Twin paradox

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In physics, the twin paradox is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin sees the other twin as moving, and so, as a consequence of an incorrect and naive application of time dilation and the principle of relativity, each should paradoxically find the other to have aged less. However, this scenario can be resolved within the standard framework of special relativity: the travelling twin's trajectory involves two different inertial frames, one for the outbound journey and one for the inbound journey. Another way of looking at it is to realize the travelling twin is undergoing acceleration, which makes him a non-inertial observer. In both views there is no symmetry between the spacetime paths of the twins. Therefore, the twin paradox is not actually a paradox in the sense of a logical contradiction. There is still debate as to the resolution of the twin paradox.

Starting with Paul Langevin in 1911, there have been various explanations of this paradox. These explanations "can be grouped into those that focus on the effect of different standards of simultaneity in different frames, and those that designate the acceleration [experienced by the travelling twin] as the main reason". Max von Laue argued in 1913 that since the traveling twin must be in two separate inertial frames, one on the way out and another on the way back, this frame switch is the reason for the aging difference. Explanations put forth by Albert Einstein and Max Born invoked gravitational time dilation to explain the aging as a direct effect of acceleration. However, it has been proven that neither general relativity, nor even acceleration, are necessary to explain the effect, as the effect still applies if two astronauts pass each other at the turnaround point and synchronize their clocks at that point. Such observer can be thought of as a pair of observers, one travelling away from the starting point and another travelling toward it, passing by each other where the turnaround point would be. At this moment, the clock reading in the first observer is transferred to the second one, both maintaining constant speed, with both trip times being added at the end of their journey.

History

Further information: History of special relativity § Time dilation and twin paradox

In his famous paper on special relativity in 1905, Albert Einstein deduced that when two clocks were brought together and synchronized, and then one was moved away and brought back, the clock which had undergone the traveling would be found to be lagging behind the clock which had stayed put. Einstein considered this to be a natural consequence of special relativity, not a paradox as some suggested, and in 1911, he restated and elaborated on this result as follows (with physicist Robert Resnick's comments following Einstein's):

Einstein: If we placed a living organism in a box ... one could arrange that the organism, after any arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had already long since given way to new generations. For the moving organism, the lengthy time of the journey was a mere instant, provided the motion took place with approximately the speed of light.
Resnick: If the stationary organism is a man and the traveling one is his twin, then the traveler returns home to find his twin brother much aged compared to himself. The paradox centers on the contention that, in relativity, either twin could regard the other as the traveler, in which case each should find the other younger—a logical contradiction. This contention assumes that the twins' situations are symmetrical and interchangeable, an assumption that is not correct. Furthermore, the accessible experiments have been done and support Einstein's prediction.

In 1911, Paul Langevin gave a "striking example" by describing the story of a traveler making a trip at a Lorentz factor of γ = 100 (99.995% the speed of light). The traveler remains in a projectile for one year of his time, and then reverses direction. Upon return, the traveler will find that he has aged two years, while 200 years have passed on Earth. During the trip, both the traveler and Earth keep sending signals to each other at a constant rate, which places Langevin's story among the Doppler shift versions of the twin paradox. The relativistic effects upon the signal rates are used to account for the different aging rates. The asymmetry that occurred because only the traveler underwent acceleration is used to explain why there is any difference at all, because "any change of velocity, or any acceleration has an absolute meaning".

Max von Laue (1911, 1913) elaborated on Langevin's explanation. Using Hermann Minkowski's spacetime formalism, Laue went on to demonstrate that the world lines of the inertially moving bodies maximize the proper time elapsed between two events. He also wrote that the asymmetric aging is completely accounted for by the fact that the astronaut twin travels in two separate frames, while the Earth twin remains in one frame, and the time of acceleration can be made arbitrarily small compared with the time of inertial motion. Eventually, Lord Halsbury and others removed any acceleration by introducing the "three-brother" approach. The traveling twin transfers his clock reading to a third one, traveling in the opposite direction. Another way of avoiding acceleration effects is the use of the relativistic Doppler effect (see § What it looks like: the relativistic Doppler shift below).

Neither Einstein nor Langevin considered such results to be problematic: Einstein only called it "peculiar" while Langevin presented it as a consequence of absolute acceleration. Both men argued that, from the time differential illustrated by the story of the twins, no self-contradiction could be constructed. In other words, neither Einstein nor Langevin saw the story of the twins as constituting a challenge to the self-consistency of relativistic physics.

Specific example

Consider a space ship traveling from Earth to the nearest star system: a distance d = 4 light years away, at a speed v = 0.8c (i.e., 80% of the speed of light).

