Because observers on and off the train experience simultaneity differently, they also experience time differently. In SR, there is no absolute time, only time as experienced from a particular reference frame. To understand how time may be different to different observers we need only accept that the speed of light is a constant in all frames of reference and be able to apply the Pythagorean Theorem. Let’s stay with Ms. Pink on the very rapidly moving train and Mr. Green on the side of the tracks. Ms. Pink has a clock mounted so both she and Mr. Green can see it. She also holds a flashlight on the floor of the train, pointing it straight upward to the ceiling, a vertical distance, x away. She turns on the flashlight and observes the path of the light. To Ms. Pink, the light travels straight up to the ceiling covering a distance of x.

 

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Mr. Green, standing by the tracks sees Ms. Pink (and he is beginning to become fond of her by now) turn on the flashlight and also watches the path of the light and the clock. To him, because the train is moving, the light travels up at an angle produced by the two motions of the light and the train, and travels back down again at the same angle.

 

The light travels a much longer distance for Mr. Green. But Einstein said that the speed of light, c, is fixed – it is the same no matter what reference frame you are in. So lets analyze this using the first equation in NASA Physics. Do you remember?

 

                                               v = d/t  (Eqn 1.1)

 

Here v = c and since c is unchanging, v for Ms. Pink must equal v for Mr. Green. So, because the distance the light travels is longer for Mr. Green, the time it takes must be longer too. For Mr. Green, the time he observes on the moving clock is stretched out, a phenomenon known as time dilation.

 

Let's call the time for the light to go from the floor of the train to the roof, as measured by Ms. Pink on the train, t_o. The equivalent time measured by Mr. Green on the side of the tracks is t. Now, rearranging Eqn 1.1 to d = vt.  The vertical distance traveled by the light, x, equals cto and the horizontal distance, the distance traveled by the train, just equals vt where the train is moving with a velocity of v. Then remembering the Pythagorean Theorem; C² = A² + B², we can derive the relationship between the two inertial reference frames, t and t_o:

 

​    t = t_o / sqrt (1-(v² / c²))

 

The factor 1 / sqrt (1-v² / c²) is called gamma (ɣ),

​ ​   so          t = ɣt_o

 

 

 

This is the relationship between the dilated time, t, (measured by Mr. Green) and the time t_o measured by Ms. Pink. But what does this mean?

 

Looking at equation 7.5, as v approaches c, the quantity v² / c² approaches 1 so the denominator approaches 0, and ɣ gets very small. So, the faster the train goes, the slower Mr. Green sees Ms. Pink’s clock running.

 

Another interesting outcome of this equation is that v can never be larger than c. When v tries to be larger than c, 1 - v² / c² becomes a negative number and, taking the square root, gamma becomes imaginary. This puts an upper limit on the speed of an object. Also notice that when v = c, ɣ= 1 / 0, which is infinite. So, time would appear to stop to an observer watching a train or spaceship pass them at the speed of light. Also, to accelerate anything to the speed of light, would require an infinite force, which is clearly impossible. So now you know why nothing can move faster than the speed of light, and why time slows down as a velocity of an object approaches the speed of light.

 

Now notice that these effects are only observed in clocks in OTHER references frames. You always see YOUR clock running at the expected rate.  What is even more strange is that while you see the clock in a speeding rocket ship moving slower than yours, the astronauts in that ship sees your clock running slower than theirs. Weird!

 

Try It!

A space ship is traveling at 0.8c with respect to you. You can see the clock inside the spaceship and can see your own clock. How much slower is the space ship clock relative to yours?

 

t = t_o / ɣ  =  t_o / sqrt (1 – v²/c²)

   =  t_o / sqrt (1 – 0.8²/1²)

   =  t_o / sqrt (1 – 0.64)

= t_o /sqrt (0.36)

   =  t_o / 0.6 = 1.67 t_o

So the clock on the spaceship ticks only once for every 1.67 ticks on the ground. Time slows down on the spaceship.

 

Try It!

Now do the same calculation for:

 

A spacecraft traveling at 0.95c.

 

Answer: t = 10 t_o; ten minutes on the ground is equivalent to 100 minutes on the spaceship

The Ageless Astronaut

You are probably familiar from science fiction stories of the differential aging of people on Earth compared to those traveling on a near light-speed spacecraft. From Eqn. 7.4 we calculate that an astronaut travelling at 95% of the speed of light for a 10 year tour of the galaxy will know no one when she returns for her friends on Earth would have aged 100 years for her 10. A long space voyager at high speed means farewell to those left behind.

 

39 by Queen

Here are the lyrics for a time dilation song, written and sung by Brian May, an astrophysicist musician.

 

Big and Littler

Length and mass are also affected by relative motion in inertial reference frames. Both can be described using the ɣ  factor. In May, 1969, the Apollo 10 service and crew modules (the fastest manned vehicles ever) managed to attain a velocity of a whopping 0.000037c! The crew - Thomas P. Stafford (Commander), John W. Young (Command Module Pilot), and Eugene A. Cernan (Lunar Module Pilot) experienced a  γ = 0.00000000068.  So from the perspective of the ground crew, the 11.03 m long module shrank by approximately 7.5 nanometers (7.5 x 10^-9m).

Space Time

Around the turn of the last century a German mathematician, Hermann Minkowski, in attempting to describe the geometry of the new SR universe, developed a space-time diagram that became known as Minkowski Space or a Minkowski Diagram. Minkowski realized that space and time are inherently linked in a quantity called space-time. Minkowski space is defined by both space and time and so represents all events past, present, and future in the visible 3D and invisible universe.

 

In the figure at right, time travels forward as you go up the vertical axis, and backward as you travel down along this axis. Space travels along a horizontal plane. We live inside the “light cone” bounded by the slanted line representing the speed of light. The place where the two cones intersect is right now. Above that plane is the future, below, the past. You can easily draw these cones because speed is just distance over time which are the two axes of this graph. So, our observable universe is represented by all space and time bounded by the speed of light. Outside that cone is beyond our ability to see or communicate with because the communication would have to happen at faster than the speed of light. Notice that to define any point on the 3D graph, you must use both a space coordinate (represented as “light seconds”) and a time coordinate (represented as seconds). Space or time by itself has no real meaning. So, the concept of space-time is appropriate.