Thought Experiments With Gravity
Some ingenious thought experiments that demonstrated the unexpected consequences of gravity on light
Even before completing general relativity, Einstein had devised ingenious thought experiments, which demonstrated the unexpected consequences of gravity on light and time. The predictions of those thought experiments were later all confirmed by observation
The most characteristic element of Einstein’s genius was probably his ability to devise thought experiments. Using imaginary trains and elevators, the German physicist managed to predict the existence of phenomena very far from common intuition. It often took many years for technologies to allow some of Einstein’s predictions to be verified by observation. Still, when those verifications finally arrived, the scientists could not help but observe the perfect agreement of the observational data with what the thought experiments had prefigured.
Exemplary cases of physical phenomena anticipated by pure thought experiments are two effects due to gravity: the bending of light and the time dilation under the action of a gravitational field. Both derive from the surprising consequences of one of the cornerstones on which Einstein’s theory of gravity is based: the equivalence principle. This is the idea that, within a sufficiently small reference system, such as the inside of an elevator or a rocket sealed off from the outside, an observer is unable to distinguish the effects of inertia from those of gravity. Without looking outside, he is unable to say, for example, whether the fact that his body and the objects surrounding him are drawn to the floor is due to the presence of a gravitational field or the thrust imparted by a constant acceleration in the opposite direction.
Gravity curves the path of light
Consider the following two cases:
- an observer locked in a motionless elevator cabin on the surface of the Earth and
- an observer who is, instead, in space, far from any gravitational field, closed inside a rocket driven by an engine that yields an acceleration of 9.8 m/s², precisely equal to the acceleration of gravity felt on Earth at sea level.
For the equivalence principle, the physical laws that the two observers experience are exactly the same. A hammer, for example, will fall on the floor with the same speed both in the elevator cabin that is motionless on Earth and in the rocket that accelerates in space. But this equivalence has curious consequences for the behavior of light. Let’s imagine that a laser beam is aimed in the rocket’s cockpit from the outside, through a porthole. As the laser beam passes through the cockpit, the rocket continues to accelerate under the thrust of its engines. For an observer who is outside the rocket and who is stationary with respect to the light source, the laser light hits the opposite wall of the cockpit at a point that, relative to the direction of the rocket’s motion, is lower — albeit very little — compared to the height from which the laser beam penetrated inside.
For this observer, the motion of the light emitted by the laser is always straight. The beam hits the opposite wall of the cockpit lower than the point it entered merely because the rocket continued to accelerate as the laser beam passed through it. But for the observer inside the rocket, things are different. For his experience, the cockpit is motionless. The light beam that passes through it describes a curve in traveling the path from the porthole to the opposite wall: the steeper the curve, the faster the speed of the rocket as the engine accelerates.
What can be deduced from this curious thought experiment? It can be inferred that, by the principle of equivalence, an observer who is in the quiet elevator cabin on Earth will see a laser beam passing through the cabin, forming the same curve seen by an observer who is in the continuously accelerating rocket. In other words, the light inside a gravitational field does not propagate in a straight line but follows a curved trajectory. Therefore, not only bodies with mass such as human beings and hammers are subject to gravity, but also photons (the quantum of light), which have no mass at all.
The most spectacular experimental confirmation of this prediction occurred in 1919 when the famous English astronomer Arthur Eddington made public the results of the analyses made on the photographs of the total eclipse of the Sun of 29 May, made during two scientific expeditions, one in Brazil, the other in Principe Island, off the coast of West Africa. The photographs showed that some stars, visible near the edge of the Sun during the eclipse, were in a slightly different position from that they usually occupied in the night sky when the gravity of the Sun did not interfere with the path of their light. The extent of the shift was in excellent agreement with the predictions derived from general relativity, published in late 1915. But the original root of the forecast was in the thought experiment that demonstrated the impossibility of distinguishing the effects of gravity from those of any acceleration. Experimental confirmation that gravity was actually capable of bending the path of light, taken up by newspapers all over the world, suddenly transformed Einstein into a public figure, the most famous scientist of the 20th century.
