A-level Physics/Cosmology/Relativity

Time Dilation

A thought experiment:
Imagine two glass train carriages on parallel railway tracks, each with a mirror along their full length facing the other train. Each train has an observer on it. The trains are travelling in opposite directions at close to the speed of light. One observer sends a pulse of light at right angles to their direction of travel, towards the other train. This pulse of light is reflected between the two mirrors over and over again.

The first part of the diagram represents what the observer who sent the light sees - the light bouncing backwards and forwards in a straight line. The other observer, however, sees the light moving in a "zig-zag" pattern. This is because he is moving away from the light as it is being sent, so after each successive reflection the light has further to travel.

Another experiment for time dilation has been carried out with muons, extremely low mass particles which decay very quickly and virtually disappear. When these particles are accelerated (by particle accelerators), their lifetimes are significantly increased, suggesting that time has been slowed down.

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Gravitational time dilation is a consequence of Albert Einstein's theories of relativity and related theories under which a clock at a different gravitational potential is found to tick at a different rate than one's own clock.

Gravitational time dilation was first described by Albert Einstein in 1907 as a consequence of special relativity in accelerated frames of reference. In general relativity, it is considered to be difference in the passage of proper time at different positions as described by a metric tensor of spacetime. The existence of gravitational time dilation was first confirmed directly by the Pound-Rebka experiment.

Definition

Background knowledge the reader may need to learn: What is a gravitational field? What is time dilation? What is spacetime? The reader may also research gravitational redshift or ordinary redshift.

Gravitational time dilation can be manifested by the presence of large mass, and the larger the mass, the greater the time dilation. In more simple terms, it is meant that observers far from massive bodies are distant observers with fast clocks, and observers close to massive bodies are time-dilated observers with slow clocks.

It can also be manifested by any other kind of accelerated reference frame such as a dragster or space shuttle. Spinning objects such as merry-go-rounds and ferris wheels are subjected to gravitation time dilation as an effect of their angular spin.

This is supported by General Relativity due to the equivalence principle that states all accelerated reference frames possess a gravitational field. According to General Relativity, inertial mass and gravitational mass are the same. Not all gravitational fields are "curved" or "spherical", some are flat as in the case of an accelerating dragster or space shuttle. Any kind of g-load contributes to gravitational time dilation.

• In an accelerated box, the equation with respect to an arbitrary base observer is $T_{d}=1-gh/c^{2}$, where
  • $T_{d}$ is the total time dilation at a distant position,
  • $g$ is the acceleration of the box as measured by the base observer, and
  • $h$ is the "vertical" distance between the observers.
• On a rotating disk when the base observer is located at the center of the disk and co-rotating with it (which makes their view of spacetime non-inertial), the equation is $T_{d}={\sqrt {1-r^{2}\omega ^{2}/c^{2}}}$, where
  • $r$ is the distance from the center of the disk (which is the location of the base observer), and
  • $\omega$ is the angular velocity of the disk.

(It is no accident that in an inertial frame of reference this becomes the familiar velocity time dilation ${\sqrt {1-v^{2}/c^{2}}}$).

A common equation used to determine gravitational time dilation is using the Schwarzschild solution, which describes spacetime in the vicinity of a non-rotating massive object. The Schwarzschild solution for time dilation for a spherically-symmetric object is:

$t_{0}=t_{f}{\sqrt {1-{\frac {2GM}{rc^{2}}}}}$, where

• $t_{0}$ is the proper time between events A and B for a slow-ticking observer within the gravitational field,
• $t_{f}$ is the proper time between events A and B for a fast-ticking observer distant from the massive object (and therefore outside of the gravitational field),
• $G$ is the gravitational constant,
• $M$ is the mass of the object creating the gravitational field,
• $r$ is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object, but is actually a Schwarzschild coordinate), and
• $c$ is the speed of light.

$2GM/c^{2}$ is the called the Schwarzschild Radius of M. If a mass collapses so that its surface lies at less than this radial coordinate (or in other words covers an area of less than $16\pi G^{2}M^{2}/c^{4}$), then the object exists within a black hole.

Consequences

If a satellite drifting in deep space is sending out laser light at n cycles per second, and an Earth-based observer sees this signal to be blueshifted, with a higher frequency of n+1 cycles per second, then the only apparent way for this situation to be sustainable (with signals being registered faster on the receiving equipment than they are being sent by the transmitting equipment, indefinitely) is if the two sets of equipment are operating differently due to their different gravitational environments.

Important things to stress

• According to General Relativity, gravitational time dilation is copresent with the existence of an accelerated reference frame.
• The speed of light in a locale is always equal to c according to the observer who is there. The stationary observer's perspective corresponds to the local proper time. Every infinitesimal region of space time may have its own proper time that corresponds to the gravitational time dilation there, where electromagnetic radiation and matter may be equally affected, since they are made of the same essence (as shown in many tests involving the famous equation $E=mc^{2}$). Such regions are significant whether or not they are occupied by an observer. A time delay is measured for signals that bend near the sun, headed towards Venus, and bounce back to earth along more or less a similar path. There is no violation of the speed of light in this sense, as long as an observer is forced to observe only the photons which intercept the observing faculties and not the ones that go passing by in the depths of more (or even less) gravitational time dilation.

If a distant observer is able to track the light in a remote, distant locale which intercepts a time dilated observer nearer to a more massive body, (putting aside the fact that a photon cannot be observed without interception with the observer) he sees that both the distant light and that distant time dilated observer have a slower proper time clock than other light which is coming nearby him, which intercept him, at c, like all other light he really can observe. When the other, distant light intercepts the distant observer, it will come at c from the distant observer's perspective.

Experimental confirmation

Gravitational time dilation has been experimentally measured using atomic clocks on aeroplanes. The clocks that travelled aboard the aeroplanes upon return were slightly fast with respect to clocks on the ground. The effect is significant enough that the Global Positioning System needs to correct for its effect on clocks aboard artificial satellites, providing a further experimental confirmation of the effect.

Gravitational time dilation has also been confirmed by the Pound-Rebka experiment and by observations of the spectra of the white dwarf Sirius B.

References

• Einstein, Albert. "Relativity : the Special and General Theory by Albert Einstein." Project Gutenberg. <http://www.gutenberg.org/etext/5001.>
• Einstein, Albert. "The effect of gravity on light" (1911), translated and reprinted in The Principle of Relativity
• Nave, C.R. "Gravity and the Photon." Hyperphysics. <http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/blahol.html#c2.>
• The Pound-Rebka-Snider Experiments