# Why black holes have positive energy?

How could Hawking get the conclusion "[...] and black holes have positive energy. That’s why empty space is stable. Bodies such as stars or black holes cannot just appear out of nothing. But a whole universe can."? It would make sense to me, if a black hole has negative energy.

Quote from book The Grand Design, page 180:

If the total energy of the universe must always remain zero, and it costs energy to create a body, how can a whole universe be created from nothing? That is why there must be a law like gravity. Because gravity is attractive, gravitational energy is negative: One has to do work to separate a gravitationally bound system, such as the earth and moon. This negative energy can balance the positive energy needed to create matter, but it’s not quite that simple. The negative gravitational energy of the earth, for example, is less than a billionth of the positive energy of the matter particles the earth is made of. A body such as a star will have more negative gravitational energy, and the smaller it is (the closer the different parts of it are to each other), the greater this negative gravitational energy will be. But before it can become greater than the positive energy of the matter, the star will collapse to a black hole, and black holes have positive energy. That’s why empty space is stable. Bodies such as stars or black holes cannot just appear out of nothing. But a whole universe can.

To understand this, you have to understand what it means to have negative or positive energy.

Following the example given by the text, the gravitational attraction of the Earth and Moon is a system with negative energy. This is because it requires an energy input to undo the system. One must exert energy to separate them, thus requiring energy input to the system to break it apart.

The matter comprising the Earth however is all positive energy. You can get an energy output from this matter and thus it must be positive. An example of energy output might be a gravity assist around the Earth, radioactive decay of nuclear materials in the Earth, or matter-antimatter annihilation of Earth particles. Any process that involves a matter interaction producing energy in the form of heat/light/whatever takes that energy out the Earth's matter and represents an energy output. Note that energy in the matter may take many forms such as heat, motion, or simply the intrinsic energy comprised in matter (via the famous $E=mc^2$ equation)

The black hole, much like the Earth, is simply comprised of matter (albeit in some unknown form at the singularity) and as such energy can be extracted from this matter, making it a system with positive energy.

• Thx for your answer. Is the negative gravitational energy of black hole greater than the positive energy of the matter particles the black hole is made of? If not, how could the energy of the universe remains zero? – lu yuan Mar 1 '17 at 12:01
• @luyuan I'm not sure there's an answer to this question. The problem is that it's wildly complicated to calculate the total negative gravitational energy of the black hole. Since the force of gravity has an unlimited range, every object in the universe is within a given black hole's gravitational field and thus contributes to the negative energy. Not to mention, the black hole itself has negative gravitational energy which you could potentially negate by inputting energy to deconstruct the black hole. But how does one break down a black hole? Is it even possible? – zephyr Mar 1 '17 at 13:48

The reason is because of the positive energy theorem in GR. See, for a brief qualitative account, wikipedia.

It's probably a little too advanced to explain in 'The Grand Design'.