But with another thought experiment (or gedànken experimént, as Einstein would have called it), it was possible to derive another incredible prediction about the effects of gravity. While moving away from a gravitational field, light increases its wavelength, which shifts towards the red end of the spectrum. It is the so-called gravitational redshift, a phenomenon whose existence the German physicist had deduced since 1907 when it was still at the beginning of the long journey that would have led him to complete its masterpiece, the theory of general relativity.
Imagine we are in the “usual” cockpit of a rocket, far from any gravitational field. The rocket travels driven by powerful and inexhaustible engines that give it uniform acceleration. A light source placed on the rocket floor emits light of a precise wavelength, for example, green light at 500 nanometers. On the ceiling of the cockpit, there is an equally accurate detector that measures the wavelength of light coming from the floor. When hit, the sensor measures a wavelength of 501 nanometers. On the way between the floor and the ceiling, the wavelength of light has therefore increased, that is, it has moved towards the red end of the electromagnetic spectrum (red light has a wavelength greater than green light).
How could something like this happen? Because of the Doppler effect, the same one that makes us perceive a decrease in the pitch of a sound, as the sound source — for example, the siren of an ambulance — moves away from us. The effect is caused by the relative motion between the source of a wave (sound or light) and the receiving device that records its passage. If the source and the receiver are moving away from each other, the perceived wavelength is greater than what is emitted; if they’re approaching, it’s lower.
In the case of the light emitted inside the rocket, the wavelength increases, that is, it shifts towards the red, because the ceiling, where the detector is, has moved away during the journey made by the light, due to the acceleration continually impressed by the rocket engines. The opposite happens if the same light emission at 500 nanometers starts from the ceiling and reaches a detector placed on the floor. This time, because of the same continuous acceleration of the rocket, the floor will have approached the source during the journey made by the light so that the wavelength recorded by the detector on the floor will be, for example, 499 nanometers. In this case, the wavelength of the light appears to be shifted towards the blue (blue light has a shorter wavelength than both red and green light).
By the principle of equivalence, the same phenomenon observed in the cockpit of a rocket accelerating in space must apply to a quiet elevator cabin located on the Earth’s surface. The light emitted from the cabin floor will appear redshifted to a detector located on the ceiling, while the light emitted from the ceiling will appear blueshifted to a sensor placed on the floor. From a physical point of view, this is a consequence of the fact that the cabin floor is closer than the ceiling to the Earth’s center of mass, which is the origin of the Earth’s gravitational field. In other words, gravity in the cabin is slightly stronger at the floor level than at the ceiling level.
Therefore, Earth’s gravity acts as a kind of brake on the light that tries to rise through the Earth’s gravitational “well.” It cannot slow it down physically, because the speed of light (in a vacuum) always remains constant, but it can take energy away from it. So light must give up part of its energy to escape from the Earth’s gravitational field. The energy loss corresponds to a wavelength increase. The longer the wavelength of the photons, the lower their frequency of oscillation, and the lower the energy they carry. On the contrary, if the light emission occurs at a point where the gravitational potential energy is higher (the ceiling of the elevator cabin located on Earth) and is directed towards a point where it is lesser (the cabin floor), then it gains energy “falling” downwards. This gain is revealed by a decrease in the wavelength and the corresponding increase in the frequency of oscillation: the light shifts towards the blue. In both cases, the situation is entirely analogous to that foreseen by the thought experiment in which the light travels between the floor and the ceiling of a rocket in constant acceleration.
This phenomenon, called gravitational redshift, was demonstrated for the first time reliably in 1954 by Daniel Popper, who measured spectroscopically the redshift of the light coming from the white dwarf 40 Eridani B. White dwarfs are the ideal “laboratories” to verify a subtle effect like gravitational redshift. They are, in fact, small and massive objects, very compact, with a strong surface gravity capable of influencing the wavelength of the light emitted to a significant extent.