To make the numbers easy, the ship is assumed to attain full speed in a negligible time upon departure (even though it would actually take about 9 months accelerating at 1 g to get up to speed). Similarly, at the end of the outgoing trip, the change in direction needed to start the return trip is assumed to occur in a negligible time. This can also be modelled by assuming that the ship is already in motion at the beginning of the experiment and that the return event is modelled by a Dirac delta distribution acceleration.

The parties will observe the situation as follows:

Earth perspective

The Earth-based mission control reasons about the journey this way: the round trip will take t = 2d/v = 10 years in Earth time (i.e. everybody on Earth will be 10 years older when the ship returns). The amount of time as measured on the ship's clocks and the aging of the travelers during their trip will be reduced by the factor ${\displaystyle \alpha =\sqrt{1-v^2/c^2}}$, the reciprocal of the Lorentz factor (time dilation). In this case α = 0.6 and the travelers will have aged only 0.6 × 10 = 6 years when they return.

Travellers' perspective

The ship's crew members also calculate the particulars of their trip from their perspective. They know that the distant star system and the Earth are moving relative to the ship at speed v during the trip. In their rest frame the distance between the Earth and the star system is α d = 0.6 × 4 = 2.4 light years (length contraction), for both the outward and return journeys. Each half of the journey takes α d / v = 2.4 / 0.8 = 3 years, and the round trip takes twice as long (6 years). Their calculations show that they will arrive home having aged 6 years. The travelers' final calculation about their aging is in complete agreement with the calculations of those on Earth, though they experience the trip quite differently from those who stay at home.

Conclusion

Readings on Earth's and spaceship's clocks

Event	Earth (years)	Spaceship (years)
Departure	0	0
End of outgoing trip = Beginning of ingoing trip	5	3
Arrival	10	6

No matter what method they use to predict the clock readings, everybody will agree about them. If twins are born on the day the ship leaves, and one goes on the journey while the other stays on Earth, they will meet again when the traveler is 6 years old and the stay-at-home twin is 10 years old.

Resolution of the paradox in special relativity

The paradoxical aspect of the twins' situation arises from the fact that at any given moment the travelling twin's clock is running slow in the earthbound twin's inertial frame, but based on the relativity principle one could equally argue that the earthbound twin's clock is running slow in the travelling twin's inertial frame. One proposed resolution is based on the fact that the earthbound twin is at rest in the same inertial frame throughout the journey, while the travelling twin is not: in the simplest version of the thought-experiment, the travelling twin switches at the midpoint of the trip from being at rest in an inertial frame which moves in one direction (away from the Earth) to being at rest in an inertial frame which moves in the opposite direction (towards the Earth). In this approach, determining which observer switches frames and which does not is crucial. Although both twins can legitimately claim that they are at rest in their own frame, only the traveling twin experiences acceleration when the spaceship engines are turned on. This acceleration, measurable with an accelerometer, makes his rest frame temporarily non-inertial. This reveals a crucial asymmetry between the twins' perspectives: although we can predict the aging difference from both perspectives, we need to use different methods to obtain correct results.

Role of acceleration

Although some solutions attribute a crucial role to the acceleration of the travelling twin at the time of the turnaround, others note that the effect also arises if one imagines two separate travellers, one outward-going and one inward-coming, who pass each other and synchronize their clocks at the point corresponding to "turnaround" of a single traveller. In this version, physical acceleration of the travelling clock plays no direct role; "the issue is how long the world-lines are, not how bent". The length referred to here is the Lorentz-invariant length or "proper time interval" of a trajectory which corresponds to the elapsed time measured by a clock following that trajectory (see Section Difference in elapsed time as a result of differences in twins' spacetime paths below). In Minkowski spacetime, the travelling twin must feel a different history of accelerations from the earthbound twin, even if this just means accelerations of the same size separated by different amounts of time, however "even this role for acceleration can be eliminated in formulations of the twin paradox in curved spacetime, where the twins can fall freely along space-time geodesics between meetings".

Relativity of simultaneity

Minkowski diagram of the twin paradox. There is a difference between the trajectories of the twins: the trajectory of the ship is equally divided between two different inertial frames, while the Earth-based twin stays in the same inertial frame.

