From experience we know and we learned in the preceding chapter that mass creates gravity. We saw that gravity bends space-time, so mass itself should bend space-time. What is the simplest possible way that mass can bend space-time?

7.1 Gravity in a Lonely Cloud

We simplify the situation as much as possible. We drive with our spaceship into some empty region of space so that no large mass is nearby, and we are in an inertial state. Then we carefully place outside the spaceship a small cloud of dust such that the dust particles are resting near each other. In Fig. 7.1 we sketched the dust particles as black balls. Then we gently move away and sit motionless near the cloud. The cloud as well as we are in an inertial state.

Fig. 7.1
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A dust cloud begins to shrink under its own gravity

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From experience we expect that gravity tends to move mass together. In fact, as soon as we leave the dust particles alone, the cloud begins to shrink . What is the simplest quantity to describe the rate of shrinking? This is how much the volume of the cloud decreases per time, that is, at which speed the volume decreases. What is the simplest quantity to describe the initial rate of shrinking? That is how much the shrinking speed starts to change from zero which means the acceleration of the volume decrease and the change of volume per time, per time. The start of the shrinking rate should depend on how much mass is in the cloud. In the simplest case, the rate should grow in proportion to the mass in the cloud. Twice as much mass in the same volume should produce twice as large shrinking rate.

Indeed, this is what nature has chosen as the law of gravity!

Let us give a numerical example of how a volume may begin to shrink. Suppose that we prepare a dust cloud of the size \(10\times 10\times 10\) meters, that is a volume of \(1000 \text{ m }^{3}\). When this cloud begins to shrink, we measure its volume each second. In Fig. 7.2, we see in the column of boxes on the left how much volume is left after 0, 1, 2, 3, 4 seconds. The rate at which the volume shrinks per time, that is from second to second, stands in the middle column of boxes. Finally, we find in the right column the rate at which this change of the volume itself is changing per second, that is the change per second. We see that the volume begins to shrink at a rate of 2 \(\text{ m }^{3}\), per second, that is \(2{\displaystyle \frac{\text{ m }^{3}}{\text{ s }^{2}}}\).

Fig. 7.2
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Example of how a volume may begin to shrink

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However, we wanted to know how mass bends space-time, not just how mass clumps. Here the equivalence principle comes again.

We know that test masses, especially small dust particles, will react to gravity in the same way, no matter what their matter consists of. Therefore we can think of the dust particles as just probing the space-time around them. While the cloud is beginning to shrink, the test masses accelerate towards each other. However, remember that nothing pulls the test masses together. The test masses accelerate towards each other because inside the cloud, space itself is beginning to shrink in time. Hence when the cloud begins to shrink, the small volume of space itself begins to shrink, as time proceeds a little bit. This is how mass is bending a small piece of space-time.

We have found the Einstein law of gravity!

7.2 Einstein Equation of Gravity

The Einstein law of gravity or under another name the Einstein equation of gravity , is:

The rate at which a small enough, resting cloud of matter begins to shrink is in proportion to the mass in that cloud. The constant of proportion is \(4\pi \) times the gravity constant .

Why not just “gravity constant” but \(4\pi \) times it? This has purely historical reasons, nothing else. The value of the gravity constant is about \(6.67\times 10^{-11}\) in appropriate units. You find it for reference in the Table A.1.

The mass density , that is the mass per volume, is nearly constant if we look at a small enough volume. It often is more practical to ask for the shrinking rate per volume, that is the relative shrinking rate . The Einstein equation of gravity in terms of the mass density , that is in terms of the mass per volume, reads

The relative rate at which a small enough, resting cloud of matter begins to shrink grows in proportion to the mass density in that cloud. The constant of proportion is \(4\pi \) times the gravity constant.

