How Gravity affects time? Many of you have probably seen the
movie "Interstellar", in which the protagonists, aboard their spaceship, have to explore a planet
that is very close to the black hole Gargantua, so close that "for every hour spent on the surface of
the planet, on Earth seven years pass”: only a few hours pass for the astronauts who come down
to the planet, while for their colleague Romilly who remains aboard the spaceship,
23 years pass! But why does this happen? To understand this we must take a step back,
to that November 1915 in which Albert Einstein, in four lectures at the Prussian Academy of Sciences,
presented the theory of general relativity: the result of over 10 years of studies,
in it the scientist tried to connect his theory of special relativity with Newton's theory of
universal gravitation. Very briefly, we remember that special relativity establishes that the
laws of mechanics, electromagnetism and optics are the same in all inertial reference systems
(i.e. those in which a mass moves in uniform rectilinear motion) and that the light propagates
in a vacuum at a constant speed, regardless of the state of motion of the source or of the observer.
The equation underlying general relativity, also called Einstein's field equation, is based on
the equivalence principle, which establishes that inertial mass and gravitational mass are
equivalent. However, the respective definitions of the two types of mass are different:
the inertial mass is proportional to the inertia of a body, which is the resistance
to the change of the state of motion when a force is applied, while the gravitational
mass is proportional to the interaction force. of a body with gravitational force.
The equivalence principle is so called because an observer jointed with the moving masses is unable
to distinguish an acceleration due to an external force from that produced by a gravitational field.
The difference is visible only from a reference system that is not jointed with the moving masses.
To try to better understand this principle,
Einstein himself proposed the elevator thought experiment.
Let's imagine we are inside an elevator on the top floor of the Empire State Building and suppose
there is no air in it. Suddenly the carrying cable breaks and the cabin begins to fall freely with
constant acceleration, equal to the acceleration of gravity, which for the Earth is 9.8 m/s2.
A person inside drops a stone and a feather. The force of gravity attracts both the two
objects and the elevator in the same way, so the relative speed between stone and feather is zero.
In other words, both the stone and the feather do not reach the bottom of the elevator,
since the latter is falling with their own acceleration. The person inside the cabin
could therefore rightly claim to be in an area of space far from the gravitational action of
stars and planets, since the two objects left to themselves remain suspended in midair.
Let us now consider the opposite situation: the cabin is placed in an area of space
where gravitational forces do not act, for example in a Space Shuttle in Earth orbit.
Let us also suppose that the elevator cabin is "pulled" upwards always with constant acceleration
by means of a cable connected to the ceiling. If the person inside the elevator leaves the feather
and the stone, they will stay in place since no force acts on them. However, the cabin floor is
moving upwards and, sooner or later, it will reach the two objects. The person inside the elevator,
ignoring the external situation, may believe he is inside a gravitational field since,
by letting go of the feather and the stone, they "fall" to the floor at the same time.
The person himself also "feels" his own weight because of the floor pushing against the feet.
It can therefore be stated that: the effects of a constant acceleration on an observer are
equivalent to those of a uniform gravitational field on the same observer supposed to be at rest.
In this consists the famous equivalence principle formulated by Einstein in 1911.
Several experimental verifications of the equivalence principle have been carried out
and the main ones take into consideration two types of experiments: the gravitational
balance and the gravitational red-shift. The degree of precision with which the equivalence
principle has been verified is so high that it is one of the pillars of all modern physics.
Einstein, following these arguments, deduced the field equations of general relativity,
which can be written in a very famous form that you may have seen on some slightly nerdy shirts:
Rμν -gμνR = -Tμν
Are you lost? Don't worry, now let's try to explain its meaning.
gμν is the metric tensor, which mathematically expresses the properties of a curved spacetime;
the indices μ and ν represent the coordinates of space-time and can assume different values:
the spatial coordinates are usually indicated by the values 1,
2, 3, while the time coordinate is indicated by the value 0.
The metric tensor also allows to calculate the distance between points in a curved spacetime.
Rμν and R take into account the curvature of spacetime;
Rμν is the curvature tensor, and also in this case the indices μ and ν
represent the three spatial coordinates and the fourth time coordinate of spacetime.
G is the universal gravitational constant
c is the speed of light in vacuum
Tμν is the impulse energy tensor, which represents the source of the gravitational interaction:
which is not only the density of matter, but also its energy and therefore also its state of motion;
in practice, the mass that generates the gravitational field is contained in Tμν.
Put simply, what do Einstein's field equations tell us?
They tell us that gravity is interpreted as a geometric effect, as the matter present
in the Universe determines the curvature of space-time. Einstein's equations simply
express the relationship between curvature on the one hand and matter and energy on the other.
The field equations also tell us that space and time are not separated as in Newton's theory,
but joined together in a four-dimensional spacetime, three spatial and one temporal.