For a moment-by-moment understanding of how the time difference between the twins unfolds, one must understand that in special relativity there is no concept of absolute present. For different inertial frames there are different sets of events that are simultaneous in that frame. This relativity of simultaneity means that switching from one inertial frame to another requires an adjustment in what slice through spacetime counts as the "present". In the spacetime diagram on the right, drawn for the reference frame of the Earth-based twin, that twin's world line coincides with the vertical axis (his position is constant in space, moving only in time). On the first leg of the trip, the second twin moves to the right (black sloped line); and on the second leg, back to the left. Blue lines show the planes of simultaneity for the traveling twin during the first leg of the journey; red lines, during the second leg. Just before turnaround, the traveling twin calculates the age of the Earth-based twin by measuring the interval along the vertical axis from the origin to the upper blue line. Just after turnaround, if he recalculates, he will measure the interval from the origin to the lower red line. In a sense, during the U-turn the plane of simultaneity jumps from blue to red and very quickly sweeps over a large segment of the world line of the Earth-based twin. When one transfers from the outgoing inertial frame to the incoming inertial frame there is a jump discontinuity in the age of the Earth-based twin (6.4 years in the example above).

A non space-time approach

As mentioned above, an "out and back" twin paradox adventure may incorporate the transfer of clock reading from an "outgoing" astronaut to an "incoming" astronaut, thus eliminating the effect of acceleration. Also, the physical acceleration of clocks does not contribute to the kinematical effects of special relativity. Rather, in special relativity, the time differential between two reunited clocks is produced purely by uniform inertial motion, as discussed in Einstein's original 1905 relativity paper, as well as in all subsequent kinematical derivations of the Lorentz transformations.

Because spacetime diagrams incorporate Einstein's clock synchronization (with its lattice of clocks methodology), there will be a requisite jump in the reading of the Earth clock time made by a "suddenly returning astronaut" who inherits a "new meaning of simultaneity" in keeping with a new clock synchronization dictated by the transfer to a different inertial frame, as explained in Spacetime Physics by John A. Wheeler.

If, instead of incorporating Einstein's clock synchronization (lattice of clocks), the astronaut (outgoing and incoming) and the Earth-based party regularly update each other on the status of their clocks by way of sending radio signals (which travel at light speed), then all parties will note an incremental buildup of asymmetry in time-keeping, beginning at the "turn around" point. Prior to the "turn around", each party regards the other party's clock to be recording time differently from his own, but the noted difference is symmetrical between the two parties. After the "turn around", the noted differences are not symmetrical, and the asymmetry grows incrementally until the two parties are reunited. Upon finally reuniting, this asymmetry can be seen in the actual difference showing on the two reunited clocks.

The equivalence of biological aging and clock time-keeping

All processes—chemical, biological, measuring apparatus functioning, human perception involving the eye and brain, the communication of force—are constrained by the speed of light. There is clock functioning at every level, dependent on light speed and the inherent delay at even the atomic level. Biological aging, therefore, is in no way different from clock time-keeping. This means that biological aging would be slowed in the same manner as a clock.

What it looks like: the relativistic Doppler shift

In view of the frame-dependence of simultaneity for events at different locations in space, some treatments prefer a more phenomenological approach, describing what the twins would observe if each sent out a series of regular radio pulses, equally spaced in time according to the emitter's clock. This is equivalent to asking, if each twin sent a video feed of themselves to each other, what do they see in their screens? Or, if each twin always carried a clock indicating his age, what time would each see in the image of their distant twin and his clock?

Shortly after departure, the traveling twin sees the stay-at-home twin with no time delay. At arrival, the image in the ship screen shows the staying twin as he was 1 year after launch, because radio emitted from Earth 1 year after launch gets to the other star 4 years afterwards and meets the ship there. During this leg of the trip, the traveling twin sees his own clock advance 3 years and the clock in the screen advance 1 year, so it seems to advance at 1⁄3 the normal rate, just 20 image seconds per ship minute. This combines the effects of time dilation due to motion (by factor ε=0.6, five years on Earth are 3 years on ship) and the effect of increasing light-time-delay (which grows from 0 to 4 years).

Of course, the observed frequency of the transmission is also 1⁄3 the frequency of the transmitter (a reduction in frequency; "red-shifted"). This is called the relativistic Doppler effect. The frequency of clock-ticks (or of wavefronts) which one sees from a source with rest frequency f_rest is

${\displaystyle f_{\mathrm{o}\mathrm{b}\mathrm{s}}=f_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}\sqrt{\left(1-v/c\right)/\left(1+v/c\right)}}$

when the source is moving directly away. This is f_obs = 1⁄3f_rest for v/c = 0.8.

As for the stay-at-home twin, he gets a slowed signal from the ship for 9 years, at a frequency 1⁄3 the transmitter frequency. During these 9 years, the clock of the traveling twin in the screen seems to advance 3 years, so both twins see the image of their sibling aging at a rate only 1⁄3 their own rate. Expressed in other way, they would both see the other's clock run at 1⁄3 their own clock speed. If they factor out of the calculation the fact that the light-time delay of the transmission is increasing at a rate of 0.8 seconds per second, both can work out that the other twin is aging slower, at 60% rate.

