The Special Theory of Relativity was introduced to the world by Albert Einstein in 1905. An enhanced and sophisticated version of Galileo's Relativity, this theory managed to be compatible with the new findings of electromagnetic radiations of the early 20th century. There was nothing but triumph and glory to it. But every new theoretical assertion in Physics is bound to be met by skepticism and ultimately, Special Relativity was not spared. The biggest and probably the most troublesome loophole in this theory was a necessity of inertial reference frames only i.e. reference frames (or observers) executing a constant linear motion. The moment one introduces even the tiniest of acceleration for any of the reference frames, the symmetry falls apart. In fact, the basic Principle of Relativity no longer holds. If one of the concerning observers is accelerating, then one can assign almost an absolute meaning to this acceleration. Einstein was starting to get annoyed by this. For, it was a subtle yet very crucial down point to his theory.
The breaking of symmetry marked a transition from an ideal world of inertial frames to a very realistic Universe of non-inertial frames existing almost everywhere. A car driver after pressing down on his accelerator, accelerates forward and imparts an opposite force on him and the passengers. Let the same car traverse a curved road and it accelerates under the influence of a centripetal force, while the passengers feel a fictional centrifugal force. It is implausible to try and use relativistic equations here because observers outside the car as well as inside shall observe the same forces. There is no equivalence of inertial frames as implied by Principle of Relativity because there are no inertial frames in the first place. Our cosmos is filled completely with matter and energy distributed in a near-isotropic fashion. The gravitational effects owing to this matter produce acceleration for any object moving through it's area of influence also known as the gravitational field. These gravitational fields permeate everywhere and must give rise to non-inertial frames nearly everywhere. Furthermore, the accepted theory of Newtonian gravitation at the time was based on instantaneous action at a distance. Newton regarded gravity as a force that always exists between any two bodies of mass. This gravitational force was believed to be instantaneous and thus travel faster than light to act upon any body. This was indeed conflicting with one of the fundamental postulates of Special Relativity that nothing can travel faster than the speed of light.
Einstein's elegant theory was on dangerous grounds. Until, he was struck by what he recalled as "the happiest thought of my life" - In 1907, when sitting at his desk in the patent office, Einstein imagined a man falling freely from the roof his house. [1] How peculiar for the ordinary to title this as one's happiest thought! Yet, for a physicist bothered since two years, this elation was quite obvious. The unfortunate man who falls from the roof of his house should momentarily experience weightlessness till he impacts with the ground. This is because the man is falling freely without the presence of any material object upon which he can exert his weight. Hence, everything which free falls with him should also appear weightless to him. Take a weight scale, glue it to his feet and toss him off the roof again with a trampoline on the ground for the sake of humanity. The man will see a zero reading on the weight scale throughout his entire free fall trajectory. The weird sensation in your stomach when you are on a roller coaster ride and it suddenly plunges down is the sensation of sudden weightlessness. Another experiment which you could try at home is to take a plastic bottle with a few holes pierced at the bottom and fill it with water. Hold the bottle up to a certain height. Initially, the water is dripping through the holes but as soon as you let go of it, you shall observe that the water no longer drips from those holes. There is no weight to the bottle as well as the water inside it.[2]
Einstein discovered the key symmetry to retain the basic Principle of Relativity. Every object in a free fall appears weightless to any other object falling with it. Thus, every observer in a freely falling reference frame should appear to be floating around as if there are no gravitational forces.[3] If the reference frame was of an observer inside a completely closed cabin in free fall, there is no way by which he could tell if the cabin is floating in free space in the absence of any external forces or falling under the influence of Earth's gravity. Consequently, if an observer is accelerating in free space inside a closed cabin then he shall feel a force pulling him in the opposite direction. This is because the floor of the cabin pushes against his feet and he feels a reaction force in the opposite direction owing to Newton's Third Law of Motion. Say, the cabin has an upward acceleration exactly equal to the value of g on Earth i.e. 9.8 m/s^2. There would be no way for the observer inside the cabin to tell apart if the cabin is sitting in the gravitational field of Earth or accelerating in free space. Some noticeable daily life examples are the g-forces experienced in a roller coaster when it suddenly accelerates up, or by a jet-fighter pilot pulling extreme maneuvers. The magnitude of these are often portrayed by 4g,7g, etc. which implies the gravitational forces experienced are 4 or 7 times the gravitational force of Earth. At high enough g-forces, the person's weight increases upto 9 times the original weight. The blood from his brain