# Physics Tutorial: Space and Time in Einstein Theory of Relativity

In this Physics tutorial, you will learn:

• Why the classical theory of relativity is not suitable to explain relativistic events?
• What contradictions of the classical theory of relativity did Einstein's Special Theory of Relativity addressed?
• What are synchronized clocks? Why clocks' synchronization is important in relativistic events?
• What happens to the time in relativistic events?
• What happens to the length of objects in relativistic events?
• In which direction do the change in time and length during a relativistic event occur?

## Introduction

Do you think time flows equally in all systems? What about length of objects?

What if you were able to perform your daily activities at very high pace? Could you do more or less activities in a day compared to situations in which you act at normal pace?

When an object moves very fast it looks thinner. Do you know why this happens?

This tutorial is focused on the relativity of time and length - two important pre-requisites to understand the Special Theory of Relativity formulated by Einstein.

## Einstein Postulates and Relativity of Simultaneity

The information available on Relativity before Einstein (including the experimental analysis and other observations made until then) led to a conceptual confusion, which lasted for many years. First, the propagation of EM waves in vacuum without the need for a material medium is a fact, as sunlight comes to our eyes after travelling a very long distance in vacuum. If the pseudo-concept of cosmic ether is used to assign a medium to the empty space, there is a big problem, as the relative motion of light to this ether cannot be detected. On the other hand, if we abandon the use of this concept, we obtain a finite speed for light, i.e. light must have the same speed in all systems of reference, the equation of which, is

c = 1/ε0 ∙ μ0

However, the Galilean transformations contradict this claim. In 1905, Einstein (only 26 years old in that time), found the solution, bringing a revolution in Physics. He started this revolution from the two postulates we discussed in the previous articles:

1. All physical phenomena (mechanical, optical, etc.) occur in the same way in all inertial systems of reference.
2. The speed of light in vacuum is the same in all inertial systems of reference.

From the first postulate, it derives that we are not able to determine whether a system is at rest or it is moving; this can neither be achieved by analyzing the mechanical phenomena nor optical ones. The speed of light is not involved in any of calculations.

The second postulate is even stranger and unexpected. The common understanding on relativity is based on the concept of a changeable speed that depends on the system of reference used to describe the motion. According to Einstein, this is valid for all material objects but not for the light. Obviously, the Newtonian concepts on space and time are not valid anymore; another approach must be used to describe this part of physics - concepts that form the framework of Einstein's theory, known as the "Special Theory of Relativity".

Let's explain this theory through the help of a "thought experiment". A bus is moving at a very high (constant) velocity V on a linear and very long railway, as shown in the figure below.

The bus has two doors D1 and D2 (one at either end), which open by means of photo-relays, i.e. they open from inside only after receiving a light signal released from the source A, which is located at centre of the bus. We consider two inertial systems of reference: S (related to the ground, the centre of which, is the point O), and S' (related to the bus, the centre of which, is the point O', i.e. the leftmost lower point of the bus).

Let's consider an observer inside the bus. From the isotropy of light - a property we discussed in the previous tutorial - two light rays emitted simultaneously from the source A in opposite direction, will reach the doors D1 and D2 at the same time, and as a result, a simultaneous opening of the two doors will take place. This is because light travels the same distance at the same speed (the source A is equidistant from each door).

As for another observer at rest outside the bus (connected to the system S), he will see another version of the event; since the bus is moving due right, the light ray emitted from the source A due left, will reach the door D1 earlier than the opposite ray emitted from A towards D2. This is because the distance from A to D1 now is shorter than the distance from A to D2, as shown in the figure. In this way, the doors do not open at the same time.

From the above experiment, it is clear that the simultaneity of the events observed is relative - a conclusion that contradicts the Newtonian concept of absolute time, which was believed to be equal for all inertial frames of reference. In fact, time simultaneity is nothing else but a special case of non-homogenous time, in which the difference of times for the same event measured by two observers in two inertial frames of reference is zero.

From this viewpoint, Einstein in his theory formulated the idea of different-flowing times in different inertial frames of reference. There is no absolute time anymore. The time flow that characterizes the chronology of events is an inner property of a given inertial frame of reference. For example, when we speak about two events occurring at a time difference (interval) of 1 s, we have not provided a complete information if the inertial system of reference related to the events is not explicitly declared.

