‘Theory of Relativity’ is a generalized term used for two different classes of theories given by Albert Einstein namely special relativity and general relativity.
General relativity was originally published by Albert Einstein in the year 1915. Before general relativity, it was believed that gravity is a “force” which acts between objects that have mass. However, Einstein’s general theory of relativity changed this view of gravity. According to general relativity, our Universe is made up of 3 Space dimensions + 1 time dimension. Together, these dimensions form a 4 dimensional continuum known as the space-time fabric. Objects having mass produce curvature in the space-time fabric. This curvature of space-time is responsible for gravity!
Bending of space-time around Earth (Image: NASA)
Einstein published the theory of special relativity in the year 1905 (before general relativity). Special relativity applies to objects that move at speeds comparable to the speed of light. It must be noted that special relativity does not involve gravity. In order to include gravity, Einstein developed his general theory of relativity. Hence, general relativity can be considered to be an extension to special relativity.
Before going any further, one must first understand that motion is relative. For example, imagine you’re standing on a footpath and see a bus passing by on the road with some constant speed ‘v’. Now, for people in the bus, each one of them is at rest with respect to one another. But for you, all of them are moving along with the bus with some speed ‘v’. A person which appears motionless to an observer in one frame of reference might not necessarily appear motionless to another observer in a different frame of reference. Hence, motion is not absolute but relative.
This holds true for every entity in the Universe except light. The speed of light in vacuum is a universal constant and remains same in all frame of references. This is one of the most important postulates in special relativity. The second postulate states that the laws of physics remain the same in all non-accelerating frame of references. These two postulates and some complex mathematics (which we’ll skip) results into some interesting consequences. Before looking at these consequences, let us first understand what is Lorentz transformation.
Since the speed of light must remain the same in all frames of references, it is necessary that the length and time co-ordinates in moving frame of references must change. The relation between the space and time co-ordinates between two frame of references in relative motion is given by Lorentz transformation.
Consider an event in a frame of reference S having space-time coordinates (x, y, z, t) and space-time coordinates (x’, y’, z’, t’) in another frame of reference S’ moving with a velocity ‘v’ in the X-direction with respect to S. Then, these coordinates are related in the following manner:
where ϒ (gamma) is the Lorentz factor given by
This will hold equally true for Y and Z directions if the motion is along the Y & Z axes respectively. Using Lorentz transformation, a number of consequences in special relativity can be obtained such as time dilation, length contraction, relativistic addition of velocities, etc.
1. Time dilation:
According to special relativity, speed of light ‘c’ is the maximum speed limit. As you approach closer to the speed of light, time would slow down in your frame of reference. This can be expressed as
Δt’ = ϒ Δt where
where Δt’ is the time between two ticks of a moving clock, Δt is the time between two ticks for a clock at rest and ϒ is the Lorentz factor whose formula is given above. One can see that as the speed ‘v’ increases, the value of ϒ also increases and hence Δt’ also increases i.e. time slows down as speed increases. Substituting v = 99.999% of speed of light, we get that the time between two ticks (one second) of a clock in the moving frame corresponds to 224 seconds!
2. Length contraction:
Similar to time, even length is affected. Length contracts in the direction of motion of the frame. This can be given as Δx’ = Δx/ ϒ.
Here, Δx’ is the length observed by an observer in relative motion to the object & Δx is the length of object in the rest frame (proper length).
Hence, as speed increases, length is shortened. For example, a spaceship moving at 86.5% of the speed of light would appear to have shrunk to half its length for a stationary observer due to length contraction (Note: Its other dimensions would not change. The dimension only in the direction of motion will be affected.)
3. Relativistic addition of velocities:
You must be knowing that when two cars move with velocities v1 and v2 respectively in the same direction, the person in car 1 would see the car 2 moving with a velocity of (v2 – v1). If cars are moving in the opposite direction, then the relative velocity is given by (v2 + v1).
