Einstein's General Theory of Relativity and Time Travel

In Einstein's Special Theory of Relativity, he addressed motion in reference frames which were in relative uniform motion with respect to each other (frames in which the laws of physics were the same and the speed of light was the same, inertial frames). However, what happens if the reference frames are accelerated with respect to each other, that is, the relative velocity between the frames is changing (as occurred in the Twin Paradox)? To handle this, Einstein developled his General Theory of Relativity. In Einstein's GTR, he allows space and time to not only be distorted as in Special Relativity, but he allows for space and time to be curved (and not flat, Minkowski space). GTR couples the Laws of Motion and Gravity (and other forms of energy).

Principle of Equivalence

    If an observer sees an object fall to the floor of his or her laboratory, the observer has a dilemma. The observer does not know if the laboratory is stationary, e.g., on the surface of something like the Earth, or if the laboratory is accelerating upward, e.g., it is carried aloft in a rocket. Einstein stated this in the Principle of Equivalence. The fact that the effects of gravity cannot be distinguished from those found in experiments performed in accelerating rockets in space (far away from any massive object) suggests that gravity is not a real force.


Einstein sought a geometric interpretation for gravity. He noted that objects on flat surfaces which are not subjected to forces roll in straight lines and that objects that roll on curved surfaces follow curved paths, even when they are not subjected to external forces. The natural motion of an unforced object could either be a straight line (as Newton envisioned) or a curved line (as would be incomprehensible to Newton) depending upon the shape of the surface upon which it was rolling.

Spacetime Curvature

Einstein proposed that mass and energy distort the shape of space-time so that falling objects are simply rolling into depressions and orbiting bodies are simply rolling >round in the depression. The objects are not undergoing forced motion; they simply follow the lay of the land. This leads to the bending of starlight, orbital motion, and, possibly, to extreme distortions of spactime where we can find Closed Spacelike Paths and/or Closed Timelike Paths.

Tests of General Relativity

Einstein became a public celebrity when Arthur Eddington and colleagues measured the deflection of light by the Sun during the solar eclipse of 1919 and found that their results agreed with the predictions of general relativity.

Positive print of a Solar Eclipse showing the bending of light

The Einstein Cross, another example of the bending of light by a gravitational lens

Data for the Solar eclipse measurements and for the time delay (Shapiro Effect) in a strong gravitational field

Deflection of Starlight

Measurements of the deflection (top) - plotted as (1 + γ)/2, where γ is related to the amount of spatial curvature generated by mass - have become more accurate since 1919 and have converged on the prediction of general relativity: (1 + γ)/2 = 1. The same is true for measurements of the Shapiro time delay (bottom). "Optical" denotes measurements made during solar eclipses (shown in red), with the arrows pointing to values well off the chart; "radio" denotes interferometric measurements of radio-wave deflection (blue); while Hipparcos was an optical-astrometry satellite. The left-most data point is the measurement made by Eddington in 1919, while the arrow just above it refers to the value obtained by his compatriot Andrew Crommelin. The best deflection measurements (green) are accurate to 2 parts in 104 and were obtained with Very Long Baseline Radio Interferometry (VLBI; see Shapiro et al. in further reading). A recent measurement of the Shapiro time delay by the Cassini spacecraft, which was on its way to Saturn, was accurate to 1 part in 105 (see Bertotti et al. in further reading).

Perihelion Advance of Mercury

Mercury is in a highly elliptical orbit about the Sun. Because of perturbations from the motions of the other planets, Mercury undergoes a slow precession, 574 arc seconds per century, as it orbits the Sun. Most of the precession (531 arcseconds per century) is understandable through Newtonian mehanics, there, however is a small part of the precession (43 arc seconds per century) which cannot be accounted for using Newtonian physics. Using General Relativity, Einstein calculated the small extra bit of precession, 43 arc seconds per century!

Gravity Waves

Just as electromagnetic radiation is produced when a charge accelerates, gravity waves are produced when a mass accelerates ( Michio Kaku video on gravity waves). As yet, there is no direct detection of gravity waves, however, there is compelling indirect evidence given by the binary pulsar, PSR 1913+16.

Binary Pulsar: PSR1913+16

A pulsar is a rapidly rotating, strongly magnetic neutron star. They are unparalleled gravity laboratories because of these properties. The binary pulsar has

Pspin = 0.059 s

Porb = 27,908 s

Orbital ecc = 0.615

Because of gravitational radiation from its orbit, its orbit has been shrinking. Its orbital period thus slowly decreases. The decrease is shown to the left! The agreement with Einstein's General Rleativity is astounding. Hulse and Taylor received the Nobel Prize in physics in 1993 for this remarkable work.

For these and other tests see http://einstein.stanford.edu/SPACETIME/spacetime3.html.

Solutions to Einstein's Equations

In Special Relativity, space is flat (Minkowski Space) and the axes of all light cones are vertical with opening angles of 45o. Time can only flow forward or backward. In General Relativity, space can be curved and the axes of light cones may assume arbitrary angles with respect to vertical, depending on the the curvature of the space ===> the flow of time can be backwards or flow in loops in highly curved spacetimes. This can have interesting consequences.

In the following sections, we consider indicate roughly how curved spacetimes may lead to closed timelike loops, and then consider several examples of spacetimes with extreme curvatures in which closed timelike loops may arise.

Spacetime Curvature

Imagine simple examples of what can happen if we curve spacetime. Suppose we have the simple 1+1 diagram (1 spatial coordinate + 1 time coordinate). What happens when we curve the spacetime?

Closed Spacelike Loop

Take the left hand edge and the right hand edge of the spacetime diagram and then wrap them around and glue them together to form a cylinder. We see that we have formed a closed spacelike loop so that we can simply move around in space, returning to the same point over and over again even if we move in one direction. However, in this diagram, if we move one direction in time, we always move into the future (or into the past).

Closed Timelike Loop

Perform the same exercise as before but now grab onto the top and bottom of the spacetime diagram and wrap the edges around and then glue them together. We see that even when you travel in one direction in time, you can return to the same point in time many times. In this instance if you move in 1 direction in space, you never return to the same point.

So, we see that there may be the possibility for closed timelike loops in General Relativity if one can think of a way to curve spacetime. As a result, there is the possibility for time travel in General Relativity. An example of this was in the solution found by Kurt Goedel in the 1940s.

Goedel Metric

Kurt Goedel (a colleague of Einstein at the IAS) found a solution of Einstein's equations for a rotating universe which was not expanding or contracting (therefore not our Universe). In Goedel's universe closed timelike loops are possible. To see this, look at the panel to the left. As one moves away from the center, because the universe spins at the same rate, objects move faster and faster. This acceleration tilts the light cones over. If one moves far away from the center and moves at speeds close to but below the speed of light, the light cones are tipped over nearly on their sides and point along the direction of motion; one can follow a closed timelike path and return to the same point in space before one has left.