Solve Twin Paradox by Applying Minkowski Space Geometry

According to Einstein’s special relativity, time inflates while length contracts in a moving object with high speed comparable to light speed. So people accepts that if one goes out to outer space he/she will be younger relative to the people on earth.

However, speed or velocity is a relative concept – we constantly fall back to Newtonian world – to think, hence, from the perspective of the outer space traveler, we people on the earth are traveling with an enormous high speed, hence, we should look much younger than the space traveler? To make the point compelling, assuming the two people are twins.

This is impossible, there only has to be one correct answer. Either the traveler looks younger or the earth twin looks younger? The correct answer is the traveler is younger, which is proved by experiments in pseudo similar environment. But the debate over this paradox as well as the explanation is chaotic and long-lasting, until Minkowski Space Geometry is applied to help deduce.

There are these three geometry – Euclid, Minkowski and Rieman Geometry. Euclid geometry is the one we get used to, many may never heard of Minkowski and Rieman Geometry, they are abstract and anti-intuitive, hence hard to grasp at the beginning, yet, powerful in understanding Einstein’s special and general relativity theory.

First, in Euclid world, the diagonal line z satisfies z^2 = x^2 + y^2. However, in the space-time world, what we observe and manifest in math/geometry is that the distance = time*speed.

No matter which reference frame we take, the above equation always holds. if we give the unit of speed light c as 1, then it can be written as

To make the life easier, let’s also simply the situation by letting the object only move on the x axis, so deltay and deltaz parts are eliminated, and we can draw the plot below, note the line with 45 degree is the light’ line because object moving on this line have deltat*speed = deltax, that means the speed has to be the unit 1, that of light. Also, we can infer the deltaS=0.

With this tool, it’s much easier to understand the length contraction. A ruler that stays still on earth can be seen as a dot on the x axis, the corresponding t value is zero. According to the Minkowski equation above, we know at line OB, the line of light, dS = 0, learning toward left side from OB to OD, to OC, the dS only can get larger, hence according to

delta x of OC must be larger than that of OD, that of OB = 0. C represents an object stays still on earth, B object of a beam of light, and D a space traveler.

Once we get this straight up – space-time line S of OC is greater than OD – very counter intuitive visually, then the next key point is to infer spacetime line is equivalent or proportional to TIME. Why?

Same, according to Minkowski’s equation:

In a still reference frame, say on earth, dx, dy, dz can all be removed, ds^2 = -dt^2, space-time distance is all contributed by the time elapsed. Hence, the longer the space-time line, the older you are.

Now we plot the twins as in space-time view

The twin brother who traveled has a smaller space-time line compared to the brother staying on earth, hence, he is younger looking than his brother.

reference changwei jun.

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