Time Dilation, Length Contraction, and the Twin Paradox (truly) Explained

 

Caveat: Misconceptions could be accrued if one hasn’t either read the post "Frame of Reference vs. Observer: Where Relativity Pedagogy Goes Wrong" before OR after reading this, or studied special relativity from other sources offering additional rigor.  

 

 

Many popular science sources have “attempted” to teach special relativistic phenomena without mathematical and conceptual rigor. Although this is achievable in theory, this may or may not be realistic in practice. Moreover, even if successful, since it’d be unlikely the student would be able to solve any relativity problems without additional learning, it would be difficult to verify, especially if required to avoid instilling misconceptions about OTHER aspects of relativity (which popular science generally seems indifferent to). Most of the explanations are pseudo-pedagogical, designed to create the illusion of understanding. This post is an (actual) attempt to achieve where most others have failed.  

 

 

Basic Axioms: 

(Note to novice: This paragraph is very important, but don’t worry if you can’t remember all the axioms off the top of your head; they’ll be repeated when necessary) 

 

One: Non-accelerative motion is relative; so long as he isn’t subjected to a net force and accelerated, an observer doesn’t perceive himself to be in motion even if he moves according to another observer--e.g., a passenger in a car traveling along a highway thinks the GROUND is in motion, not himself. Two: Usually, a frame of reference is merely an observer(s), typically a non-accelerating one, with some location in space; but this post will have its own definition, provided in the next paragraph. Three: The laws of physics are the same according to all inertial (non-accelerating) observers/frames of reference. Four: The speed of light is always measured to be C (in vacuum) by all inertial frames. 

 

(Note to laymen: The paragraph below is mostly a caveat to avoid misconceptions; do read, but don’t worry over the details). 

 

The exclusive “observer” for this post (and almost all relativistic calculations) is what I call the frame measurer, which can be thought of as a frame of reference itself. This is an “observer(s)” standing stationary perpendicular to a moving frame, such as a car or spaceship, at equal distance between the events of interest. This observer is unencumbered by (nonrelativistic) light delays. If two events happen simultaneously from his point of view, another observer standing at rest with respective to him (e.g., another stationary observer on the ground) located closer to one event won’t SEE them happen simultaneously, but that’s only because the light (conveying the image) coming from the other event will have to travel further to reach that observer, causing a “delay.” This is what I call a “nonrelativistic delay” due to the finite speed of light, not relativity. The frame of reference itself doesn’t deal with these delays.

 

Relativistic effects occur when the “constraint of C” violates classical kinematics. For example, in classical physics, when an observer throws a ball across a speeding vessel, the velocity of the ball from the point of view of the ground is simply the velocity from the point of view of the vessel plus the speed of the vessel; if you replaced the ball with a light pulse, however, the pulse would still travel at C, violating classical physics. The velocity of the ball couldn’t be calculated classically, either. 

 

Imagine a speeding spaceship with a light pulse clock hanging from the ceiling. A pulse of light is shot straight down from the point of view of the passenger in the ship. From their point of view, the light goes down, reflects off a mirror at the bottom and returns. This represents one “tick of the clock.” The passenger thinks he’s stationary and the GROUND is moving, and the ship’s clock runs no differently than normal, moving straight up and down, ticking away. 

 

But the ground frame can’t see it this way. The ship is moving incredibly fast, and the mirror at the bottom is speeding AWAY from the pulse. The pulse will reach it eventually, but the ground frame will see the light move diagonally toward the mirror then diagonally back up after reflecting off it. The length of the two diagonal paths is longer than the verticals. Both frames agree on the VERTICAL component of the light’s path, but only the ground frame thinks there’s a horizontal one. And there’s the constraint of C: The speed of the pulse must be agreed upon. Thus, because the ground frame thinks the light travels more distance at the same speed, a tick of the ship’s clock will take longer from the ground frame; in other words, the ship’s clock will be “constricted” from his point of view. Since the passenger thinks the ground is moving at the same speed, the situation would be symmetrical from the ship frame with respect to the ground’s clocks. 

 

THE LEAST YOU NEED TO KNOW ABOUT TIME DILATION: The light pulse in a light pulse clock in the ship must travel more distance at the same speed according to the ground, and this causes the ground frame to measure the ship’s clocks as slowed or “constricted,” as I call it. The situation is symmetrical from the point of view of the passenger with respect to the ground’s clock. This is time dilation or time constriction, which leads directly to length contraction. 

 

If you’re interested in deriving the time dilation formula, Y, use the Pythagorean theorem to show that time from the ground frame, ∆T, as a function of ∆T’, the ship frame, is T’/(1-V^2/C^2)^1/2, with the latter being Y. 

 

Length contraction.