Giving that now the existence of cosmic ether is already rejected, it is meaningless to speak about an absolute space. Indeed, the existence of a space without any object or process in it used as reference frames for position coordination (up-and-down, left-and-right, back-and-forth), is unthinkable. Likewise, an absolute time, without objects and chronological references (earlier-or-later) cannot be imagined. Therefore, the only acceptable theory is the relativistic approach of Einstein, known as the Special Theory of Relativity.

## Inertial System in the Special Theory of Relativity and Synchronization of Clocks

Let's consider again the concept of inertial system, especially in regard to the measurement of time interval between two physical events. A physical event is an event that occurs somewhere in space, at a given (time) instant. These two concepts (space and time) have a meaning only when we provide a sufficient information about the frame of reference in which the questions "Where?" and "When?" get answer. Based on the first postulate, all inertial systems are equivalent.

Like in the classical theory of relativity, we have a three-dimensional of origin O and three perpendicular Cartesian axes X, Y and Z, which allow us to determine the position of any object by measuring the three basic positions using a standard of measurement (a known unit). Obviously, the same physical event occurs in different coordinates to the system S when measured in another inertial system S'. Now, we cannot say anymore that the time of event occurrence is the same in both systems, as in the case of the Newtonian system. Hence, a procedure for measuring the time that is typical for a given system, is necessary. This time is t for the system S and t' for the system S' where t' ≠ t.

In Einstein's theory, the system of synchronized clocks is used to describe one or more relativistic event/s. These are perfect imaginary clocks, placed at every point of space (of the universe); yet, every inertial system has its own local clocks, which are stationary relative to the system itself. In this sense, we must imagine that the clock systems of two inertial frames of reference intertwine with each other during the relative motion. When we say a particle is "born" somewhere, physically this means the particle is generated in the point of space of coordinates (x, y, z) for the system S and (x', y', z') for the system S'. It is obvious that in the place where the even occurs, there are two standard clocks: one is the clock of the system S and the other is the clock of the system S' which record the times t (for S) and t' (for S') for the same event. All the fixed clocks of the system S show the same time t, as well as those of the system S' (which all show all the time t'), as all clocks within the same system are synchronized. The question that arises here is: "How the synchronization of clocks inside the same inertial system is done?" Let's use the following figure to answer this question.

In the figure above, a central clock fixed at centre O of the system S and another clock at the point P(x, y, z) are shown. All clocks of the inertial system S (including the two clocks shown in the figure), are stationary relative to each other.

The central clock contains a mechanism which when operated (starting from t = 0) emits a light signal in all directions. On the other hand, the clock at the point P is settled forward to show the time τ = l / c prior to the event occurrence. This clock contains a mechanism that makes it start working when receiving a light signal (in this case, the light signal comes from the origin O). It is obvious that the clock located at P start recording time (from the value τ) when the light signal moves by τ seconds from the origin. The two clocks will henceforth show the same value (we say they are synchronized). This procedure is used to synchronize all clocks of the system S, as well as those of the system S' (using a central clock located at O').

Everything discussed so far is simply imagination and theory; however principally they have a great importance to explain the time difference between two events or the same event when viewed from two inertial frames of reference. We should keep in mind that all events are absolute as objective facts, however, their time (and place) of appearance is different in different inertial frames of reference. Thus, the same event is shown at coordinates (x, y, x, t) in the system S and (x', y', z', t') in the system S'.

### Example 1

The closest star to the Earth is Proxima Centauri, which is 4.26 light years (a unit of distance showing the distance travelled by the light in one year through vacuum) away from us. Suppose that just in the instant you are reading this text, we observe a process in this star. What time would a clock located at the surface of Proxima Centauri show at the instant of event's occurrence if this clock is synchronized with our clock?

### Solution 1

The process we observe now, has occurred in the given star 4.26 light years ago (the time needed to the light to reach our sight). Since our clock shows the value 0 at the occurrence of the event, this value is also shown by the clock located at the star. However, to achieve this, the clock located at the star must be settled forward by

τ = distance/speed of light = 4.26 years

In other words, when the event occurs, the clock at the star must show the value t' = - 4.26 years (minus 4.26 years).

This means the time measurement using the clock at Proxima Centauri shows the value 0 when light reaches the Earth (4.26 years after the event's occurrence).

