to ask is not ‘ Who is really moving relative to the other?’ (cause they both claim that they are the moving twin – and they would both be right).

Rather, the correct question is:

Who is really moving relative to the Universe?

Once we ask this latter question, their relative motion becomes unimportant – and a few answers start emerging.  The earth twin’s clock is essentially keeping the same time as all other clocks on earth – since they are in the same relationships  to the rest of the universe; i.e. for all purposes, the earth stays stationary relative to the rest of the Universe. It stays stationary for the entire trip of the traveling twin.

The same is NOT TRUE of the traveling spaceship. A clock on the spaceship does not keep the same time as a clock on earth – because the spaceship MOVES relative to the Earth – and relative to the Universe!

Now, one could reverse things and argue that – well, so – if we treat the spaceship as stationary, then it is EARTH that moves and EARTH that executes the ‘turnaround’. And that is true – except for one important distinction. When the EARTH moves, the entire UNIVERSE moves with it!

That is, from the ship’s reference frame, the earth and the Universe are part of the same inertial system – that either stays fixed – or moves as one unit.

Now, it should be clear that the spaceship is NOT an inertial frame – at least for the start – and the turnaround portions of it’s journey (since it is accelerating w.r.t the Universe).  And Einstein taught us that, in non-inertial frames (in accelerating frames),  clocks slow down. In fact, even the simple act of being in a stronger gravitational field (non-inertial frame) slows clocks down.

So – a clock runs slower on the moon that it does on earth. And it runs slower at the bottom of Mount Everest than it does on top (since gravity is slightly stronger at earth’s surface than it is on top of Everest).

Given Einstein’s explanation of clocks slowing down in non-inertial frames, it should make sense that the astronaut’s clock slows down – at least during two special periods – that of journey initiation – and that of turnaround.

This is one way to look at it.

### Worldlines

Another way to look at it is in terms of what are known as ‘wordlines’ – which are simply the path an object follows in spacetime (as opposed to just space). When the astronaut executes her turnaround, she shifts her worldline. The shift is such that her resulting worldline is SHORTER than the one of the stay at home twin. This shorter worldline is exactly shorter by the same amount of TIME that is the difference in the aging of the twins.

### But, But, you just said that Acceleration is not the real reason, and now you are saying acceleration IS the real reason for the time difference ?

Acceleration, or more appropriately, the switching to a non-inertial frame, IS the real reason that the moving twin’s clock slows down. However, to revisit the scenario where there isn’t any acceleration – and the traveling twin simply ‘jumps ship’ to get back to earth, there is still an important thing the traveling twin does.  In order for her to execute her ‘jump’ to the earth-bound ship, she has to again switch inertial frames! There isn’t any alternative. So – her case is no different from the twin that simply traveled out and returned on the same ship. The switch in inertial frames needs to occur in both cases, in order for her to get back to earth.

And this switch is what slows the clock of the traveling twin down!

### What if the spaceship does NOT turnaround?

This is an interesting question – and the answer is ‘The question doesn’t make any sense’. To ask which twin aged how much when there is no ‘comparison’ at the end of the story in meaningless. They each age according to their own world clock – and in this scenario, they would both age the SAME amount (as per the equations of relativity). However, there is no way to measure this SAME amount – since there is no reference frame in which this comparison can be made (since they never re-unite).

### Mach’s Hypothesis – What if there were NO objects in the Universe?

A more interesting question can be asked ‘What if we take the Universe away’?

What if there is NOTHING else in the Universe – except for these two twins (one of them on a moving spaceship and the other sitting inside a stationary pod)? Now, what would happen to their respective ages as the traveling twin traveled out to space – and back again to meet the twin the pod?

Well – according to Mach (a compatriot of Einstein), there is no way to tell what would happen. As per Mach, inertia (our understanding of inertia) only results because there is such as thing as ‘the rest of the Universe’. Without any objects to create gravity (and hence, create spacetime),  objects would feel no inertia. Hence , it doesn’t make sense to talk about the inertial or non-inertial effects on an object in the absence of a Universe.

### What use can you put this result to?

Suppose you fall for someone half your age (say you are 40 and she is 20) ; All you have to do is travel close to ‘c’ for a couple of years – and return to earth. You would be simply 42, whereas she would be closer to 40 !  Again, this all sounds fantastic, but is all completely true – and has been measured over and over again (usually, using something called the Mossbauer effect).

### What WOULD happen if the spaceship actually attained the speed of light? What would the Universe look like to an astronaut inside the ship?

Again, I would like to stress that this is not possible. The closer the ship gets to ‘c’, the more massive it becomes (it’s relativistic mass grows larger and larger). This implies that more and more energy is required to propel it by the same amount as before.  Just before approaching the speed of light, an INFINITE amount of energy would be required to propel it to reach ‘c’. That makes it theoretically and practically impossible for the ship to ever attain the speed ‘c’.

However, let us, for the sake of a thought experiment, assume that we succeeded in getting the ship moving at ‘c’.  What would spacetime look like to an astronaut in the ship?

• The spaceship would appear to have infinite mass and zero length (as measured by someone on earth).
• Time (Universal Time) would be at a standstill (i.e. nothing in the Universe would ever age).
• Every star, every galaxy would appear to have zero length – that is – would appear ‘flattened’ (this has been pictured in Star Trek etc. fairly well).
• The astronauts themselves, in the ship, feel no different – they age normally, they do not actually reduce to ‘zero length’ etc.

As a related paradox (and one popularized in Back to the future), read this post on the Grandfather paradox.

1. Solution 1 to the grandfather paradox (from the post above) – This involves the son (let’s call him Marty) becoming his own grandfather. After going back in time and killing his young father (George), Marty then sires a child who later becomes his father, another George.
2. Solution 2 – The second solution works if the father also has the ability to travel in time.  In 1954 Marty’s father George travels forward in time one year to 1955, when he impregnates Marty’s mother Lorraine before immediately returning back to 1954 – just as his future son, Marty, arrives and kills him.

### Summary

This post is an attempt to clarify some of the most puzzling aspects of the special theory of relativity – as highlighted by one of it’s most famous thought experiments – that of the asymmetric aging of twins.  I am done debating this issue (with friends)….and would like to get back to my regular worldline now

If you have something you would like to add to the conversation, please feel free to comment…

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