Einstein Field equations for dummies

In summary: The idea is that for every point on the path, there is a unique point on the path that is an extremal point.This is a bit more complex than the definition for extremal points, but it's still pretty simple.In summary, extremal proper time is maximal aging, and extremal path is the coordinate system used by the Universe.
  • #1
alecrimi
18
0
Hi all!
When we talk about the Einstein Field equations.
What do we mean with "extremal proper time" or "extremal path"?
Why "extremal" ?
and why "proper" ?

and why do we need to introduce the concept of "geodesic" ?
Cheers
 
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  • #2
Why "extremal" ?
and why "proper" ?
They are jargon words. What else would you call them?

Space-time is tricky to talk about - these concepts help. Remember, you now have four dimensions ... eg. what would a "maxima" in 4D mean?
https://www.physicsforums.com/archive/index.php/t-488751.html

and why do we need to introduce the concept of "geodesic" ?
Same reason we have any kind of coordinate system - only the geodesic is the coordinate system used by the Universe... which makes it special.
 
  • #3
In the context of proper time "extremal" mostly means maximal in practice. And proper time is what you measure with a wristwatch, or by aging. So you can think of "extremal proper time" as "maximal aging", usually.

However, there is a significant difference between "mostly means in practice" and "always means".

To understand more precisely the definition of extremal, first consider a function of a single variable, y = f(x)

An an extremal point, dy/dx = 0, i.e. the slope is horizontal.

Extremal points can be either a maximum, a minimum, or a saddle points.

If all this isn't a review, or if you lack calculus, you may need to do some further reading and research to fully understand this. (I'm sorry, but I don't know your background).

A quick example might help. Rather than draw graphs, which is the clearest, I'll use some well known simple equations:

y=x^2 has a minimum at x=0
y = -x^2 has a maximum at x=0
y = x^3 has a saddle point at x=0

We can apply similar definitions to functions of more than one variable, in which case we require all the derivatives to vanish to have an extremal point.

In the case of an extremal path, we do a bit of fancy mathematical footwork to extend the same basic definition we use with one variable to an infinite number of variables.
 

1. What are the Einstein field equations?

The Einstein field equations are a set of ten equations that describe the relationship between the curvature of space-time and the energy-momentum content of the universe. They were developed by Albert Einstein as part of his theory of general relativity.

2. Why are the Einstein field equations important?

The Einstein field equations are important because they provide a mathematical framework for understanding the nature of gravity. They have been extensively tested and have accurately predicted a wide range of phenomena, such as the bending of light by massive objects and the existence of black holes.

3. What do the Einstein field equations tell us about the universe?

The Einstein field equations tell us about the structure and dynamics of the universe. They show how space-time is curved by the presence of matter and energy, and how this curvature affects the motion of objects in the universe. They also provide a way to understand the expansion of the universe and the formation of large-scale structures such as galaxies and clusters of galaxies.

4. How can I understand the Einstein field equations?

Understanding the Einstein field equations can be challenging, as they involve complex mathematical concepts. However, a basic understanding can be gained by studying the principles of general relativity and the concepts of space-time curvature and energy-momentum. There are also many online resources and books that offer simplified explanations and visual representations of the equations.

5. Can the Einstein field equations be simplified?

The Einstein field equations are already a simplified version of the full theory of general relativity. However, there are certain situations where they can be further simplified, such as in the case of a universe with a high degree of symmetry. Additionally, various approximations and numerical techniques can be used to make the equations more manageable for specific applications.

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