However, mass is energy, divided by the square of the speed of light. In terms of the energy density, the Einstein equation of gravity for energy reads

The relative rate at which a small enough, resting cloud of matter begins to shrink grows in proportion to the energy density in that cloud. The constant of proportion is \(4\pi \) times the gravity constant, divided by the square of the speed of light.

7.3 Enter Pressure

Fig. 7.3
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A dust cloud inside a gas. We sketched the gas particles as small black disks and we show their momentary speed by small arrows. The dust cloud contains also the energy coming from the disordered moving particles in the gas. This energy also creates gravity

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We assumed that we can always place our test masses inside the cloud, such that, at least at the beginning, they rest with each other. What happens if our cloud does also contain pure energy , that is light? Pure energy moves always at the speed of light, so we cannot place it like our masses. This light behaves much like a gas. Suppose we put our masses inside a gas. The simplest case is when at least near the cloud the gas looks everywhere the same. Therefore the gas particles, or light in the case of pure energy, are constantly entering and leaving the cloud which we marked with test masses. The number of gas particles which leave the cloud per second is the same as the number of particles which enter the cloud during this second. Still, the particles of the gas always bump against each other so that they never rest relative to the test masses in the cloud. You can see this in Fig. 7.3. Because just as many gas particles enter the cloud per second as they are leaving the cloud, we can imagine that for every gas particle that is about to leave the cloud, a gas particle from the outside bounces against it to keep the gas inside the cloud together. In other words: the outside gas presses from any of the three directions of space. But pressure is energy density, as we saw in Sect. 1.12. This means that we have to add the sum of the three pressures in the three directions of space to the energy density in the cloud.

As a result, we get the complete Einstein equation of gravity :

The relative rate at which a small enough, resting cloud of matter begins to shrink, grows in proportion to the energy density plus the pressures in each of the three directions in that cloud. The constant of proportion is \(4\pi \) times the gravity constant, divided by the square of the speed of light.

However, inside the cloud, the gas particles are not only moving around but also bumping against our test masses. If pressure is constant in all directions, the test mass will not move on the average. However, under a microscope we see the test mass vibrating under the bombarding small gas particles. Hence we cannot place our test masses perfectly at rest. The very concept of pressure does only make sense if we do not look too closely, that is if our cloud of masses is not too small.

Einstein himself was aware of thisFootnote 1:

We know that matter is built up of electrically charged particles but we do not know the laws which govern the constitution of these particles. In treating mechanical problems, we are therefore obliged to make use of an inexact description of matter which corresponds to that of classical mechanics. The density [. . . ] of a material substance and the hydrodynamical pressures are the fundamental concepts upon which such a description is based.

With the exception of Sect. 9.9, we deal in the following sections with matter whose pressure is so small that we can ignore it so that we can still use the model of the dust cloud of the Sect. 7.1.

7.4 Enter Speed

The law of how mass creates gravity should fit in with the theory of special relativity. That is: if we are in an inertial state and the cloud is passing us free-falling at some speed, then the Einstein equation of gravity should not change. However, we know that all kinds of things change. First of all, the cloud has more mass by a factor \(1/\gamma \).

Fig. 7.4
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A moving dust cloud begins to shrink

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Therefore it creates more gravity! However, also the reaction of the volume changes. The length of the cloud in the direction of the speed is now smaller by this \(\gamma \) factor, while its size vertical to the speed does not change, as we can see by comparing the Figs. 7.1 and 7.4. Hence the volume of the moving cloud is smaller by this factor \(\gamma \). What is more, the volume begins to shrink in less time. We know from Sect. 2.1 that our time runs faster than the proper time of the moving cloud by the inverse factor \(1/\gamma \). Hence the cloud shrinks faster by this amount and it begins to shrink even at the faster rate of \(\left( 1/\gamma \right) \times \left( 1/\gamma \right) \). In total the volume of the cloud begins to shrink faster by the factor

$$ \underbrace{\gamma }_{\text{ Volume } \text{ shrinks }}\times \underbrace{\left( 1/\gamma \right) \times \left( 1/\gamma \right) }_{\text{ shrinking } \text{ begins } \text{ faster }}=1/\gamma $$

which is precisely the amount by which the mass in the cloud became larger. Hence the Einstein equation of gravity holds also for free-falling clouds.

