# General Relativity

Newton’s laws tell us upon applying a force on an object (say, we a kick a ball), the object acquires an uniform linear motion, which means that the object will follow a straight trajectory at a constant speed (I’m supposing that there isn’t any other forces applied on the object). For instance, if a planet is orbiting a star, if the star would “disappear”, following Newton’s laws, the planet would leave immediately its elliptic orbit and take a straight path.

However, Einstein postulated that nothing can propagate faster than light. If this postulate is general, then Newton had to be wrong, i.e., the planet couldn’t “know” instantly that its host star had disappeared, therefore, its orbit couldn’t change immediately. In the case of the Earth and the Sun, if the Sun disappears, according to Einstein, the Earth would only “feel” this event after, at least, approximately 8 minutes (the time light takes to travel from the Sun to the Earth). For 8 minutes, the planet Earth would continue to orbit the point where the Sun was (in fact, the center of mass of the system).

This means that there is something like an “information network”, which “connects” all bodies. In this network, information cannot travel at an infinite speed. This “network” is the gravitational force itself, which obeys relativity.

There is another problem with the Newton’s law of Universal Gravitation, which led Einstein to believe that this theory had to be wrong, or at least incomplete. (It’s important to note that Newton’s laws had been confirmed and demonstrated beyond question at the time Einstein thought about this; there weren’t any experimental clues to believe that Newton could be wrong about gravitation, so Einstein was very audacious.) The problem is that Newton’s law doesn’t explain “what” gravity is, it only describes “how” it works (note that Science is about explaining “how” and, sometimes, “what”, not “why”, of course, therefore this objection could easily be neglected as irrelevant).

The first big step of Einstein to go beyond Newton’s law was to consider that the acceleration due to the gravitational force is completely equivalent to an acceleration observed in any accelerated movement. In other words, the force that hold us to the ground is “equal” to other forces, like the one which push us against the car seat when we accelerate the car. This is the so-called Equivalence Principle, which was expressed by Einstein as follows:

“We will go therefore to assume complete physical equivalence between a gravitational field and the corresponding acceleration of a reference system. This hypothesis extends the principle of special relativity for systems of reference uniformly sped up.”

This principle implies, for example, the possibility to travel around space maintaining my weight constant! According to Newton’s law it would be impossible, because my weight would decrease as my distance from our planet increases (and it would be back to “normal” when I would come back). That’s not the case, because Einstein was right, and so, theoretically speaking, it’s possible to make a spaceship, which is able to accelerate to compensate the decrease of the gravitational acceleration (and also increase, when returning). This finding may not look like a big deal, but it was crucial for Einstein, and it opened the way for General Relativity. The concept of gravitation was not so mysterious anymore: it was just another accelerated motion, something more “concrete” and easier to study than the abstract concept of gravity.

An accelerated motion is characterized by the variation of the velocity vector. (Basically, a vector is a virtual arrow, which tell us the magnitude and the direction of something. In this case, the velocity vector tells us about the magnitude of the speed, and also about the direction in space of this movement.) I stress “vector”, because it’s possible to have an accelerated motion with constant speed, where only direction is changing. For instance, in uniform circular motion, the object describes a circle with constant speed. Since the trajectory is a circle, this implies that the direction of the velocity is always changing. The acceleration of this motion is well known: it’s the ratio between the square of the speed and the radius of the circle. If we consider that the speed is high enough on this circle, then we have to take into account relativistic effects, but not special relativity, because that theory is only applicable to non-accelerated motion. Notwithstanding, we can expect some similar effects, like space distortion. It is reasonable to ask what space gets distorted? It is the space of the trajectory! In the case of circular motion, this means that the perimeter will be different at different velocities! Does this mean that the radius of the circle is changing (since the perimeter is usually equal to 2πR, R=radius)? No! The radius is a distance which is not affected, because the motion is not occurring in this direction, i.e., the radius is always perpendicular to the instantaneous velocity, so it’s not affected by relativistic effects. Therefore, is the equation P= 2πR wrong? It’s not wrong, it’s just not applicable in this case! This formula supposes that the space is flat (Euclidean space), however, in General Relativity, the space doesn’t have to be flat, it’s actually curved (this is called a Riemann space). Note that a circle drawn on a curved space will have a perimeter that is not given by the equation above. Similarly, the sum of the internal angles of a triangle it’s not 180º in general, it depends on the curvature of the space in which the triangle is drawn. In other words, Einstein’s theory brings some incredible implications: an accelerated movement distorts the space around! Consequently, the gravitation itself manifests through the deformation which massive objects make (all bodies produce a deformation around them, though only massive objects are able to create a “strong” and “noticeable” deformation).