Then the ship turns back toward home. The clock of the staying twin shows "1 year after launch" in the screen of the ship, and during the 3 years of the trip back it increases up to "10 years after launch", so the clock in the screen seems to be advancing 3 times faster than usual.

When the source is moving towards the observer, the observed frequency is higher ("blue-shifted") and given by

${\displaystyle f_{\mathrm{o}\mathrm{b}\mathrm{s}}=f_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}\sqrt{\left(1+v/c\right)/\left(1-v/c\right)}}$

This is f_obs = 3f_rest for v/c = 0.8.

As for the screen on Earth, it shows that trip back beginning 9 years after launch, and the traveling clock in the screen shows that 3 years have passed on the ship. One year later, the ship is back home and the clock shows 6 years. So, during the trip back, both twins see their sibling's clock going 3 times faster than their own. Factoring out the fact that the light-time-delay is decreasing by 0.8 seconds every second, each twin calculates that the other twin is aging at 60% his own aging speed.

Light paths for images exchanged during trip
Left: Earth to ship. Right: Ship to Earth.
Red lines indicate low frequency images are received, blue lines indicate high frequency images are received

The x–t (space–time) diagrams at left show the paths of light signals traveling between Earth and ship (1st diagram) and between ship and Earth (2nd diagram). These signals carry the images of each twin and his age-clock to the other twin. The vertical black line is the Earth's path through spacetime and the other two sides of the triangle show the ship's path through spacetime (as in the Minkowski diagram above). As far as the sender is concerned, he transmits these at equal intervals (say, once an hour) according to his own clock; but according to the clock of the twin receiving these signals, they are not being received at equal intervals.

After the ship has reached its cruising speed of 0.8c, each twin would see 1 second pass in the received image of the other twin for every 3 seconds of his own time. That is, each would see the image of the other's clock going slow, not just slow by the ε factor 0.6, but even slower because light-time-delay is increasing 0.8 seconds per second. This is shown in the figures by red light paths. At some point, the images received by each twin change so that each would see 3 seconds pass in the image for every second of his own time. That is, the received signal has been increased in frequency by the Doppler shift. These high frequency images are shown in the figures by blue light paths.

The asymmetry in the Doppler shifted images

The asymmetry between the Earth and the space ship is manifested in this diagram by the fact that more blue-shifted (fast aging) images are received by the ship. Put another way, the space ship sees the image change from a red-shift (slower aging of the image) to a blue-shift (faster aging of the image) at the midpoint of its trip (at the turnaround, 3 years after departure); the Earth sees the image of the ship change from red-shift to blue shift after 9 years (almost at the end of the period that the ship is absent). In the next section, one will see another asymmetry in the images: the Earth twin sees the ship twin age by the same amount in the red and blue shifted images; the ship twin sees the Earth twin age by different amounts in the red and blue shifted images.

Calculation of elapsed time from the Doppler diagram

The twin on the ship sees low frequency (red) images for 3 years. During that time, he would see the Earth twin in the image grow older by 3/3 = 1 years. He then sees high frequency (blue) images during the back trip of 3 years. During that time, he would see the Earth twin in the image grow older by 3 × 3 = 9 years. When the journey is finished, the image of the Earth twin has aged by 1 + 9 = 10 years.

The Earth twin sees 9 years of slow (red) images of the ship twin, during which the ship twin ages (in the image) by 9/3 = 3 years. He then sees fast (blue) images for the remaining 1 year until the ship returns. In the fast images, the ship twin ages by 1 × 3 = 3 years. The total aging of the ship twin in the images received by Earth is 3 + 3 = 6 years, so the ship twin returns younger (6 years as opposed to 10 years on Earth).

The distinction between what they see and what they calculate

To avoid confusion, note the distinction between what each twin sees and what each would calculate. Each sees an image of his twin which he knows originated at a previous time and which he knows is Doppler shifted. He does not take the elapsed time in the image as the age of his twin now.