But to complicate matters, all the stars, including white dwarfs, also have a radial velocity with respect to Earth: i.e., they approach us or move away from us by a few km per second. This motion is detected as a spectral lines’ shift towards the red or the blue caused by the Doppler effect. Therefore, it is necessary to separate, in the overall value of a white dwarf’s redshift, which part is due to its intrinsic radial velocity and which part is due, instead, to the force with which its gravity “holds” the light emitted in the direction of the Earth.
Luckily, in some cases, decisive help is available to solve the problem. This help comes in the form of a binary companion, as happens with the white dwarfs 40 Eridani B and Sirius B. The study of their orbits around the center of mass of their respective binary systems allowed to determine, using the rules of Newtonian gravity, which part of the redshift measured spectroscopically was to be attributed to the radial velocity and which part, instead, to the surface gravity of the star. In this way, Popper was able to determine in 1954 that the spectra of 40 Eridani B had a gravitational redshift corresponding to an increase of 7 parts per 100,000 of the wavelength of 42 spectral lines of hydrogen — a value in good agreement with general relativity’s predictions. Therefore, also this time, a real physical phenomenon had been anticipated by a thought experiment based on the principle of equivalence.
Gravitational time dilation
But the fact that gravity can change the wavelength (i.e., the oscillation frequency) of the light hid another, surprising implication: time must slow down in the presence of an intense gravitational field.
Imagine an observer on the surface of a massive and compact body with a mighty gravitational field. From there, the observer emits a beam of green light at 500 nanometers. The oscillation frequency of light of this wavelength is 600 THz (terahertz), i.e., 600,000 billion cycles per second. Another observer, very distant from the first, located in space far from the influence of any gravitational field, records that light emission with a special detector but receives it as a red light at 750 nanometers, i.e., a light whose oscillation frequency is 400 THz, that is 400,000 billion cycles per second.
What happened to that light? To escape the gravity of the massive and compact body from whose surface it was emitted, what appeared to the first observer as green light gradually lost its energy, reducing its oscillation frequency until it became red light for the distant observer. But the oscillation frequency is in itself a clock, which indicates the rhythm with which time passes. In our imaginary scenario, a second spent on the surface of the massive body from which the green light was emitted lasts 600,000 billion oscillation cycles. In contrast, a second spent by the observer in space, which receives that same light that has become red in the meantime, only lasts 400,000 billion cycles. It is, therefore, not the same second. The first observer’s second, measured at the center of a powerful gravitational field, is longer than the second measured by the observer floating in space (to be precise, 1.5 times longer). In other words, time flows all the more slowly the stronger the gravitational field in which you are immersed.
It is possible to demonstrate the need for this effect, called gravitational time dilation, with another thought experiment based on the principle of equivalence. Let’s imagine that an unfortunate scientist is plummeting from a very high height towards the ground, accelerating during the fall, according to Earth’s gravity (about 9.8 m/s²). While he is falling, he is in a free fall. Therefore, the laws of physics that apply in its immediate vicinity are the same as those that apply in any inertial system, such as inside a train car traveling at a constant speed. During the fall, the scientist brings with him the most precise of watches: a device that bounces the light between two mirrors so that each bounce corresponds to the passage of an exact fraction of a second.
When he is still many kilometers from the ground, the scientist, falling, whizzes past an observer inside an aircraft suspended in the air who has an analogous “photonic” clock. For both the scientist and the observer in the plane, the light of their own watch bounces back and forth, always following the same trajectory, similar to a tennis ball that someone bounces between the hand and the floor. But, if each of them looks at the other’s watch, they will instead see the light bounce obliquely between the mirrors, forming triangles. In fact, due to the relative motion between the precipitating scientist and the observer in the floating plane, the light rays emitted by the other’s clock have a horizontal motion (between the two mirrors) and a vertical motion. The sum of the two components of the motion is precisely an oblique trajectory with each rebound.
But what does all this mean? Since light always moves at a constant speed, and since the triangular trajectory traveled by light in the watches that appear moving is longer than the horizontal path traveled in the watches that appear motionless, it inevitably follows that a second beat by a moving clock is longer than a second beat by an unmoving clock. In other words, the time for a moving observer flows more slowly than the time for a stationary one, by an amount proportional to the velocity of his/her motion.