This would mean that if two spaceships are moving with speeds of 99% of the speed of light in opposite directions then the person in one spaceship must see the other spaceship moving with a velocity of (v2 + v1) i.e. (99% + 99%) of the speed of light which is equal to 198% of the speed of light. This is not possible because the speed of light cannot be exceeded. Special relativity solves this problem. According to special relativity, the relative velocity between two frame of references is given as:
Substituting 0.99c (99% of speed of light) for both v1 and v2, we get the relative velocity as u = 0.99995 c which is less than the speed of light. When v1 and v2 are much lesser as compared to the speed of light the denominator vanishes and only the numerator (v2 + v1) remains. But at speeds closer to speed of light, the denominator has a significant value and cannot be ignored.
4. Mass-Energy equivalence:
Another important consequence of special relativity is that energy and mass are equivalent. An object at rest with a mass mo has an energy content of moc2 where ‘c’ is the speed of light. Similar to the mass-energy relation (Eo=moc2), the energy-momentum relation relates the total energy of a body to its rest mass (mo) and momentum (p) as follows:
E2 = p2c2 + (moc2)2
For particles like photons (particles of light), whose rest mass mo is zero, the equation reduces to E =pc which means that the energy associated with photons is because of their momentum.
These are some of the most important consequences of special relativity. Now, lets move on to general relativity.
General relativity describes gravity as the curvature of space-time. General relativity predicted the existence of black holes and their properties even before they were discovered. General relativity is based on Einstein field equations which are non-linear and very difficult to solve. According to general relativity, objects having mass bend the space-time fabric. Greater is the mass, more is the bending. General relativity results into a number of consequences which are discussed below.
1. Time dilation:
Time dilation occurs not only due to high speeds but also in presence of gravity. Time slows down in the presence of gravitational field. Time dilation due to due to gravity can be given as:
Here To is the time measured in the gravitational field & T is the time measured far away from the gravitational field. The above equation shows that greater is the mass of a body, more is the slowing down of time. It is amazing to note that, time dilation at a black hole is so strong that time stops at its event horizon. The slowing down of time in a gravitational field also leads to a phenomena known as gravitational redshift. Light traveling outside a gravity well is found to be redshifted (it’s wavelength increases) when observed from a point in a lower gravitational field.
2. Gravitational lensing:
We know that the space-time curves around massive objects. According to general relativity, light follows the curvature of space-time. As a result, light bends around massive objects. The bending of light around heavy objects like galaxies, quasars, galaxy clusters, etc makes them behave like a lens.
Gravitational lensing by a galaxy cluster (Image: STScI, Source: Wikimedia Commons)
As seen in the above figure, the white arrow shows the path of light coming from a background galaxy. The light coming from the background galaxy bends around the galaxy cluster between Earth & the background galaxy (similar to a lens) and we get to see numerous images of the background galaxy. The orange arrow shows the apparent position of the background galaxy. This phenomena is called as gravitational lensing.
3. Gravitational waves:
Gravitational waves are ripples in space-time that travel at the same speed as the speed of light. Einstein predicted their existence 100 years before they were finally detected in the year 2016 by the Advanced LIGO team coming from two merging black holes 1.3 billion light years away.
Numerous other observations of gravitational waves coming from merging black holes as well as merging neutron stars were made after the first one thus confirming their existence.
General relativity has been tested a number of times and has proved to be successful in majority of the cases. General relativity can explain several observations that Newton’s law of gravitation cannot such as Mercury’s anomalous perihelion shift as it orbits around the Sun, existence of black holes, singularities, etc. Hence, it is the most accepted theory of gravitation today.
In this manner, theory of relativity proves to be one of the most beautiful theories of all times. The theory is incomplete and has certain limitations yet it makes some very bold predictions about our Universe that seem to agree with the observations.
This article gives a simplified explanation to Einstein’s theory of relativity and covers most of the important topics without involving much mathematics. If you have any queries, feel free to ask them in the comments section.