## Dilation of Time and Contraction of Length

### a. Relativistic Dilation of Time

In the previous paragraph, we saw that the simultaneity of events is relative; two events that may be synchronous in one system may not be synchronous in another inertial system (the thought experiment with the bus). In addition, we explained that the time itself is not absolute but relative (t ≠ t').

Now, we will compare time intervals between two events in two different inertial frames (systems) of reference. For this, we use the time measurements taken in a physical event described below.

A wagon is moving at constant velocity due right. We denote the inertial system connected to the wagon by S'. There is a light source A (for example a laser) at bottom-middle of wagon and a receiver R and a standard clock O' very close to the position of the light source.

A mirror M is placed at the ceiling of wagon, at the height H from the floor. When a light signal is emitted vertically upward by the source A, it is reflected back by the mirror M and falls to the receiver R. During this process, the time interval Δt' measured by the clock O' is

∆t' = 2H/c

where c is the speed of light.

Now, let's see what measures an observer outside the wagon, who is at rest with the tracks (the inertial system S). If the train (including the given wagon) moves due right at velocity (V), the light path is not anymore only vertical, as due to the horizontal displacement of the train. Obviously this velocity must be large enough to be considered, i.e. it must be comparable to the speed of light.

In the following figures, there are three instants included: the first figure shows the instant in which the light signal is emitted by the source A, the second figure shows the instant at which the light signal reaches the mirror M and the third figure shows the instant in which the reflected light reaches the receiver R.

Since the x-coordinate (in the system S) of source, receiver, and mirror (including the clock inside the wagon) change with time in the direction of motion, the two events - the emission of light signal and its receive from the receiver - do not occur at the same point of the space (in the system S). Therefore, the light path is not vertical despite it is still moving at speed c. This new path is longer than the (vertical) path followed by the light ray when viewed from inside the wagon (in the system S'). The time interval (recorder by the three synchronized clocks shown under the wagon) for the light ray to travel the path source A - mirror M - receiver R is Δt.

Since the train is moving at uniform velocity V, we have:

∆t = 2L/V

where 2L is the horizontal distance travelled by the train during this event. On the other hand, from Pythagorean Theorem, it is clear that the distance (path) d travelled by the light ray during this process is:

d = √L2 + H2

We are considering the same event; so the time interval Δt can also be expressed as

∆t = 2d/c

Combining the last three equations, we obtain (giving that H = c · Δt'/2)

∆t = 2 ∙ √L2 + H2/c
= 2 ∙ √(V ∙ ∆t/2)2 + (∆t' ∙ c/2)2 /c

Raising both sides of the above equation in power 2 to remove the square root, we obtain

(∆t)2 = 4 ∙ [V2 ∙ (∆t)2/4 + c2 ∙ (∆t')2/4]/c2
(∆t)2 = V2 ∙ (∆t)2 + c2 ∙ (∆t')2/c2
(∆t)2 = V2 ∙ (∆t)2/c2 + c2 ∙ (∆t')2/c2
c2 ∙ (∆t)2 = V2 ∙ (∆t)2 + c2 ∙ (∆t')2
c2 ∙ (∆t)2 - V2 ∙ (∆t)2 = c2 ∙ (∆t')2
(∆t')2 = c2 ∙ (∆t)2/c2 - V2 ∙ (∆t)2/c2
(∆t')2 = (∆t)2 - V2 ∙ (∆t)2/c2
(∆t')2 = (∆t)21 - V2/c2
∆t' = ∆t ∙ √1 - V2/c2

Obviously, the expression within the square root is less than 1 because V < c and V2/c2 is a number between 0 and 1. This means the value of square root itself is a number between 0 and 1 as well. This means Δt' < Δt. In other words, the time measured by the clocks inside the wagon (system S') measure a smaller time (work slower) for the same event when compared to the clocks placed outside the wagon (those of the system S). In other words, any time interval that measures the duration of an event in an inertial system considered at rest, is 1 - V2/c2 times shorter than the corresponding time interval for the same event when it occurs in an inertial system moving at velocity V relative to the system at rest. Therefore, we say the time dilates (expands) when the event duration is measured in an inertial system moving at velocity V in respect to the system considered at rest.

Einstein gave a funny explanation when repeatedly asked about the relativity of time. He said: "When you sit with a nice girl for two hours you think it's only a minute, but when you sit on a hot stove for a minute you think it's two hours. That's relativity."

