# If energy in a flat space is zero, are we saying that a flat space is isolated? [closed]

UPDATED: AT LEAST 5 PEOPLE SAVED FROM EXISTENTIAL CRISIS

In classical thermodynamics, an isolated system is one where internal energy sums to zero, since energy is zero in a space with a minkowsi metric, are minkowki spaces considered isolated?

I am surprised about the how difficult it is for people to understand a simple question. As it is well known, an isolated system is a physical system that does not interact with its surroundings. It is also one where the internal energy of an isolated system is constant.

Now, taken together, by definition, there is no way for an isolated system to provide any information to its surroundings, so there is no way that the isolated system could contribute to the total energy of whatever surroundings it is in, for that would require some sort interaction with the surroundings. So it should be intuitive that in the context of its surroundings, the energy of an isolated system is zero.

Since there can not be a change in internal energy, and the internal energy of a system is the sum of all the kinetic and potential energies, then it is a perfectly true statement to say that all the internal energy sums to zero. A fact that is true by definition (whether people abuse the term isolated system is an altogether different issue).

This is the first sense that the question makes sense. In the context of GR, if mass-energy creates curvature, and an isolated system has no mass-energy contribution to its surroundings, then it seems natural to understand that system as not creating curvature and in fact being defined as flat (e.g. having a Minkowski Metric).

This concept should not be new to anyone, and in fact has been discussed by individuals as famous as Hawking within the context of the zero energy universe. Since the universe is proposed as perhaps being the only hypothetical example of a physically isolated system, this represent the second sense that the above question makes sense. In fact, we find out:

"Due to quantum uncertainty energy fluctuations such as electron and its anti-particle a positron can arise spontaneously out of nothing but must disappear rapidly. The lower the energy of the bubble, the longer it can exist. A gravitational field has negative energy. Matter has positive energy. The two values cancel out provided the universe is completely flat. In that case the universe has zero energy and can theoretically last forever."

We can also find this whole issue being relevant in any theory that proposes that matter and energy is in fact trapped at the event horizon of a black hole. In the context of Birkhoff's theorem, there is a corallary that "the metric inside a spherical cavity inside a spherical mass distribution is the Minkowski metric". Since the Minkowski metric is a metric of spacetime, it is related to an earlier question I asked: "Is the observable universe enclosed by an infinitely dense shell?" from which it is tempting to conclude that any observer in a universe with a Hubble radius will always see a flat space because infinite mass in a thin, nearly spherical shell must create a pretty flat space inside the observed cavity. This is the third sense in which the question is relavent.

If we go back an earlier question on the positive mass conjecture and read the paper by Witten, we find the following:

"Although there is no satisfactory way to define the local energy density when gravity is present, one can define the total energy of a gravitating system. The total energy (and momentum and angular momentum) of a gravitating system can be defined in terms of the asymptotic behavior, at large distances, of the gravitational field. However, it is far from obvious that the total energy so defined is always positive.

It is an old conjecture that this total energy is in fact always strictly positive, except for flat Minkowski space, which has zero energy. This matter has been studied by a variety of means."

We also know from Beckenstein and Hawking that black holes (objects that occur in GR, Georg, but not really in chemistry ;-) ), are in fact thermodynamic systems. So the fact that they do have a temperature, and are observable is at least of some interest to people who think about metrics and spacetime, which is the fourth sense that this question is relevant.

So I guess in summary, I don't really understand why this question is hard for anyone to understand...although if it turns out that in some very unlikely situation that somehow I rocked the world of physics so bad that I shook the house of cards to the ground, just let me know.

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I'm not sure how to make any sense of this terminology, there are both flat and not flat vacuum solutions to the Einstein field equations of general relativity. If you re-interpret them as systems coupled to a heat bath or something similar, what is your heat bath? And what is the reason to ask this question in the first place? – Tim van Beek Feb 2 '11 at 11:45
-1 because this doesn't make any sense. Try to elaborate and make the question clearer. In particular isolate the parts that are just statements (like the statement about TD) from the actual question. – Marek Feb 2 '11 at 13:07
"Classical Thermodynamics" does not know about Minkowski metrics! But also, my knowledge of internal energy of a isolated system is different of Yours. There is a rather good page on this: en.wikipedia.org/wiki/Internal_energy -1 – Georg Feb 2 '11 at 13:08
oh dear lord...I come home and at least 5 people are having a their own personal existential crisis – Humble Feb 3 '11 at 1:52

## closed as not a real question by kakemonsteret, QGR, pho, mbq♦Feb 3 '11 at 8:45

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