You may be already wondering that this phenomenon may not affect only space, because Special Relativity connects space and time: they are inseparable. So time is also affected!

How can we understand “distorted time”? It’s not easy to “visualize” it, because we don’t “see” time, we just feel its “flux”. Returning to the previous example, let us consider that the object describing circles is a clock. The moving clock will count, like in Special Relativity, a smaller “amount” of time than a clock at rest. However, if the radius is changed, let’s say that it becomes smaller, then, according to the equation mentioned above for the acceleration (which is not exactly correct in this case, but provides a qualitative idea), the acceleration is larger, even if the velocity is kept unchanged, therefore the relativistic effects will be stronger. (It’s equivalent to the case when an object gets closer to a strong gravitational field.)

In books and documentaries, it’s frequent to show this relativistic conceptualization of gravity by showing space as a bi-dimensional membrane: one can consider space as a stretched bed sheet where we place objects (which can represent stars and planets). It’s clear that those objects will deform the sheet according to their mass. It’s a good way to visualize the space deformation (related with gravity), however we should not forget that this analogy might lead us to wrong interpretations of the theory. First, we live in a tri-dimensional world, so the deformation is more complex: it is deformed in itself (not like the sheet, because the deformation also occurs in an additional dimension, which we cannot visualize, because for that we would need a forth dimension). Second, time is also deformed. Third, it’s the mass that deforms space, not weight, like in the sheet, because weight is exactly the consequence of the space distortion (it cannot be simultaneously the cause and the consequence). There are representations for the time distortion, but they are so complex that I think they don’t provide any helpful understanding of the phenomenon, so I won’t present them here. Once again, I should remind you that the relativistic effects are difficult to observe in our quotidian, both from Special and General Relativity. Einstein, as any good physicist, wanted to confirm his theory with experiments. He proposed the following: during a solar eclipse, the Moon obstructs sunlight, preventing light of reaching Earth, and so we can measure how much is a light beam (coming from a distant star) deflected because of the presence of the Sun. This light beam, of course, has to pass very close to the Sun, in order to be deflected by a measurable angle. The role of the Moon is to “turn off” the sunlight, otherwise it would be impossible to distinguish a very tenuous light coming from a distant star within the strong sunlight. Why the Sun? Any other object would be too far or not massive enough to make a detectable deflection. The measured angle was about 0.00049º, which is like the angle of a coin (in the vertical position) seen from a distance of approximately 3 km. The experiment was successfully done and it confirmed the fantastic theory of Einstein.

I must also say that there were a few difficulties concerning the experiment: in the first solar eclipse after the theory was published, the weather was not good (it was cloudy), and the World War I was also an impediment.  In the second solar eclipse, Eddington did the experiment, but the scientific community was not completely sure about the veracity of Eddington’s results, for he had supplementary (pacifistic) motivation to confirm Einstein’s theory (though now it is believed that Eddington didn’t adulterate his data). So only in the third solar eclipse came the awaited confirmation. Notwithstanding, the precision of the instruments was not very good for such a high demanding precision experiment, so, only after Einstein death, the theory predictions were verified beyond doubts.

If you have any questions, comments, remarks, feel free to post them below. I also remind you that English is not my native language, so I’ll appreciate any corrections to the text.

Marinho Lopes

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