The Maths of General Relativity (7/8) - The Einstein equation

[Music] welcome back to science clique today the mathematics of general relativity part 7 the einstein equation our mathematical model is finished on the one hand we know how to describe a space-time geometry its curvature and to deduce how objects naturally move thanks to the geodesic equation for this we have developed a whole set of mathematical tools such as the metric tensor the christopher symbols the riemann curvature tensor the richie tensor and the richie scalar on the other hand we also know how to describe the energy content of the universe and thus the bodies that it contains along with their motion using a single tool the energy momentum tensor the most powerful idea in general relativity will be to equate these two notions using a single relation we will write an equality between curvature and content this equation which reads as an equality between the geometry and the content of the universe is called the einstein equation in our mathematical model the einstein equation is the only one that cannot be proven it is inferred from observations postulated because it describes very well the world around us this equation contains all the predictive quality of general relativity it allows us to directly relate the abstract mathematical model to concrete predictions about the universe usually the einstein equation takes the following form all terms to the left of the equation represent the geometry of space time they involve the richy tensor the richie scalar and the metric tensor the term to the right of the equation corresponds to the contents it is simply the energy momentum tensor multiplied by a certain constant which determines the intensity of gravity in the universe this constant depends on the speed of light and on newton's gravitational constant it is chosen so that relativity is compatible with older theories and recovers newtonian gravity when the pull is weak it is important to understand that the einstein equation is extremely hard to solve as we saw previously the richie tenser and the richie scalar which appear on the left of the equation contain complex calculations involving derivatives sums and products of the metric tensor moreover the two sides of the equation are closely related the content of the universe tells space-time how to bend but conversely the curvature of space-time dictates how objects move solving such an intricate problem on paper is almost impossible in most cases it is necessary to make approximations and computer simulations to solve such [Music] problems in the einstein equation the curvature of space-time is embodied by the rich tensor and the richie scalar which is its contraction but in a four-dimensional space time like ours the equation has a very interesting symmetry it remains valid if we swap the energy momentum tensor with the richie tensor and invert the gravitational constant this new equation is completely equivalent and we will see that in some cases it can be easier to use [Music] einstein's equation is very complex but it still accepts a number of exact solutions the simplest and most practical is the following let us imagine an empty universe in which we place a massive body of mass capital m we suppose that it is spherical static that is to say that it does not evolve over time and that it has no electric or magnetic properties this body may represent a star like the sun or a planet like the earth our goal is to determine the metric tensor and therefore the geometry of space time at a point outside the object first let's choose a coordinate system we decide to observe the point from a very far distance and we measure time on our clock t the distance between the point and the center of the body r and two angles theta and phi our point sits in a vacuum therefore the energy momentum tensor is zero using the second version of einstein's equation we deduce that the rich utensil at this point must also be zero to determine the metric from the rich tensor the calculations are tedious and require solving multiple differential equations but thanks to the symmetries that we postulated for the problem it is still possible to obtain an exact expression this is the schwarzschild metric [Music] the schwarzschild metric depends on the mass of the object if the mass is equal to zero we recover the minkowski metric describing an empty space-time but around the body like the earth the schwarzschild metric indicates that space-time is curved and the closer we get to the planet the greater the curvature becomes in the case of a planet its surface prevents us from going too close to the center and inside the planet the schwarzschild metric is no longer valid but in the case of a sufficiently compact object whose surface is very close to its center we would be able to get close enough in order to reach a distance at which the temporal term of the metric with these coordinates is zero we have reached a point of no return time is infinitely stretched compared to the outside this is the schwarzschild radius the horizon of a black hole you

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Length: 7min 29sec (449 seconds)

Published: Tue Jan 05 2021