# Curved Space-time

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A better version of this article is — THEORY OF RELATIVITY IN BRIEF which is also a featured article on our blog. So, we recommend you to read that article instead of this, and, also that article describes RELATIVITY completely.

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The Principle of equivalence is the foundation of general relativity and it states that –

The effects in a gravitational field are same as the effects in accelerated frames of reference

or in other words you can’t be 100% sure that if you are at rest in a gravitational field or you are accelerating uniformly. We can understand it by the given two cases:

1. A man is in a rocket at rest on Earth, he drops a ball in the rocket
1. The man, now, is in the same rocket without any contact with a gravitational field and the rocket is accelerating in ‘empty space’ at acceleration of 9.8 m/s2 (m/s2 is unit in which acceleration is measured.) (9.8 m/s2 is the acceleration due to gravity on Earth.) Now, again he drops a ball.

Einstein thought similarly like the above cases to develop the principle of equivalence as a FOUNDATION of his 1915 general relativity. He will see the ball fall in the same way in both cases. So, we can conclude that the effects in the accelerated rocket are same as the effects in a gravitational field.

So, now, we know that equivalence principle is the foundation of general relativity. But, how general relativity describes gravity? Let us take an example to find out the answer to the above question. Consider a rocket (that isn’t have any contact with a gravitational field) which is 300,000 kilometers long, that means, light takes one second to reach from top to bottom (and vice versa) of the rocket and according to special relativity, light’s speed will remain constant no matter what the speed of rocket. Consider, currently it is at rest or moving with a constant (or uniform) speed. Now consider two people are standing at the top and bottom of the rocket. The man at top sends two pulses or waves (or rays) of light to the person standing at bottom of the rocket at a time gap of one minute, that is, he sends the second pulse of light after 1 minute of sending the first pulse of light. The person, at bottom, with no doubt, receive these two pulses at a time gap of 1 minute. Now, the rocket starts to accelerate uniformly. Now, if the person at top sends two pulses of light at a time gap of 1 minute, the person standing at bottom will receive the second pulse in less than 1 minute. Why does it happen? The answer is because light has to travel less distance in this case as the person standing at bottom is moving towards the light rays. This can be understood by given image: Image © MADHUR SOROUT and MADDYZ PHYSICS.

The person at bottom is moving towards light ray and light ray is moving towards person and because speed of light is absolute, the distance (or space), light has to travel is decreased (or contracted). So, the person at bottom will receive the two pulses in a time interval, less than one minute.

This means that space has contracted (or shortened) for the person at bottom and time has dilated (or lengthened) for him.

What can we conclude from this? According to equivalence principle, the effects in accelerating bodies are same as the effects in a gravitational fields. Now, we can conclude that in a gravitational field space contracts and time dilates.

In other words, near an object having mass (or energy), space contracts and time dilates.

Now consider a simple example of an apple falling on Earth, WHY IT FALLS? It falls because Earth has contracted the space around it and when the space contracts, it pushes everything in its path, it means that space is pushing the apple to the ground.

This ‘gravitational’ Contraction of Space and Dilation of Timeis called ‘Curving of Space-time’.

There is also something special about this contraction of space — space pushes the objects in some definite paths, the straight path between any two points in 4-dimensional space-time. This straight path between two points is called ‘geodesics’. All the objects in four-dimensional space-time follow straight lines in 4-dimensional space-time. Because these paths are straight, so these are the shortest paths. These shortest or straight paths are called ‘GEODESICS’. This concept seems somewhat difficult to grasp. Let us understand this concept with a simple example. The surface of Earth is two-dimensional and curved because it is a sphere. Now, on a globe of Earth, mark two points far enough and join them, do you get a straight line after joining them? The answer is no. The straight line or shortest path between these two points is curved. Now, if a rocket flies over Earth in a straight path, its shadow on Earth, that is, its path on two-dimensional surface will be curved. This means that the rocket is moving straight in three-dimensional space but its path seems to be curved on two-dimensional surface of Earth. This can be understood by given image: Source: A BRIEFER HISTORY OF TIME by STEPHEN HAWKING

This effect is same in 4-dimensional space-time and 3-dimensional space. For example, space is pushing Earth and other planets to follow the geodesics, that is, the straight paths in 4-dimensional space-time but they seem to move in curved paths in 3-dimensional space. But when an apple falls, it moves in straight path in 3-dimensional space also, that is, the apple is not moving in curved paths in 3-dimensional space when it falls. Why? Let us take an example again and find out. When you will mark two very nearby points on a very large globe of Earth, and join them, you will find that this line or shortest path is STRAIGHT. This means that the distance between the the two points in space (which the apple covers) is too short that its path in 3-dimensional space is also straight. Now, what we can conclude from above discussion? What is actually GRAVITY? The conclusion is —

Gravity is not a force like other, but, it is a consequence of curvature of space-time. The presence of mass and energy curves space-time that means near a mass, space contracts and time dilates and away from the mass, space is less contracted and time is less dilated. All the objects follow the shortest path, that is a straight path in 4-dimensional space-time. These paths are known as geodesics and space is pushing the objects to follow these geodesics (straight paths), but, the objects seem to move in curved paths in 3-dimensional space.