• If he wants to calculate when his twin was the age shown in the image (i.e. how old he himself was then), he has to determine how far away his twin was when the signal was emitted—in other words, he has to consider simultaneity for a distant event.
• If he wants to calculate how fast his twin was aging when the image was transmitted, he adjusts for the Doppler shift. For example, when he receives high frequency images (showing his twin aging rapidly) with frequency ${\displaystyle f_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}\sqrt{\left(1+v/c\right)/\left(1-v/c\right)}}$, he does not conclude that the twin was aging that rapidly when the image was generated, any more than he concludes that the siren of an ambulance is emitting the frequency he hears. He knows that the Doppler effect has increased the image frequency by the factor 1 / (1 − v/c). Therefore, he calculates that his twin was aging at the rate of

${\displaystyle f_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}\sqrt{\left(1+v/c\right)/\left(1-v/c\right)}\times \left(1-v/c\right)=f_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}\sqrt{1-v^2/c^2}\equiv ϵf_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}}$

when the image was emitted. A similar calculation reveals that his twin was aging at the same reduced rate of εf_rest in all low frequency images.

Simultaneity in the Doppler shift calculation

It may be difficult to see where simultaneity came into the Doppler shift calculation, and indeed the calculation is often preferred because one does not have to worry about simultaneity. As seen above, the ship twin can convert his received Doppler-shifted rate to a slower rate of the clock of the distant clock for both red and blue images. If he ignores simultaneity, he might say his twin was aging at the reduced rate throughout the journey and therefore should be younger than he is. He is now back to square one, and has to take into account the change in his notion of simultaneity at the turnaround. The rate he can calculate for the image (corrected for Doppler effect) is the rate of the Earth twin's clock at the moment it was sent, not at the moment it was received. Since he receives an unequal number of red and blue shifted images, he should realize that the red and blue shifted emissions were not emitted over equal time periods for the Earth twin, and therefore he must account for simultaneity at a distance.

Viewpoint of the traveling twin

During the turnaround, the traveling twin is in an accelerated reference frame. According to the equivalence principle, the traveling twin may analyze the turnaround phase as if the stay-at-home twin were freely falling in a gravitational field and as if the traveling twin were stationary. A 1918 paper by Einstein presents a conceptual sketch of the idea. From the viewpoint of the traveler, a calculation for each separate leg, ignoring the turnaround, leads to a result in which the Earth clocks age less than the traveler. For example, if the Earth clocks age 1 day less on each leg, the amount that the Earth clocks will lag behind amounts to 2 days. The physical description of what happens at turnaround has to produce a contrary effect of double that amount: 4 days' advancing of the Earth clocks. Then the traveler's clock will end up with a net 2-day delay on the Earth clocks, in agreement with calculations done in the frame of the stay-at-home twin.

The mechanism for the advancing of the stay-at-home twin's clock is gravitational time dilation. When an observer finds that inertially moving objects are being accelerated with respect to themselves, those objects are in a gravitational field insofar as relativity is concerned. For the traveling twin at turnaround, this gravitational field fills the universe. In a weak field approximation, clocks tick at a rate of t' = t (1 + Φ / c²) where Φ is the difference in gravitational potential. In this case, Φ = gh where g is the acceleration of the traveling observer during turnaround and h is the distance to the stay-at-home twin. The rocket is firing towards the stay-at-home twin, thereby placing that twin at a higher gravitational potential. Due to the large distance between the twins, the stay-at-home twin's clocks will appear to be sped up enough to account for the difference in proper times experienced by the twins. It is no accident that this speed-up is enough to account for the simultaneity shift described above. The general relativity solution for a static homogeneous gravitational field and the special relativity solution for finite acceleration produce identical results.

Other calculations have been done for the traveling twin (or for any observer who sometimes accelerates), which do not involve the equivalence principle, and which do not involve any gravitational fields. Such calculations are based only on the special theory, not the general theory, of relativity. One approach calculates surfaces of simultaneity by considering light pulses, in accordance with Hermann Bondi's idea of the k-calculus. A second approach calculates a straightforward but technically complicated integral to determine how the traveling twin measures the elapsed time on the stay-at-home clock. An outline of this second approach is given in a separate section below.

Difference in elapsed time as a result of differences in twins' spacetime paths

Further information: Hyperbolic motion (relativity)

Twin paradox employing a rocket following an acceleration profile in terms of coordinate time T and by setting c=1: Phase 1 (a=0.6, T=2); Phase 2 (a=0, T=2); Phase 3-4 (a=-0.6, 2T=4); Phase 5 (a=0, T=2); Phase 6 (a=0.6, T=2). The twins meet at T=12 and τ=9.33. The blue numbers indicate the coordinate time T in the inertial frame of the stay-at-home-twin, the red numbers the proper time τ of the rocket-twin, and "a" is the proper acceleration. The thin red lines represent lines of simultaneity in terms of the different momentary inertial frames of the rocket-twin. The points marked by blue numbers 2, 4, 8 and 10 indicate the times when the acceleration changes direction.

