Einstein Tensors and Energy-Momentum Tensors as Operators

In summary, the conversation discusses whether tensors can be seen as operators on two elements, specifically two four-dimensional vectors. It is mentioned that these tensors are functions that take two four-dimensional vectors as input and give a scalar as an output. However, it is noted that it would be unconventional to call them 'operators' as the term is usually used for functions with one input that returns a result of the same type as the input. The conversation then explores which four vectors can be used for the tensors to operate upon in order for the resulting scalar to have a physical meaning. It is clarified that for the energy-momentum tensor, any two vectors can be used and it gives the rate of flux in a certain direction. For the Einstein tensor, it
  • #1
Alain De Vos
36
1
Can these tensor be seen as operators on two elements.
So given two elements of something they produce something, for instance a scalar ?
 
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  • #2
Yes, they are both functions that take two four-dimensional vectors as input and give a scalar as an output.
Although AFAIK there are no strict rules about this, it would be unconventional to call them 'operators'. From my observation, 'Operator' seems to usually be used to mean a function with one input that returns a result of the same type as the input (eg vector in and vector out, or scalar in and scalar out).
 
  • #3
I wander, which four vectors do you have to choose for the tensors to operate upon in order for the resulting scalar to have an physical meaning.
Does it work on any two covariant vectors, only two and the same displacement vectors, the 4-speed vector ?
 
  • #4
For the energy-momentum tensor ##T##, any two vectors can be used. ##T(\vec v_1,\vec v_2)## gives you the rate of flux, in direction ##\vec v_1##, of the energy-momentum item defined by ##\vec v_2##.
For the Einstein tensor it will also be any two vectors, but somebody else will need to tell you what the physical significance is.
 

What are Einstein tensors and energy-momentum tensors?

Einstein tensors are mathematical objects used in Einstein's theory of general relativity to describe the curvature of space-time. Energy-momentum tensors are mathematical objects that describe the distribution of mass, energy, and momentum in a given space-time.

How are Einstein tensors and energy-momentum tensors related?

Einstein tensors and energy-momentum tensors are related through Einstein's field equations, which describe how the curvature of space-time is related to the distribution of mass, energy, and momentum. In other words, energy-momentum tensors are used to calculate the Einstein tensors, which in turn determine the curvature of space-time.

What is the significance of using tensors as operators in this context?

Tensors are used as operators in this context because they allow for the mathematical representation of the symmetries and dynamics of space-time. This allows for a more accurate and comprehensive understanding of the effects of mass, energy, and momentum on the curvature of space-time.

How are Einstein tensors and energy-momentum tensors used in practical applications?

Einstein tensors and energy-momentum tensors are used in practical applications such as predicting the behavior of objects in strong gravitational fields, such as black holes, and in understanding the expansion of the universe. They are also used in developing technologies such as GPS, which relies on the principles of general relativity.

Are Einstein tensors and energy-momentum tensors still relevant in modern physics?

Yes, Einstein tensors and energy-momentum tensors are still highly relevant in modern physics. They are an essential part of Einstein's theory of general relativity, which is still one of the most widely accepted theories for describing the behavior of gravity. They also play a crucial role in the ongoing research in cosmology and the study of the universe.

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