7.5 Enter Outside Masses

In Sect. 7.2 we described how mass creates gravity for a carefully prepared small cloud of dust, far away from other large masses. In reality there are stars, planets and the like outside the cloud. Let us consider them. Because our cloud is small enough, it moves like a test mass under the gravity of outside masses, that is, it free-falls as you can see in Fig. 7.5. Based on the equivalence principle, the cloud is free-falling and reacts as if there would be no large mass nearby. The shrinking rate does only depend on how much mass is inside the dust cloud. The law of gravity does not change at all!

The outside masses can only change the form of the small free-falling cloud of mutually resting test masses, as we already saw in Fig. 5.12.

Fig. 7.5
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A shrinking small cloud acts like a test mass and free-falls under gravity of a nearby mass

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7.6 Local and Global Space-Time

The Einstein law of gravity is formulated for a small enough piece of space (our “cloud”) and a small enough period of time in which the cloud “begins” to shrink, that is altogether in a small enough piece of space-time, thus a local piece of space-time. How do we get the global picture of what happens under gravity? Take for example the sun. We know that the sun will bend space-time and that far away from it, space-time will be flat. We insert here and there small clouds of test masses to probe space-time. Then we patch these local pictures to a smooth map of space-time. It is a little like building a globe of patches showing parts of the surface of Earth. Only if we have connected them smoothly do we see that Earth indeed has the form of a globe. We get the global picture of how the globe is bending.

However, in space-time, things are more involved than for a surface. If the volume begins to shrink near a mass, then space-time must bend to connect the nearby space where there is no mass. What is more, not only does the volume of a cloud shrink in time, but time itself depends on the relative speed of the free-falling masses in a cloud. So we have to follow how space and time evolve.

7.7 Bended Space-Time and Tensors

For a bended surface in an ordinary three-dimensional space, it is enough to know one number per surface point, the Gauss curvature understand how a surface bends. This curvature basically measures how much the ratio of the perimeter and the diameter of a small circle around a point differ from \(\pi \), as in Fig. 7.6. In other words, it measures to what extent the Euclidean geometry is wrong. However, space-time has more directions in which it can bend so that one number is not enough. Careful thinking shows that for each point in space-time we need twenty numbers to render the bending rate.

Fig. 7.6
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Measuring how much the surface of the Earth is bending near the North Pole. Draw a circle around the North Pole, on the surface of Earth. The perimeter of the circle is \(\pi \) times the black dashed diameter which runs inside the Earth. However, the diameter measured on the bended surface of Earth is the gray arc spanning from left to the right, and this is longer than the dashed straight line. Hence on the surface of the Earth, the ratio of the perimeter and the diameter of a circle is less than \(\pi \), similar as in Fig. 6.8 for a circle around a perfect ball

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Fig. 7.7
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A material bends

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It was Riemann who generalized the theory of bended surfaces of Gauss to three and more dimensional spaces. The twenty numbers are named after him, the Riemann curvature tensor . For instance, for our cloud of dust particles of Sect. 7.1 which at first are resting near each other, the Riemann curvature tensor describes how the form and the volume of the cloud is beginning to change as time proceeds.

The Einstein equation of gravity describes only how the volume of the cloud is beginning to change but not how its form does. In order to express the Einstein equation of gravity in terms of tensors, we need therefore a simplified version of the Riemann curvature tensor, namely the Ricci tensor .

The mathematical toolbox for such calculations is tensor analysis . “Tensor” is a word of Latin origin and means “tension”. In fact, engineers use tensors to describe how materials of bridges and the like bend under tension. Materials bend inside space-time which we can easily show, as in Fig. 7.7. However, it is much harder to imagine how space-time itself bends. We will show in Sect. 9.7 that there is a fundamental difference between a bending material and a bending space-time.