The following paragraph shows several things:

• how to employ a precise mathematical approach in calculating the differences in the elapsed time
• how to prove exactly the dependency of the elapsed time on the different paths taken through spacetime by the twins
• how to quantify the differences in elapsed time
• how to calculate proper time as a function (integral) of coordinate time

Let clock K be associated with the "stay at home twin". Let clock K' be associated with the rocket that makes the trip. At the departure event both clocks are set to 0.

Phase 1: Rocket (with clock K') embarks with constant proper acceleration a during a time T_a as measured by clock K until it reaches some velocity V.

Phase 2: Rocket keeps coasting at velocity V during some time T_c according to clock K.

Phase 3: Rocket fires its engines in the opposite direction of K during a time T_a according to clock K until it is at rest with respect to clock K. The constant proper acceleration has the value −a, in other words the rocket is decelerating.

Phase 4: Rocket keeps firing its engines in the opposite direction of K, during the same time T_a according to clock K, until K' regains the same speed V with respect to K, but now towards K (with velocity −V).

Phase 5: Rocket keeps coasting towards K at speed V during the same time T_c according to clock K.

Phase 6: Rocket again fires its engines in the direction of K, so it decelerates with a constant proper acceleration a during a time T_a, still according to clock K, until both clocks reunite.

Knowing that the clock K remains inertial (stationary), the total accumulated proper time Δτ of clock K' will be given by the integral function of coordinate time Δt

${\displaystyle \mathrm{\Delta }\tau =\int \sqrt{1-(v(t)/c{)}^2}dt}$

where v(t) is the coordinate velocity of clock K' as a function of t according to clock K, and, e.g. during phase 1, given by

${\displaystyle v(t)=\frac{at}{\sqrt{1+{\left(\frac{at}{c}\right)}^2}}.}$

This integral can be calculated for the 6 phases:

Phase 1 ${\displaystyle :c/a\text{arsinh}(a{T}_a/c)}$

Phase 2 ${\displaystyle :T_c\sqrt{1-V^2/c^2}}$

Phase 3 ${\displaystyle :c/a\text{arsinh}(a{T}_a/c)}$

Phase 4 ${\displaystyle :c/a\text{arsinh}(a{T}_a/c)}$

Phase 5 ${\displaystyle :T_c\sqrt{1-V^2/c^2}}$

Phase 6 ${\displaystyle :c/a\text{arsinh}(a{T}_a/c)}$

where a is the proper acceleration, felt by clock K' during the acceleration phase(s) and where the following relations hold between V, a and T_a:

${\displaystyle V=a{T}_a/\sqrt{1+(a{T}_a/c{)}^2}}$

${\displaystyle a{T}_a=V/\sqrt{1-V^2/c^2}}$

So the traveling clock K' will show an elapsed time of

${\displaystyle \mathrm{\Delta }\tau =2T_c\sqrt{1-V^2/c^2}+4c/a\text{arsinh}(a{T}_a/c)}$

which can be expressed as

${\displaystyle \mathrm{\Delta }\tau =2T_c/\sqrt{1+(a{T}_a/c{)}^2}+4c/a\text{arsinh}(a{T}_a/c)}$

whereas the stationary clock K shows an elapsed time of

${\displaystyle \mathrm{\Delta }t=2T_c+4T_a}$

which is, for every possible value of a, T_a, T_c and V, larger than the reading of clock K':

${\displaystyle \mathrm{\Delta }t>\mathrm{\Delta }\tau }$

Difference in elapsed times: how to calculate it from the ship

Twin paradox employing a rocket following an acceleration profile in terms of proper time τ and by setting c=1: Phase 1 (a=0.6, τ=2); Phase 2 (a=0, τ=2); Phase 3-4 (a=-0.6, 2τ=4); Phase 5 (a=0, τ=2); Phase 6 (a=0.6, τ=2). The twins meet at T=17.3 and τ=12.

In the standard proper time formula

${\displaystyle \mathrm{\Delta }\tau ={\int }_0^{\mathrm{\Delta }t}\sqrt{1-{\left(\frac{v(t)}{c}\right)}^2}dt,}$

Δτ represents the time of the non-inertial (travelling) observer K' as a function of the elapsed time Δt of the inertial (stay-at-home) observer K for whom observer K' has velocity v(t) at time t.

