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### Origin of the “Special Theory of Relativity” and the concept of “spacetime”

Einstein’s Special Theory of Relativity (Special Relativity) and General Theory of Relativity (General Relativity) are most known concepts of physics. Even a non-Physics person is likely to know the equation E = mc².

This post discusses the origin of the word “Relativity” and uses it to understand the concept of “spacetime”.

Definitions

Laws of motion are those that describe the laws of motion (movement) of bodies. For instance, Newton’s equation F = ma is a law of motion because it describes the motion of the body.

A reference frame is a frame from which we observe a body. If we observe a falling apple, the ground is the reference frame here (assuming that we are standing on the ground while observing the apple) and apple is the body we are observing. A reference frame is called an inertial reference frame if it is at rest or moving with constant speed.

### Galileo

Galileo first said that laws of motion are independent of the inertial reference frame. It means that the same equation of laws of motion are applicable to a person observing the object by staying still and another person observing the object while moving with constant speed.
Laws of motion are the same in all inertial frames.
This is called Galilean Relativity (1632).

### Newton

Decades later, Newton (1643–1727) gave the formula for the law of motion: F = ma. This equation satisfies the Galilean Relativity.

[For those comfortable with mathematics, you can do this transformation and check for yourself.

In a reference at rest, Newton's equation of motion of a body is d²x/dt².

If a body is in an inertial reference frame moving with a constant velocity k,  then its x coordinate becomes x+kt. So, we replace "x" with "x+kt" in Newton's law of motion. Doing that we get d² (x + kt)/dt².  We notice that this is equal to d² (x)/dt² which is the equation of motion for a body at rest.

We call this replacement of "x" with "x+kt" as a transformation. Newton's laws of motion do not change with this transformation. In technical words, it is invariant under this transformation.]

But Newton had a deeper question — “relative to what?”. You are at rest with respect to your neighbour. But you both are moving with respect to a person in the solar system (because earth is moving). For a person outside the solar system, the whole solar system is moving and so on.

Newton said that there is something called “absolute space”, a static one, relative to which we can measure the motion of bodies. [Read “Newton’s bucket” to understand how Newton demonstrated it.]

### Measuring relative to space

When we talk about motion of a body relative to a stationary observer or relative to an observer moving with constant speed, note that we are referring to the notion of space here. Laws of motion don’t change relative to space.

Mathematically, we can see that we use the terms “x” and “x+kt”, which essentially refer to space.

Laws of motion are hence invariant to space of inertial reference frames.

### Maxwell

All was fine till Maxwell came.

In the 19th century, Faraday discovered that one can generate electricity by moving a conductor in a magnetic field and similarly generate a magnetic field by passing electricity in a conductor.

Maxwell gave theoretical formulation to this principle. Maxwell gave 4 sets of equations that describe electricity and magnetism.

Two things follow from Maxwell’s equations.
1. Galilean Relativity or simply Relativity doesn’t hold to these equations. It means that if we replace “x” with “x+kt” in these equations, we get different equations, unlike Newton’s law where we got the original equation back. Thus, Maxwell’s equations are not invariant to space.
2. One can derive the speed of the wave with electric and magnetic components. It turned out that the speed of this wave is precisely equal to the speed of light.
People inferred three conclusions from these equations
1. Galilean Relativity or simply Relativity applies only to laws of motion, not to laws of electricity and magnetism.
2. From the fact that the wave with electric and magnetic components described by Maxwell’s equation travels at the speed of light — Maxwell postulated that the electromagnetic wave described by his equations is light itself. Light is an electromagnetic wave.
3. One could measure the speed of light (electromagnetic wave described by Maxwell’s equations which M called as light) from Maxwell’s equation but the problem was that the equation doesn’t tell — speed relative to which reference frame? Maxwell hypothesised that there exists a medium called ether through which light passes and speed of light given by Maxwell’s equation was relative to this medium. Just like Newton hypothesised an absolute reference frame relative to which objects move, Maxwell’s ether is the absolute reference frame relative to which light’s speed is measured.
Experiments followed to check the presence of ether. The famous Michelson-Morley experiment proved the non-existence of Ether.

In summary, there were two lingering questions
1. How do we explain the speed of light given by Maxwell’s equations without a reference point?
2. Why don’t Maxwell’s equations follow the principle of Relativity?
Einstein solved these questions with his Theory of Special Relativity.

### Einstein and Spacetime

In 1905, Einstein published a paper called “On the electrodynamics of moving bodies”, where he tried to answer the above two questions. Einstein’s answers were

1) We are not able to find the medium, in reference to which speed of light is as measured by Maxwell’s equations — because we don’t need such medium at all, to define the speed of light.

In other words, you don’t require a reference point or a medium relative to which speed of light can be talked about. Speed of light is the same in all reference frames.

It’s a radical thought. It profoundly changed the field of Physics.

