What is the meaning of "Energy is locally never negative"?

[Cosmology2015]

Hello to all members!
I heard on a documentary about general relativity and singularities that "energy is locally never negative". However, I was not able to get grasp the meaning of this term. Could someone explain to me the meaning of this term in the framework of general relativity. I really appreciate any response .

 

[PAllen]

It is hard to be sure without the full context, but my guess would be that in general relativity physics must be locally identical to special relativity. In classical special relativity there is no negative energy.

A caveat is that quantum field theory (a nonclassical special relativistic theory) allows local negative energy (e.g. the Casimir effect); and Hawking radiation can be interpreted as requiring the existence of negative energy. Thus, non-classically such a blanket statement is dubious. Nonetheless, many physicists believe that at macroscopic scales, such effects will be bounded so as to prevent a-causal effects (in general relativity, the existence negative energy on large scales would allow several types of time machine to be built).

 

[George Jones]

Adding to what PAllen wrote, this is known as the weak energy condition. If $T$ is the stress energy tensor, and $u$ is the 4-velocity of of an arbitrary observer (i.e., is timelike and future-directed), the weak energy condition is expressed mathematically as

$$T_{\mu\nu} u^\mu u^\nu \ge 0.$$

 

[Cosmology2015]

Hello to all members!
First, I would like to thank George Jones and PAllen for the quick response! With the answer that I was offered, although I do not have the mathematical training to study this subject at the moment, I could find out more about the subject. Wald's classic book on general relativity provides a continuation of the response known as the strong energy condition:

$$T_{ab}u^{\mu }u^{\nu }\geq -\frac{1}{2}T$$
At this point, another question arises for me. Why this condition combined with Einstein's field equations leads us to the fact that the universe had a beginning (Big Bang)? Again, I really appreciate any response .

 

[PAllen]

$$T_{ab}u^{\mu }u^{\nu }\geq -\frac{1}{2}T$$
At this point, another question arises for me. Why this condition combined with Einstein's field equations leads us to the fact that the universe had a beginning (Big Bang)? Again, I really appreciate any response .

That is complex theorem requiring sophisticated math - it was the pinnacle of Hawking and Penrose work circa 1970. It is possible someone can describe the thought process of the theorem at a more elementary level, but I can't (especially as I've never worked through the theorem's proof at all). I believe this proof is presented in "The Large Scale Structure of Spacetime", by Hawking and Ellis.

 

[marmoset]

I'm no expert but a sketch of the arguments leading to the singularity theorems is given in chapter one of Hawking and Penrose's book 'The nature of space and time', which is the content of a series of lectures they gave together with a relatively easy going style (lots of easier sketches of much harder arguments). Steven Hawking's lectures from the book can be found here: https://arxiv.org/abs/hep-th/9409195.

According to this book (around page 8 of the arxiv article above), intuitively the energy conditions come into the singularity theorems to show that particle worldlines leaving a point can meet again, and from here show that "there can be particles whose history has a beginning or end at a finite time". There is an equation telling you how much nearby worldlines come together or move apart (called the Raychauduri-Newman-Penrose equation in the book), and the energy conditions let you show that a term in this equation is always positive (that is making the worldlines converge as opposed to diverge).

 

[Cosmology2015]

Hello to all members!
First of all, again, I would like to thank all the answers. These answers are very important to me as this is a matter that although I do not have the proper mathematical training, I have a strong desire to learn. I intend to study hard all references to try to understand what is currently possible considering my mathematical training. The explanation provided by Marmoset seems to be the ideal starting point. I hope to succeed in understanding this topic. If anyone has more references, please write on this thread. Again, I really appreciate any response .

 

[PAllen]

Hello to all members!
First of all, again, I would like to thank all the answers. These answers are very important to me as this is a matter that although I do not have the proper mathematical training, I have a strong desire to learn. I intend to study hard all references to try to understand what is currently possible considering my mathematical training. The explanation provided by Marmoset seems to be the ideal starting point. I hope to succeed in understanding this topic. If anyone has more references, please write on this thread. Again, I really appreciate any response .

I glanced at Marmoset's link, and it is great. I doubt you could find any more accessible treatment leading up to the singularity theorems. You should be able to get a qualitative sense of the arguments without following the math. Good luck!