To calculate the elapsed time Δt of the inertial observer K as a function of the elapsed time Δτ of the non-inertial observer K', where only quantities measured by K' are accessible, the following formula can be used:

${\displaystyle \mathrm{\Delta }{t}^2=\left[{\int }_0^{\mathrm{\Delta }\tau }{e}^{{\int }_0^{\overline{\tau }}a({\tau }^{\prime })d{\tau }^{\prime }}d\overline{\tau }\right]\left[{\int }_0^{\mathrm{\Delta }\tau }{e}^{-{\int }_0^{\overline{\tau }}a({\tau }^{\prime })d{\tau }^{\prime }}d\overline{\tau }\right],}$

where a(τ) is the proper acceleration of the non-inertial observer K' as measured by himself (for instance with an accelerometer) during the whole round-trip. The Cauchy–Schwarz inequality can be used to show that the inequality Δt > Δτ follows from the previous expression:

${\displaystyle \begin{array}{rl}\mathrm{\Delta }{t}^2& =\left[{\int }_0^{\mathrm{\Delta }\tau }{e}^{{\int }_0^{\overline{\tau }}a({\tau }^{\prime })d{\tau }^{\prime }}d\overline{\tau }\right]\left[{\int }_0^{\mathrm{\Delta }\tau }{e}^{-{\int }_0^{\overline{\tau }}a({\tau }^{\prime })d{\tau }^{\prime }}d\overline{\tau }\right]\\ & >{\left[{\int }_0^{\mathrm{\Delta }\tau }{e}^{{\int }_0^{\overline{\tau }}a({\tau }^{\prime })d{\tau }^{\prime }}{e}^{-{\int }_0^{\overline{\tau }}a({\tau }^{\prime })d{\tau }^{\prime }}d\overline{\tau }\right]}^2={\left[{\int }_0^{\mathrm{\Delta }\tau }d\overline{\tau }\right]}^2=\mathrm{\Delta }{\tau }^2.\end{array}}$

Using the Dirac delta function to model the infinite acceleration phase in the standard case of the traveller having constant speed v during the outbound and the inbound trip, the formula produces the known result:

${\displaystyle \mathrm{\Delta }t=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\mathrm{\Delta }\tau .}$

In the case where the accelerated observer K' departs from K with zero initial velocity, the general equation reduces to the simpler form:

${\displaystyle \mathrm{\Delta }t={\int }_0^{\mathrm{\Delta }\tau }{e}^{\pm {\int }_0^{\overline{\tau }}a({\tau }^{\prime })d{\tau }^{\prime }}d\overline{\tau },}$

which, in the smooth version of the twin paradox where the traveller has constant proper acceleration phases, successively given by a, −a, −a, a, results in

${\displaystyle \mathrm{\Delta }t=\frac{4}{a}\sinh (\frac{a}{4}\mathrm{\Delta }\tau )}$

where the convention c = 1 is used, in accordance with the above expression with acceleration phases T_a = Δt/4 and inertial (coasting) phases T_c = 0.

A rotational version

Twins Bob and Alice inhabit a space station in circular orbit around a massive body in space. Bob suits up and exits the station. While Alice remains inside the station, continuing to orbit with it as before, Bob uses a rocket propulsion system to cease orbiting and hover where he was. When the station completes an orbit and returns to Bob, he rejoins Alice. Alice is now younger than Bob. In addition to rotational acceleration, Bob must decelerate to become stationary and then accelerate again to match the orbital speed of the space station.

No twin paradox in an absolute frame of reference

Einstein's conclusion of an actual difference in registered clock times (or aging) between reunited parties caused Paul Langevin to posit an actual, albeit experimentally undetectable, absolute frame of reference:

In 1911, Langevin wrote: "A uniform translation in the aether has no experimental sense. But because of this it should not be concluded, as has sometimes happened prematurely, that the concept of aether must be abandoned, that the aether is non-existent and inaccessible to experiment. Only a uniform velocity relative to it cannot be detected, but any change of velocity .. has an absolute sense."

In 1913, Henri Poincaré's posthumous Last Essays were published and there he had restated his position: "Today some physicists want to adopt a new convention. It is not that they are constrained to do so; they consider this new convention more convenient; that is all. And those who are not of this opinion can legitimately retain the old one."

In the relativity of Poincaré and Hendrik Lorentz, which assumes an absolute (though experimentally indiscernible) frame of reference, no twin paradox arises due to the fact that clock slowing (along with length contraction and velocity) is regarded as an actuality, hence the actual time differential between the reunited clocks.

That interpretation of relativity, which John A. Wheeler calls "ether theory B (length contraction plus time contraction)", did not gain as much traction as Einstein's, which simply disregarded any deeper reality behind the symmetrical measurements across inertial frames. There is no physical test which distinguishes one interpretation from the other.