### Example 2

A spaceship moving at 0.4 c goes to a star which is 10 light years away from Earth and turns back without staying any second there. Calculate:

1. The time of this trip measured by a clock placed on Earth surface
2. The time measured for the same event by a clock placed inside the spaceship

### Solution 2

1. The event measured by the clock on the Earth surface lasts for
∆t = 2d/V = 4 l.y. + 4 l.y./0.4 c = 8/0.4 = 20 years
2. When the same event is recorded by using the clock inside the spaceship, we obtain for the duration of this event
∆t' = ∆t ∙ √1 - V2/c2
= 20 yrs ∙ √1 - (0.4 c)2/c2
= 20 yrs ∙ √1 - 0.16 c2/c2
= 20 yrs ∙ √1 - 0.16
= 20 yrs ∙ √0.84
= 20 yrs ∙ 0.9165
= 18.33 years
As you see, the time flows in different pace in two inertial systems moving at speed relative to each other.

### b. Relativistic Contraction of Length

Now we will see that the length of objects or paths measured in two inertial systems of reference is also relative, unlike in the classical theory of relativity. Recall the Galilean transformations, in which the distance between two points (or the length of an object) doesn't change, whatever inertial system you choose to take the measurements - transformations that now look outdated.

Let's consider again a wagon moving at velocity V along a horizontal path (tracks) as we did in the previous paragraph. At the left end of wagon there is a light source A which emits light rays in the horizontal direction. At the right of wagon there is a mirror M, which reflect the rays back to the source A.

Obviously, the y-direction here is useless as the light source emits signals only in the horizontal direction. Any observer inside the wagon (in the system S') measures a length of L' that corresponds to the distance AM. It is clear that the distance travelled by the light ray viewed from inside the wagon is 2L', where L' is the length of wagon measured from inside it. In addition, all clocks inside the wagon show the time Δt' for the event that regards the emission of a light signal and its return to the receiver (located at A) after being reflected by the mirror M.

Now, let's consider an observer at rest outside the wagon (in the system S). If we synchronize the clocks of the two systems to show the values 0 at the moment of the light ray emission, we have the value xA(0) for the initial position of light source and xM(0) the initial position of mirror. Therefore, the length of wagon viewed from outside the wagon is L = xM (0) - xA(0). When the light signal goes to the mirror M (after a time interval Δt1), the new x-coordinates of source and mirror will be xA(Δt1) and xM(Δt2) respectively.

After the time t' has elapsed, the source is at the new coordinate xA(Δt1) and the mirror is at xM(Δt1). For the observer connected to S, the distance travelled by the light signal during its motion from source to mirror is L + V · Δt1. Therefore, we have

c ∙ ∆t1 = L + V ∙ ∆t1

or

∆t1 = L/c - V

Now, let's denote by Δt2 the time (measured from S) needed for the light source to move from mirror to receiver (adjacent to source). Using a similar reasoning as for Δt1, we obtain

∆t2 = L/c + V

Thus, the entire interval Δt needed for the light ray to move from the source to the mirror and back to the source again, is

∆t = ∆t1 + ∆t2
= L/c - V + L/c + V
= L ∙ (c + V) + L ∙ (c-V)/(c-V) ∙ (c + V)
= 2L ∙ c/c2 - V2

Dividing both numerator and denominator by c2, we obtain

∆t = 1/c2 ∙ 2L ∙ c/1/c2 ∙ (c2 - V2 )

Hence,

∆t = 2L/c ∙ 1 - V2/c2

Now, let's compare this value with the time interval Δt' obtained for the same event but viewed from inside the wagon (in the system S'). Obviously, this time interval is

∆t' = 2L'/c

In the previous paragraph, we found that

∆t' = √1 - V2/c2 ∙ ∆t

Therefore, we have

2L'/c = √1 - V2/c2 ∙ ∆t
2L'/c = √1 - V2/c22L/c ∙ 1 - V2/c2
2L'/c = 2L/c ∙ √1 - V2/c2
L' = L/1 - V2/c2

Thus, we obtain

L = L'·√1 - V2/c2

As you see, the length of a moving wagon viewed from outside is shorter than when it is measured from inside. For this reason, the above formula is known as the formula of relativistic contraction of length. This contraction is not because of any force of compression exerted inwards; rather, it is related only to relativistic effects caused due to the motion of wagon at very high speed, comparable to the speed of light.