2) If the speed of light is the same in all reference frames, it leads to some logical problems. Suppose if I stand on a car moving at 50 km/hr and throw a ball in the direction of the car with 10 km/hr, the speed of the ball with respect to earth is 60 km/hr (50+10). It’s because when the ball is thrown, both the ball in the hands of the person and the person are already moving with the car, at 50 km/hr. Throwing it with 10 km/hr gives it an additional push of 10km/hr and hence it becomes 60 km/hr.

This notion goes haywire as per Einstein’s above postulate that the speed of light is the same in all reference frames. By this, it means that if I put a light source on top of a moving car and make the source to emit light — the speed of light won’t become “c + speed of car”, as it happened in case of car and ball. Speed of light just remains “c”, irrespective of the motion of the car.

How’s that possible? Einstein’s spacetime is the answer.

Einstein said that when we are discussing the example of car and ball where we add their velocities, we are considering the speeds relative to only space (reference frame of the ground). We are assuming that there exists a universal time common to both the observer on the ground and observer in the car.

Einstein said that considering only space as a variable is wrong. To account for the constant speed of light in all reference frames, we have to consider spacetime where both space and time change as per reference frames.

In other words, the observer on the ground and the observer in a car not only have a different notion of space (distance covered by object etc.), they also have different notions of time. Clocks tick slower in moving reference frames.
All of us not just have our own positions (x,y,z), we also have our own time. We carry our own time, like we carry our own positions.
We used only space (x,y,z) while discussing relative motion using Newton’s laws. Einstein says that we have to use spacetime (x,y,z,t) instead. Time is not the same in all reference frames.
Note the transition. Newton considered an absolute reference frame of space relative to which bodies move. For Maxwell, ether was the absolute reference frame relative to which light moves. Einstein discarded both. For Einstein, spacetime is the absolute reference frame relative to which we have to make the observations. In this spacetime framework, speed of light is constant in all inertial reference frames.
Einstein showed that if we consider spacetime instead of just space, Maxwell’s equations remain invariant with respect to the frame of reference.

Mathematically, it means that we don’t just replace adjusted “x” coordinates in Maxwell’s equations, we also replace adjusted “t” (time) coordinates. If we do that, Maxwell’s equations apply to both stationary observer and those moving with constant speed.

It means that Maxwell’s laws are applicable to all inertial reference frames in spacetime framework. Thus, Einstein solved the second question above — the problem of (lack of) Relativity of Maxwell’s equations.

It is hence said that Einstein restored the relativity of Maxwell’s equations.

The name Relativity flows from this fact — it restored the relativity of Maxwell’s equations. Actually, Einstein didn’t call his theory as Special Relativity. As noted above, the title of his paper was “On the electrodynamics of moving bodies”. Another famous Physicist Max Planck gave the name “relative theory”. It was later given the name “Theory of Relativity” by Alfred Bucherer. This became Special Theory of Relativity after General Theory of Relativity was proposed because this theory considers only a special case of constant speed, while General Theory of Relativity applies even to accelerating frames.

Finally, Einstein’s two postulates of Special Relativity are:
1. Laws of physical systems don’t change in inertial reference frames.
2. Speed of light is the same in all reference frames. Further, nothing in space can travel more than the speed of light.
There — my dear ladies and gentlemen — goes the story of Special Theory of Relativity and spacetime.

### Intuitive understanding of spacetime

We are usually accustomed to the notion of just space. Hence, it might be difficult to comprehend spacetime.

For example, if you are standing, you might think that you are rest. Einstein says that you are not. You are thinking that you are at rest because you are using only the notion of space (x,y,z). If you use the notion of spacetime (x,y,z,t), you are not at rest. You are moving because “t” is increasing.

In other words, in the framework of spacetime, you are always moving. Even when you are at rest in space (x,y,z), you are moving at the rate of 1sec in time dimension as per spacetime (x,y,z,t).

Remember the equation E = mc²? It says that even a body at rest has the energy of mc². We learnt that the body gets energy when it moves. Where did energy come from if a body is at rest (not moving)?

Here again, when we say that the body is at rest, we are using the notion of space (x,y,z). In Einstein’s spacetime, the body is not at rest. It is moving along the time dimension. The energy mc² can be thought of as the energy due to motion along the time dimension.

### FAQ

Q1: In this process of explaining Special Relativity and Spacetime, we didn’t properly explain the conundrum in the car-ball example. If a light is emitted from a moving car, how come the speed of the car doesn’t get added to the speed of light?

A: Speed of light is denoted by c.

c = distance/time = space/time.

"c" remains constant because as a body gets into motion, length contracts and time slows down (dilates), keeping the ratio of length and time (speed) constant.
“c”, the speed of light is the thread that binds “space” and “time” and makes “space-time”.
Q2: How did Einstein arrive at the conclusion that the speed of light is constant in all reference frames? Does it directly follow from Maxwell’s equations?

