Space Time & Relativity

Intuitive approach

Everything we see is a perspective, not the truth

The electromagnetic waves (visible light) have finite speed, and there is always some delay, by the time light reaches from an object to an observer.

An observer close to an object will see it almost at real time. Whereas, another observer at a very far distance ( say, one light year away) will see the same object as it used to be in the past (i.e. a year ago).

Thought Experiment

Color changing object

Consider one magical object has property to change its color at every one second. It chooses one unique color each time from the infinite color bands.

An observer standing in front of it will see the object changing color - an unique color at every one second.

Rainbow effect

Consider that magical object is now stretched to a very long size. Its length becomes a light year long ( yes, it's magical!).

Light from the far end will take one year to reach to the observer. Different points along the object length will take different time to reach to the observer.

At any given moment, observer will see different color ( light from different timestamps) at different points on the object. Though, entire object has only one color at any given time, the observer will find rainbow pattern on the object.

Visible length

Consider another magical object that grows in length at the speed of light (almost!). In one year, its length will become 1 light year.

An observer can only see half the object ( half light year long). Lights from other half of the object have not yet reached to the observer.

Time dilation

Clock at rest

Depending upon the interval size between each tick-tock, time appears to move slow or fast.

Clock at rest (with respect to observer's reference frame) will appear ticking at normal rate

Clock in motion

The perceived interval between each tick-tock will increase when clock is in motion.

The Clock will tick at one position and then tock at another position. Observer will have to wait a little longer for the tock, as light has to travel some extra distance to reach to the observer.

That will make the observer feel that time is ticking at slower rate in the moving clock.

Length Contraction

Object at rest

An observer perceives length of an object when lights from both the ends reach. It does not matter even if lights, starting at the same time from both the ends, reach at different time. Continuous flow of light from same points help visualizing the length of the object.

Object in motion

Object in motion appears shorter to the observer in the direction of motion.

Consider an object of length L1 (when at rest) is moving right to left. The object is at position T1 (shown one end as green and other end as red). A few moments later, new position is T2 (shown both end in green and red)

What length will observer visualize when object is at T2?

When lights from near end (green part at T2) reach to observer, lights from far end (red part at T2) are still travelling and have not yet reached. But, there would be some lights from the past (say T1), coming from the far end (red part at T1), which are just reaching. Hence, the observer will see lights from green part at T2 and red part at T1 together. This will make the object appearing shorter (length L2).

Asynchronous nature of clock

Clocks synchronization at rest

Clock at center resets its time to 0 and fires light signal towards both the clocks. Light takes T ticks to reach at both the clocks. As signal arrives, both clocks reset its time to T. This synchronizes all the three clocks.

The clocks at rest are synchronous.

All the clocks at rest have the same time at different location (x)

The clocks in motion lag behind

The clock in motion ticks slower than the clock at rest due to time dilation.

At top, A clock is moving through a path ticking at slower rate. At bottom, there are many stationary clocks (synchronous) shown along the path. They are ticking faster.

Clocks synchronization in motion

All the three clocks are moving at speed V. Clock at center resets its time to 0 and fires light signal towards both the clocks.

Light reaches left clock before time as clock is moving towards the light. Light has to travel lesser distance. Left clock resets itself and it becomes ahead of center clock.

Light reachs right clock much later as clock is moving away from the light. Light has to travel more distance. Right clock resets itself and it becomes behind of center clock (and left clock too). All the three clocks are out of sync.

Summary

The clocks in motion are asynchronous and slower than the stationary clock

The leading clock (in direction of motion) lags behind. The clock trailing the motion are ahead in time.

On top of that, all of them are behind the stationary clock

The big picture

When one observer is in motion from second observer's point of view, first observer's clock will move slow. From first observer point of view, second observer is in motion and hence, second observer's clock will move slow. Both can claim that other's clock is running slow. Contraction?

Example

On a race track of length L, superman (S) is running with speed V from left to right. Two observers (R1, R2) are there, one at starting position and other at the finishing line.

How much time will it take to complete the race from each person's clock?

How much time will each person think, it would take from other person's clock?

V = (12/13)c

L = 156 light-minutes

R1 clock from R1 perspective

R1 is stationary. S is in motion with speed V and runs for distance L.

Time lapsed in R1 clock = L/V = 169 minutes

R2 is also stationary from R1 point of view. R2 clock will also tick 169 minutes at the end of the race.

S clock from R1 perspective

From R1 point of view, S is in motion. Hence, S clock moves slower and it should take less than 169 minutes.

Time elapsed in S clock (as per R1) = 65 minutes

S clock from S perspective

S is stationary and rest of the world (R1, R2, race track etc) is rushing by S.

Object in motion is length contracted. From S point of view, Track length will be shorter (L'). Track will appear to be moving with speed V

Time elapsed in S clock = L'/V = 65 minutes

R1 clock from S perspective

R1 is in motion from S perspective. Hence, R1 clock moves slower and it should take less than 65 minutes

Time elapsed in R1 clock (as per S) = 25 minutes

R2 clock from S perspective

Like R1, R2 is also in motion from S perspective. Hence, time elapsed in R1 clock should be 25 minutes.

Wait! There is more.


R1 and R2 both are in motion from S perspective. Clocks in motion are asynchronous.

From S point of view, R1 & R2 are moving right to left. Leading clock (R1) lags behind and trailing clock (R2) is ahead of time.

time difference between R1 & R2 ( as per S) = 144 minutes.

At the beginning of the race, R2 clock was already at 144 minutes ahead.

When R1 clock ticks 25 minutes, R2 clock will tick 144 + 25 = 169 minutes.