is pulled down due to these forces and he/she passes out momentarily. This resembles the state of two inertial frames moving relative to each other. There is no way for them to tell which one's moving. In other words, both reference frames are equal and this constituted the first postulate of the equivalence of inertial frames in Special Relativity. For an observer inside a completely closed cabin moving with a constant linear velocity, there would be no way for him to determine whether he's at rest or in motion Similarly, the equivalence of an inertial reference frame in the absence of any forces and an inertial frame in free fall or the equivalence of acceleration of a reference frame in free space and acceleration due to gravitational field constituted the first ever principle of General Relativity. This principle became widely known as the "Equivalence principle". It states that in a small enough region, gravitational and inertial forces are often of the same nature and indistinguishable.
The constraint of a "small enough region" or in other words a 'local' region is introduced to account for the tidal forces. A general observation regarding any object under the gravitational influence of our Earth for example is that the object always falls towards the center of mass or "barycenter" of the Earth. Due to this, two objects in a free fall won't fall parallel towards the surface of Earth. They would converge towards a common point. If the two objects are inside a closed cabin in free fall, then it shall appear as if they are moving (accelerating) towards each other. This renders the cabin as a non-inertial reference frame and violates the Equivalence principle. Therefore, the equivalence principle requires a local or small enough region of a reference frame.
In this way, Special Relativity was retained once again. Summarizing everything : a body freely falling in a gravitational field is equivalent to a body moving with constant velocity in deep space (weak equivalence principle) and a body sitting in a gravitational field is equivalent to a body uniformly accelerating in deep space (strong equivalence principle). It was also known that the acceleration with which any object falls in a gravitational field is independent of its mass. If this wasn't true then objects of different mass inside a freely falling reference frame would accelerate with different magnitudes. Such a reference frame would then be deemed as non-inertial. From all these observations, Einstein was able to infer that gravity is not a consequence of the material composition of an object but rather the nature of the spacetime around it. In formal language, a metric theory of gravity was born which was compatible with Special Relativity and its postulates.
The consequences of this theory were immediately recognized. The most prominent consequence being the effect of gravity on electromagnetic radiations. Earlier, light was known to be composed of massless particles. Newtonian Gravity asserted that the attractive forces between any two material bodies due to gravity is directly dependent on their mass. Thus, it was believed that light and particularly electromagnetic radiations shall remain unaffected by any gravitational influence. But, General Relativity suggested a different picture. Imagine, a completely enclosed cabin fitted with tremendously powerful rocket engines. Inside the cabin is an observer holding a light torch parallel to the cabin floor and pointed at one of the walls of that cabin. Now, the rocket engines are fired, imparting a large vertical acceleration to the cabin. As the cabin continues to accelerate up, this observer turns on the light torch. We ask ourselves the following question - "Where do we observe the light to be incident on the cabin wall?". At first glance, one may think that the light spot should be visible exactly in the direction where the torch is pointed . A little investigation reveals this to be untrue. There are two well established facts at our disposal - first the cabin has a large acceleration upwards and secondly (most importantly) , light always travels at a constant speed. Therefore, as light leaves the torch and starts travelling towards the wall, the spot at which the torch was pointed moves up ever so slightly because the cabin is accelerating. By the time, the light approaches the wall, the original spot A would have moved up by a significant extent and the light will fall upon a different lower spot B. This is the most intuitive outcome provided the two given facts. If the cabin were not accelerating, then the light ray would have fallen straight on spot A, or if the speed of light were infinite then the ray would have instantaneously reached the wall. Neither is the case, and hence the light ray rather traces a parabolic curve throughout its journey. Just like water flowing out of a garden hose. The water moves straight horizontally with a constant velocity but gravity causes it to trace a parabolic curve. Similarly, the light moves straight horizontally with a constant speed but the acceleration causes it to trace a parabola. But behold! The Strong Equivalence Principle states that one cannot distinguish by any means between a constantly accelerating frame (cabin) and a frame (cabin) sitting in a gravitational field. If light bends down in an upwards accelerating cabin then it should also bend in the gravitational field of a massive body. In fact, our observer inside the cabin will declare and quite rightly that his cabin is sitting inside a gravitational field which pulls down the light ray coming out from his torch. Like, water being pulled down from a garden hose.