When we did the same experiment emitting the light vertically, there was no change in height H of the wagon. This is because the wagon's motion was horizontal. Hence, we conclude that the relativistic effect of length contraction is observed only in the moving direction.

Another thing to point out here is the shape of objects during a relativistic event. Thus, any object in such an event still looks in the original shape despite the "pressing" effect caused by the motion at very high speed.

### Example 3

What is the length of the wagon discussed in the theory section measured from a fixed point outside the wagon, if the wagon is moving at 0.8 c and the light signal needs 0.02 s to move from the source to the mirror and back to the source again when the event is viewed from inside the wagon? (The values may not be very realistic; they are only for illustration purpose.)

### Solution 3

We have the following clues:

V = 0.8 c
Δt' = 0.02 s
L = ?

First, we work out the length L' of wagon in the system S' connected to the wagon. We have

∆t' = 2L'/c
L' = c ∙ ∆t'/2
= (3×108 m/s) ∙ (0.02 s)/2
= 3×106 m

Thus, the length L of wagon measured from outside it, is

L = L'·√1 - V2/c2
= L' · √1 - (0.8 c)2/c2
= L' · √1 - 0.64
= L' · √0.36
= L' · 0.6
= 0.6 ∙ 3×106 m
= 1.8×106 m

## Summary

The old theory of relativity led to many contradictions, which led to the formulation of the modern theory of relativity by Einstein, based on his two famous postulates.

From the first postulate, it derives that we are not able to determine whether a system is at rest or it is moving; this can neither be achieved by analyzing the mechanical phenomena nor optical ones. The speed of light is not involved in any of calculations.

The second postulate is even stranger and unexpected. The common understanding on relativity is based on the concept of a changeable speed that depends on the system of reference used to describe the motion. According to Einstein, this is valid for all material objects but not for the light. These findings led to the formulation of "Special Theory of Relativity".

According to this theory, the simultaneity of the events observed is relative - a conclusion that contradicts the Newtonian concept of absolute time, which was believed to be equal for all inertial frames of reference. From this viewpoint, Einstein in his theory formulated the idea of different-flowing times in different inertial frames of reference. There is no absolute time anymore. The time flow that characterizes the chronology of events is an inner property of a given inertial frame of reference.

Since the time of event occurrence is not the same in two inertial systems, as in the case of the Newtonian system, a procedure for measuring the time that is typical for a given system, is necessary. This procedure is known as clocks' synchronization. All events are absolute as objective facts, however, their time (and place) of appearance is different in different inertial frames of reference. Thus, the same event is shown at coordinates (x, y, x, t) in the system S and (x', y', z', t') in the system S'.

The time interval of an event's occurrence when measured from a reference frame connected to the moving object, is

∆t' = ∆t ∙ √1 - V2/c2

where V is the speed of moving object (of S') and Δt is the time interval of event's occurrence when measured from a fixed point outside the moving object (in S). This formula is known as the formula of time dilation in relativistic events.

On the other hand, the length of an object contracts during a relativistic event as a counterbalance towards the dilation of time. The formula of length contraction in relativistic events is

L = L'·√1 - V2/c2

where L is the length of moving object when it is measured from a fixed point on the ground (in S) and L' is the length of object measured from any point connected to it (in S').

Any object in a relativistic event still looks in the original shape however, despite the "pressing" effect caused by the motion at very high speed.

## Physics Revision Questions for Space and Time in Einstein Theory of Relativity

1. The time measured by a clock inside a spaceship moving at 0.5 c shows 36 days from the taking off to the landing instant. How many days did the mission last when viewed from Earth?

1. 48 days
2. 45 days
3. 41.6 days
4. 31.2 days

2. A star is moving at 0.6 c in a direction which is perpendicular to our sight. What is the radius of this star (if measured from a point on its surface) if the thickness calculated by an observer at rest on Earth surface is 4 × 106 km?

1. 2.5 × 106 km
2. 3.2 × 106 km
3. 5.0 × 106 km
4. 3.2 × 106 km

3. A 20 m long and 5 m high wagon is moving very fast, at 0.3 c along horizontal tracks. What is the height of wagon when viewed from a fixed point outside the wagon?

1. 19 m
2. 4.75 m
3. 5 m
4. 5.49 m