A: It’s a perfectly valid question. How did Einstein arrive at the conclusion that the speed of light is constant in all reference frames? Isn’t it a simple assertion without any base?

Note that Einstein came up with the concept of spacetime to address the inconsistencies arising out of the constant speed of light. So, in a related question, one may also ask — why do we have to force fit our theories to satisfy the assertion that the speed of light is constant?

Well, the answer is that Einstein postulated the constant speed of light based on reasoning. One has to admit that Maxwell’s equations don’t directly throw out the fact of constant speed of light. There is some amount of leap of reasoning involved in deducing it.

But, the theory built around this assertion gives a nice framework that explains many things. To start with, it made Maxwell’s equations apply in all inertial frames of reference — it restored the relativity of Maxwell’s equation — which we think is important because “application of laws in inertial reference frames is a treasured principle in Physics”. However, it needn’t mean that any transformation that manipulates Maxwell’s equations to make them applicable in inertial frames of reference, has to be considered. Experiments play a significant role at this stage.

After Einstein’s Special Theory of Relativity, several experiments proved that the speed of light is constant in all inertial reference frames. Experiments also proved the effects of constant light speed, like length contraction and clocks ticking slowly in moving reference frames, which were theoretically deduced from Einstein’s equations. So, we believe it.

Some also argue that Maxwell’s equations were derived with some specific assumptions, applicable only to particular medium etc. Hence, deducing the constant speed of light from this, building theories based on this and extending it into many other areas is troublesome.

It is true that deductions from Maxwell’s equations with assumptions (that are not applicable everywhere) needn’t hold true for all cases.

Note that the deductions needn’t be applicable elsewhere, it doesn’t mean that they shouldn’t be applicable.

For instance, if a body is covered in cloth, and only a portion is visible. You see the colour of that small portion of the cover and assert that the colour of the body is “x”. You build some theory around it. But then, later the cover is removed and the colour of the whole body turns out to be the same as that of the small portion observed initially.

At this stage, it doesn’t matter if the initial deduction of the colour of the body is based on a true version or a distorted version of the body. Once, we realize that the whole body is of the colour we initially deduced, the source of initial deduction doesn’t matter. The initial source just gave us a hint.

It’s something similar here. We made the observation of the constant speed of light using Maxwell’s equations and extended to many other areas. But the point is that the constant speed of light was later proved experimentally. Hence, we accept the assertion of the constant speed of light.

In summary, we can say that constant speed of light was inspired or inferred from Maxwell’s equations but since it is experimentally proven, the theories built on it are true and applicable for all cases, not just to those cases where assumptions of Maxwell’s equations are applicable.

It’s the way science progresses. You observe something, hypothesise something from it and build a theory based on it. The theory is then put to test. If it turns out to be true, we find the hypothesis on which theory is built on, as acceptable. In this case, both the hypothesis (constant speed of light) and theory (special relativity) are put to the experimental test and both turned out to be true. So, we believe them.

Q3: Why can’t we travel at the speed of light?

We didn’t discuss here but Einstein’s equations show that mass increases as velocity increases. Note that energy is mass (E = mc²). So, as energy increases due to speed, mass also increases. It means that the energy you need to apply increases as the speed increases.

Einstein’s equations show that such required energy increases exponentially at near the speed of light. Theoretically, one needs infinite energy (infinite mass) to reach the speed of light. So, it’s not possible to reach the speed of light.

A few years ago, there was news that scientists found a neutrino that apparently travelled at faster than the speed of light. But later it was found that there was a measurement error — neutrino didn’t cross the speed of light.

But, note that the constancy of the speed of light applies only when it is travelling through space/vacuum. Its speed can change if you add a medium to it. Thus, light travels slower in air, water etc., from where the concept of refraction index emerges.

Q4: What is the difference between Special Relativity and General Relativity?

In the above discussion, we noted the effects of constant speed of light for reference frames at rest and reference frames moving at a constant speed. This is called Special Relativity.

We have to update it when we add acceleration to it.

In 1915, 10 years after the publication of Special Relativity (1905), Einstein added acceleration to this analysis and published his famous General Theory of Relativity.

Q5: Is this all theoretical fancy physics stuff OR are there any applications?

Relativistic correction of time is a crucial component of GPS. Without it, our GPS would show huge errors, making it useless.

GPS works as follows. It sends a signal to the person and calculates the time taken for the signal to return. Based on this time, it estimates the distance. So, timekeeping is a crucial part of GPS’s accuracy. Even minute differences can throw up large errors.

Since GPS satellites are in motion relative to the person on earth, time in GPS satellites’ frame would be slower than that on earth. This relativistic time correction has to be added to the final calculation for the estimation of distance.

We all owe Einstein for our GPS.

[Technically, GPS requires two corrections — slowing of time due to motion (special relativity) and for the presence of satellite in earth’s gravitation (general relativity). We will deal with general relativity in another post]