If you like to see the mathematical expression for the Einstein equation of gravity, in terms of the Ricci tensor, please have a look at Appendix A.5.

7.8 How to Solve the Einstein Equation of Gravity

Not only obtaining the global picture is difficult but focusing on one small cloud is difficult as well. What happens after that cloud has begun to shrink? The test masses inside have just begun to move against each other, free-falling. To follow them, we must look into even smaller parts of the cloud, adjust our speed and time, look for the shrinking rate and so on. That is because the masses in the cloud act on space-time at the same time by bending it, and react on space-time by free-falling. We cannot easily separate action and reaction of mass. In other words, not every pattern of masses about which we think can be realized in space-time. That makes gravity unique among the interactions in nature. That’s just what makes solving the Einstein equation of gravity so difficult.

In fact, we know only a few exact solutions to the Einstein equation of gravity. In order to solve it exactly, we must place mass in a balanced way so that we can handle the self-interaction.

  1. 1.

    For the perfect ball of mass, as in Sect. 6.5, the same amount of mass is sitting in every direction from its center. Hence mass balances out. Gravity depends only on the distance to the center of the ball, not the direction. All that remains to be done is to find the \(\gamma \) factor, that is at which speed test masses will vertically free-fall at a certain distance from the center. We will do so in Chap. 8 and get the Schwarzschild exact solution . The Newton law of gravity will follow from it for weak gravity. This case is most important because stars and planets are perfect balls to a considerable degree.

  2. 2.

    Observations show that over large enough distances, there is more or less the same amount of mass everywhere in the universe. Then again mass is balanced because the bending space-time must be the same everywhere in space and can only depend on time. We thus will see in Sect. 9.8 how the big-bang of the universe comes about.

And basically, that’s that! There are some slightly more general solutions for a perfectly rotating ball, or a perfect ball with electric charges, or even a rotating perfect ball with electric charges, and the like. However, for an arbitrary collection of masses, we have to use tensor analysis and have to solve approximately the Einstein equation of gravity on a computer.

Make no mistake, we cannot prove that the law of gravity must be the Einstein equation. We only said that the Einstein gravity law is the simplest possible one. Physicists constructed other theories in which mass changes the space-time around it in a more complicated way. However, then it becomes more and more difficult to accommodate the law of gravity to the equivalence principle. In other words: it is hard to construct a theory which correctly describes how mass creates gravity and how mass reacts on gravity. What is more, experiments and observations showed again and again that only the Einstein gravity law plus the equivalence principle seem to lead to the correct law of gravity.

Now look how beautifully everything falls into place:

  1. 1.

    We fixed positions and speeds for test masses inside a small cloud such that they rest relative to each other and then let them loose. The simplest way mass can bend space-time is the rate at which the volume begins to shrink, while free-falling.

  2. 2.

    This shrinking rate does only depend on the mass inside the volume.

  3. 3.

    It does so in the simplest manner, that is in proportion to the mass inside.

  4. 4.

    The gravity law fits in with the theory of special relativity because it does not change even if we free-fall relative to the free-falling cloud.

  5. 5.

    One physical quantity, that is to say the mass inside the cloud, determines how another geometrical quantity, that is to say its volume, changes with time. In other words: the Einstein equation of gravity does not regulate directly how the form of the small volume changes with time!

  6. 6.

    This is done by masses outside the cloud. They deform the cloud but do not change the volume to fit the global picture of the bending space-time.

The theory of using the Einstein equivalence principle and the Einstein gravity law is the theory of general relativity .

Next we want to see the gravity law in action. How does the motion of the planets around the sun follow from it? How does this law fit in with the classic Newton way of looking at gravity, where the sun seems to “pull” Earth around it? And what new effects can we find?