A new way to visualize General Relativity

Elastic sheet representation 00:13

• General relativity is a theory that describes freefall more accurately than Newton's law of gravity.
• The most commonly used representation of general relativity is an elastic sheet with massive objects placed on it.
• This representation is not rigorous and has a large number of problems.
• The elastic sheet representation is misleading because it suggests that objects are placed on space-time like marbles, while in reality, space-time is the fabric of the universe that contains objects.
• It is preferable to flatten objects onto the surface of space-time to show that they are contained within it.

Curvature of space-time 01:10

• In 1915, Albert Einstein proposed a new theory that made it possible to describe freefall more accurately by stating that massive objects distort the fabric of the universe itself and drag objects into a fall.
• Curvature of space-time is responsible for the phenomenon of gravity.
• Objects seem to attract each other when they are following the curve geometry in straight lines.
• Imagine the surface of a sphere on which two ants would head north. The two trajectories will meet at the north pole because of the curved geometry of the sphere on which straight lines tend to get closer to one another.

Issues with elastic sheet representation 02:22

• The elastic sheet representation sort of explains gravity by using gravity outside space-time to explain gravity inside it, which is not acceptable.
• The representation completely ignores the time dimension of space-time.
• The diagram for the elastic sheet representation ignores the time dimension, which is not very intuitive.
• A four-dimensional geometry cannot be faithfully rendered.
• It is necessary to find a trick to represent the dimension of time in addition to the three dimensions of space.

Representing the dimension of time 05:49

• To represent the dimension of time, we need to remove one dimension of space.
• The time component of curvature explains gravity.
• The apple moves in a straight line, but the curvature of space-time rotates the orientation of this straight line between time and space.
• If the apple falls towards the ground, it is because it started with a speed through time.
• The curvature of space-time generated by the earth has converted the apple's temporal speed to a spatial one.

Slicing the diagram 08:23

• The world lines diagram where objects form tubes over time is not very intuitive.
• The last step is to slice the diagram to form an animation, which will make it more intuitive.

Curvature of the Universe 08:37

• The curvature of the universe causes straight lines to dive into the earth, and it becomes a movement of contraction.
• The rate of this contraction is constant and perpetual because the curvature of space-time, which depends only on the mass of the earth, is always the same.
• It's important to understand that the geometry does not really contract, but the fact that straight lines get closer together gives this impression of contraction.
• The phenomenon is quite similar on the surface of a sphere, where the curvature is constant, but straight lines seem to be perpetually getting closer together.

Visualizing General Relativity 09:28

• The representation of the shrinking grid is the most appealing for visualizing general relativity.
• The earth, because it is very massive, deforms space-time, giving it a curvature.
• For us, the curvature of space-time appears as an endless contraction of the grid.
• In technical terms, the volume contained between geodesics shrinks over time because of the curvature.

Inertial Frames 09:58

• The grid that shrinks represents what we call inertial frames in freefall.
• With respect to this grid, a body that is not subject to any force will conserve its movement.
• If we drop an apple with no initial velocity, as no force acts upon it, it will remain motionless relative to the grid.
• But as the grid contracts, the apple will fall.

Acceleration of the Planet 10:27

• With this image of relativity, it's easy to see that the surface of the planet is constantly accelerating upwards because it's always going against the natural movement of the grid.

Object in Motion 10:41

• Finally, if we throw an object sideways with an initial velocity, no force is applied to it, and it will, therefore, continue in a straight line within the grid.
• But as the grid contracts, the object is constantly pulled back towards the earth.
• That's exactly how the moon orbits the earth and the sun.
• The surface of the planet is constantly accelerating upwards because it's always going against the natural movement of the grid.