The bending of light near a massive body is termed as - "Gravitational Lensing"[4] and was one of the first experimental confirmations of Einstein's Theory of Relativity. Einstein suggested that by this phenomenon, it might be possible for light to bend when it passes close enough to the Sun. It may be possible that those stars which are usually behind the Sun, can be observed from Earth by this bending effect.
However, the light coming from Sun during daytime is intense enough to obscure any feeble light emitted by the stars behind it. Except, during a total solar eclipse, when most of the solar disk is covered by Moon. This creates a time window of sufficient darkness, around the Sun enabling any stars near it to be visible through a telescope. An expedition of astronomers was summoned to observe the solar eclipse of 29 May 1919. It consisted of two teams headed towards two different locations. One of the teams stationed at Principe in West Africa comprised of the famous astronomer Sir Arthur Eddington.[5] Their motive was to observe the gravitational lensing effect predicted by Einstein's Relativity. The results obviously were in favour of Einstein. General Relativity passed its experimental test with flying colours. In a dinner held by the Royal Astronomical Society, Sir Arthur Eddington beautifully and quite comically, described his results in poetic verses -
"Oh leave the Wise our measures to collate. One thing at least is certain, light has weight. One thing is certain and the rest debate. Light rays ,when near the Sun, do not go straight."
The results were considered as a breakthrough in the history of Physics and made it to the frontpage of major newspapers. The news also created a significant impact on the popularization of Physics in the common household. Einstein gained fame and celebrity status as people started to know him as the physicist who discovered a new theory of gravity after Newton. Many experimental confirmations followed thereafter which included gravitational time dilation, the precession of the perihelion of mercury, black holes, gravitational waves and so on. Although, the period of almost ten years between the discovery of Equivalence Principle in 1907 till the finally correct version of General Relativity published in 1916 was a gruesome struggle for Einstein. The struggle was to formulate a precise mathematical model to encompass his theoretical principles. The behavior of spacetime and objects moving through it is accurately described by Differential Geometry i.e. the geometry of curved surfaces. With the help of his friends and colleagues, Einstein learnt this exceedingly difficult field of mathematics and managed to formulate a unified tensor equation known as the "Einstein Field Equations". Even today the equations are incredibly complex to solve and astronomers adhere to the simple Newtonian Law of Gravitation for majority of astronomical calculations. Nevertheless, the impact of General Relativity on modern physics was unprecedented. In the next part of this blog, I shall make a mediocre attempt to touch the mathematics of Differential Geometry and Tensor Calculus with its applications in General Relativity.
- Thank You.
References -
[1] https://einsteinpapers.press.princeton.edu/vol7-trans/151
[2] https://www.youtube.com/watch?v=0jjFjC30-4A&t=257s
[3] https://www.youtube.com/watch?v=FO_Ox_dH0M8
[4] https://hubblesite.org/contents/articles/gravitational-lensing
[5] https://www.nature.com/articles/d41586-019-01172-z