Now let’s complete the thought experiment, considering a second close encounter. The precipitating scientist, continuing his fall towards the ground, now whizzes past another observer who stands on the top of a hill and has a “photonic” clock quite similar to the two already seen. The phenomenon described for the previous encounter also happens this time. Both the scientist in free fall and the observer on the hill see the light making horizontal bounces between the mirrors of their own watches. In contrast, they see the light in the other’s watch following oblique trajectories. The difference between the two situations is that now the scientist, being closer to the ground, has a higher velocity since the Earth’s gravitational field has accelerated continuously his fall.
The consequence of the falling scientist’s higher velocity is that the triangles described by the light in the photonics watches that appear in motion now have longer sides than previously observed. What can we deduce from such a difference? We can infer that time measured lower in a gravitational field flows slower than time measured higher. It is like saying that our feet age slower than our head (albeit by a negligible amount) because they are generally closer than the head to the center of mass of the Earth, from which the Earth’s gravitational field originates. So here is what the gravitational dilation of time consists of: the stronger the gravity, the slower the time flows.
Even this phenomenon has been experimentally proven. The first to succeed in this accomplishment were two physicists from Harvard University, Pound and Rebka, in 1959. Using an experimental apparatus based on the transitions between different energy levels of a ⁵⁷Fe atom, they showed that time flows more slowly, albeit by very little, at the base of a building 22.5 meters high rather than at its top. The experiment outcome was consistent with the predictions of general relativity. About half a century later, in 2010, other researchers, using very precise atomic clocks, managed to demonstrate the gravitational time dilation on height differences of less than 1 meter! As further proof of this phenomenon, GPS has been using for many years methods for correcting the times beat by atomic clocks onboard the satellites in orbit, to compensate for their higher speed with respect to the clocks located on the ground. In fact, at the altitude at which those satellites operate, time flows faster than at sea level. Without this constant correction, the positions provided by the GPS would accumulate an error of more than 10 km per day.
If someone was thinking of moving to the deepest mines to age more slowly than those who remain at sea level or those who live in the mountains, know that it would not be a good idea. Given the short duration of human life, the Earth’s gravity is far too weak to affect aging significantly. It has been calculated, in this regard, that living at low altitude guarantees a gain of just 90 billionths of a second over a 79-year life span. Even if we take into consideration the entire history of the Earth, that is, a 4.6 billion-year-long temporal abyss, the gain guaranteed by the gravitational time dilation would be modest, to say the least. A clock located at sea level would be just 39 hours behind a clock located on the summit of Everest, the highest mountain in the world. We have to go down to the level of the Earth’s core to have a slightly more noticeable time slowdown. A study published in 2016 calculated that the Earth’s core is about two and a half years younger than the Earth’s surface, thanks to gravitational time dilation. But 2.5 years out of 4.6 billion are practically nothing.
Even the Sun does not have enough gravity to produce a considerable time slowdown, despite being over 330,000 times more massive than Earth. On the Sun’s surface, the time saving compared to a body located outside the solar system is just 1.337 seconds per year. It means that if a hypothetical indestructible being had spent all 4.6 billion years since the formation of the solar system on our star’s photosphere, it would have accumulated a total of 195 years of savings compared to an analogous being located in space, for example, at the distance of Pluto. The time slowdown becomes a little more noticeable if we approach the Sun’s center near the edges of the solar core. There, the time savings would be 113.658 seconds per year, making a total of 16,579 years considering the entire lifespan of the Sun from its formation until now. Therefore, without considering the internal mixing of matter due to convection, we could say that the Sun’s core is more than 16,000 years younger than the Sun’s surface. Even here, little thing, if the whole time horizon is 4.6 billion years.
In conclusion, the only places in the Universe where it is possible to experience a truly remarkable gravitational time dilation are compact objects such as white dwarfs, neutron stars, and black holes, whose gravity is far stronger than that of main-sequence stars and planets. In any case, it is truly remarkable that a phenomenon so elusive and contrary to intuition was predicted with the sheer force of abstract thought, through ingenious thought experiments based on the principle of equivalence and the constancy of the speed of light.