Not only are space and time no longer absolute but they are also intrinsically connected! Therefore,
each point in spacetime, which defines an “event”, can only be defined through three spatial and one
temporal coordinates. What makes this theory particularly difficult is a kind of "feedback
loop". That is to say: the curvature of spacetime is determined by the distribution of matter and
energy distributed in the Universe, which in turn is governed by the curvature of space-time itself!
Einstein's theory gives an interpretation of gravity including Newton's theory, which remains
valid as a particular case where the masses that generate gravity (for example in the solar system)
are not too large. The effect of any material body is to "curve" the space around it.
This "curvature" causes the bodies to deviate from their rectilinear trajectory
thus causing what we call gravitational attraction.
Another consideration: if a body is subject to acceleration due to gravity, it means that it no
longer moves with uniform rectilinear motion, but with accelerated motion, and therefore Euclidean
geometry (in short, the one that tells us that the sum of the angles of a triangle is 180 degrees),
again for the equivalence principle, it is no longer valid in the presence of a
gravitational field. In the presence of a gravitational field, space-time is curved:
this implies that, in addition to space, time is also distorted, that is, closer we get to
the source of a gravitational field, slower time flows. And it is precisely for this reason
that for the Interstellar astronauts who descend on the planet in orbit around Gangantua,
time flows much slower than it does for the colleague who awaits them aboard the spaceship!
Precisely because the gravitational field generated by a hole is very intense,
the black hole will produce a high curvature in the spacetime around it, and therefore the effect
of time dilation will be more conspicuous. Would you like to travel close to a black hole like in
Interstellar to experience this effect? Write it in the comments! But don't go too close to it,
because you risk being inexorably swallowed up by its gigantic gravitational field. In the case of a
planet like the Earth, its mass is much smaller than that of a black hole, and therefore the
curvature produced by our planet in spacetime is small: it follows that the dilation of time will
also be very small, so point that to observe it we had to wait for the advent of atomic clocks,
characterized by sensitivity of the order of a billionth of a second!
In October 1971, Joseph Hafele and Richard Keating, to measure the relativistic dilation
of time, boarded airliners with previously synchronized cesium atomic clocks. The clocks
made two turns around the Earth, in opposite directions. Upon return, the measured times were
compared with each other and with the time marked by a clock left on the ground: compared to this,
the clock that had traveled east was delayed by about 60 nanoseconds, while the clock that
had traveled to west was 270 nanoseconds ahead. The detected time differences were in excellent
agreement with the relativistic predictions taking into account both the gravitational dilation of
time, due to the different altitude, and the kinematic dilation of time, due to the motion
of the clocks (in the flight to the east the speed of the plane is added to that of rotation
of the Earth and the kinematic effect prevails over the gravitational one, of opposite sign).
In 1976 the Italian physicists Luigi Briatore of the University of Turin and Sigfrido Leschiutta
of the National Electrotechnical Institute "Galileo Ferraris" (today INRiM, National
Institute of Metrological Research) directly verified gravitational expansion by comparing
the times of two atomic clocks at rest, one located in Turin (height 250 m above sea level),
the other on Plateau Rosa (height 3500 m), which differed by about 34 nanoseconds per day,
in accordance - within the experimental uncertainty - with the theoretical prediction .
In the pioneering experiments of the 1970s, the observed slowdown effects were on the order of
one part in 1012, at the limit of the accuracy of the atomic clocks of the time
(which were wrong by 0.1 microseconds per day, i.e. one second every 30,000 years!).
The most sophisticated clocks currently available - optical atomic clocks - have reached an accuracy
of around one part in 1018, which means they are wrong by one second every 10 billion years,
roughly the age of the universe! These watches would allow you to observe the different passing
of time even at low speeds and on gradients of just a few tens of centimeters. Would you like
to have such a precise watch? And what could you use it for? Write it in the comments!
It should not be forgotten that the relativistic effects of time have also entered daily life.
GPS, the positioning system we use to locate a point on the earth's surface with an error
of about ten meters, works by converting the travel times of radio signals exchanged between
satellites in orbit (at twenty km altitude) and a terrestrial receiver. Since the clocks on the
satellites are in motion respect to those on the ground and feel a weaker gravitational field,
the time they measure is different from the time measured by the receiver.
There are two corrections: one of 7 microseconds per day,
due to the kinematic dilation of time (satellite clocks go slower than terrestrial clocks)
and another, of greater magnitude and opposite sign, of 45 microseconds per day, due to the
gravitational dilation of time (satellite clocks go faster than terrestrial clocks).
If these effects were not taken into account, the daily error on localization would be greater than
10 km. The fact that the GPS leads correctly to the destination therefore represents a
continuous verification of the theory of general relativity. The relativistic link
between time and gravity has other interesting application implications. Ultra-precise clocks
can act as highly sensitive gravity sensors, which can be used to measure the inhomogeneities
of the earth's crust (which determine small local variations of the gravitational field)
or even to monitor the magmatic movements of volcanoes. These studies design a new field
of research, on which many metrological institutes around the world are engaged.