 

Suppose the ship moves along a million-mile ruler on the ground. A million miles is the REST length of the ruler, which you know, but this not the length the ship frame will measure—and this length you DON’T know. The clock in the front of the ship starts at zero and reads a certain time at the end of its trip across the length of the ruler. Suppose the ground frame thinks the ship clock moves slow by a factor of three. There is, therefore, a disagreement about how LONG it took the ship to travel between origin and destination, or each end of the ruler. The ground frame sees the ship’s clock count a third of the time as his clock—in other words, the ground thought the ship’s travel time was triple what the ship’s clock says. But relative speed is equal. Thus, the passenger must see a million miles on the ground as approximately 333,333 miles. 

 

Now I’ll reverse the situation. Last time, you knew the ruler’s rest length, but you wanted to find its length from the point of view of the ship (in standard relativity terms, L prime, or L’, as function of L). Now you’re dealing with a new length between origin and destination—in this case, length of the ship and its bow and stern--and you want to find out the length of the ship from the point of view of the ground (in standard relativity terms, L as a function of L prime, or L’). The ship would see the ground’s clock at the front of the ship moving slowly by a factor of three while moving at the relative speed and would, likewise, realize that the ground sees the length of the ship as contracted.  

 

(You can skip this paragraph if you want to move onto the twin paradox). You can use length contraction to understand the relativity of simultaneity: When two events at separate locations along the direction of relative motion are simultaneous in one frame, they can’t be simultaneous in another. If you replace the ruler with one that has a contracted length equal to the rest length of the ship, such that the passenger thinks ruler and ship are equal length, and repeat the thought experiment above, the passenger sees the alignment of the ship at each end of the new ruler simultaneously. If lightning struck both ends of the ship from their point of view at that exact moment, it couldn’t strike the ship’s ends simultaneously from the ground frame. The ground thinks the ruler’s longer than the ship. Lightning will have to strike the back of the ship from their point of view when the back end passes the beginning of the ruler, which is the only time it can; but since the ruler is longer, the ship will still have to travel the difference in the rest length of the ruler and the contracted length of the ship before the front gets to the location on the ground when the second lightning bolt strikes.  

 

THE LEAST YOU NEED TO KNOW ABOUT LENGTH CONTRATION: The travel time between origin and destination is always longer from the frame they are set/at rest in; resultantly, the frame that sees the origin and destination in motion always measures the length/distance between them as contracted or “constricted” in proportion to time dilation. (Skip if you want to move on to twin paradox) Length contraction leads to the relativity of simultaneity because in order for two events to occur simultaneously in one frame along the direction of motion, the contracted distance between the events in the other frame must equal the rest length of the distance between the events—making the two lengths unequal in the other frame and rendering simultaneity impossible from their point of view.   

 

Differences in the length of the trip, or distance between origin and destination, will be the approach I’ll take to the twin paradox, the idea that someone can’t go on a space voyage at relativistic speeds without returning much older than his/her twin. A summary will follow it. 

 

 

Twin Paradox

The “markers” in this thought experiment are the Earth and a star 10 light-years away. The origin and destination, Earth and star, are set/at rest in the Earth frame, so the distance between the markers will be contracted from the ship frame. This necessarily makes the trip less distance and time from the passenger’s point of view, since speed is equal. And since the ship must accelerate and decelerate, the passenger is irrefutably “the traveler”; in other words, the passenger can no longer claim the ground “took a trip” because acceleration/deceleration is NOT relative (though its MAGNITUDE differs from frame to frame).

 

But isn’t the ship no longer in an inertial frame during these periods of non-relative motion? Yes. But no matter how fast a ship moves or accelerates and regardless of whether it accelerates or decelerates, if you cut the trip up into, say, one-inch intervals along the Earth frame, the velocity during each interval will be nearly constant and, therefore, in an inertial frame (half the speed of light covers 94,000 miles in a second, and there’s 5,280 feet in a mile and twelve inches in a foot). The length between each of the sub-markers will always be less from the ship frame, and, therefore, the time to traverse them will, likewise, be less.     

   

Summarized: 

One: Moving clocks and “lengths” are “constricted”; the latter is determined by the frame the origin and destination WEREN’T set/at rest in and which frame was subjected to a net force, making its frame/observer(s) the undisputed traveler(s). Two: Although the undisputed traveler will accelerate, lengths/distances can be cut up into arbitrarily small subunits to ensure velocity stays approximately constant during these intervals. Three: Since relative velocity is equal and all the length intervals are smaller from the traveling frame, the trip must be shorter from his point of view. 

 

 

Hopefully, this at least gave you an intuitive sense of how basic special relativity works. Following Einstein’s maxim, I tried to make it as simple as possible—but no simpler. The latter may have rendered my objective infeasible—you can judge for yourself. If you had any success reading this, to ensure you didn’t misconstrue anything or want to learn more, look at "Frame of Reference vs. Observer: Where Relativity Pedagogy Goes Wrong."