In 2005, Robert B. Laughlin (Physics Nobel Laureate, Stanford University), wrote about the nature of space: "It is ironic that Einstein's most creative work, the general theory of relativity, should boil down to conceptualizing space as a medium when his original premise [in special relativity] was that no such medium existed ... The word 'ether' has extremely negative connotations in theoretical physics because of its past association with opposition to relativity. This is unfortunate because, stripped of these connotations, it rather nicely captures the way most physicists actually think about the vacuum. ... Relativity actually says nothing about the existence or nonexistence of matter pervading the universe, only that any such matter must have relativistic symmetry (i.e., as measured)."

In Special Relativity (1968), A. P. French wrote: "Note, though, that we are appealing to the reality of A's acceleration, and to the observability of the inertial forces associated with it. Would such effects as the twin paradox exist if the framework of fixed stars and distant galaxies were not there? Most physicists would say no. Our ultimate definition of an inertial frame may indeed be that it is a frame having zero acceleration with respect to the matter of the universe at large."

Timeline of special relativity and the speed of light

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Timeline_of_special_relativity_and_the_speed_of_light 

Albert Einstein and Hendrik Lorentz in 1921 in Leiden

This timeline describes the major developments, both experimental and theoretical, of:

• Einstein’s special theory of relativity (SR),
• its predecessors like the theories of luminiferous aether,
• its early competitors, i.e.:
  • Ritz’s ballistic theory of light,
  • the models of electromagnetic mass created by Abraham (1902), Lorentz (1904), Bucherer (1904) and Langevin (1904).

This list also mentions the origins of standard notation (like c) and terminology (like theory of relavity).

Criteria for inclusion

Task Force One, the world's first nuclear-powered task force. Enterprise, Long Beach and Bainbridge in formation in the Mediterranean, 18 June 1964. Enterprise crew members are spelling out Einstein's mass–energy equivalence formula E = mc² on the flight deck.

Theories other than SR are not described here exhaustively, but only to the extent that is directly relevant to SR – i.e. at points when they:

• anticipated some elements of SR, like Fresnel’s hypothesis of partial aether drag,
• led to new experiments testing SR, like Stokes’s model of complete aether drag,
• were disproved or questioned, e.g. by the experiments of Oliver Lodge.

For a more detailed timeline of aether theories – e.g. their emergence with the wave theory of light – see a separate article. Also, not all experiments are listed here – repetitions, even with much higher precision than the original, are mentioned only if they influence or challenge the opinions at their time. It was the case with:

• Michelson and Morley (1886) repeating the experiment of Fizeau (1851), contradicting Michelson’s interpretation of his 1881 experiment;
• Michelson–Morley (1887), more conclusive than the original experiment by Michelson (1881) and difficult to reconcile with their experiment of 1886, or other first-order measurements;
• Kaufmann’s 1906 repetition of his 1902 experiment, because he claimed to contradict the model of Einstein and Lorentz, considered consistent with the data from 1902;
• Miller (1933) or Marinov (1974), with results different than Michelson–Morley.

For lists of repetitions, see the articles of particular experiments. The measurements of speed of light are also mentioned only to the minimum extent, i.e. when they proved for the first time that c is finite and invariant. Innovations like the use of Foucault's rotating mirror or the Fizeau wheel are not listed here – see the article about speed of light.

This timeline also ignores, for reasons of volume and clarity:

• the long story of spacetime and the concept of time as the fourth dimension; e.g. the ideas of Lagrange and Wells;
• mathematical innovations that influenced the formalism of SR, e.g. the introduction of fibre bundles;
• indirect evidence for SR, through the evidence for relativistic theories like general relativity or relativistic quantum mechanics;
• publication of countless textbooks and popular science books or articles, even very influential classics like Mr Tompkins by George Gamow;
• the cultural impact of SR, e.g. publication of documentaries or commemorations of SR during the World Year of Physics 2005;
• new, untested theories modifying SR like Doubly special relativity or Variable speed of light.

Before the 19th century

A redrawn version of the illustration from the 1676 news report. Rømer compared the apparent duration of Io's orbits as Earth moved towards Jupiter (F to G) and as Earth moved away from Jupiter (L to K).

• 1632 – Galileo Galilei writes about the relativity of motion and that some forms of motion are undetectable; this would be later called the relativity principle, essential for special relativity as one of its postulates.
• 1674 – Robert Hooke makes his observations of the Gamma Draconis star, or γ Draconis for short. He proves a variation in its position on the sky, which would be later identified as stellar aberration.
• 1676 – Ole Rømer gives the first piece of evidence that the speed of light is finite, through his observation of the moons of Jupiter; the discovery divides scientists of his time.
• 1690 – Christiaan Huygens gives the first estimate of the speed of light in air or vacuum, based on Rømer’s work. The result is equivalent to about 2×10⁸ m/s in modern units, correct only to the order of magnitude.
• 1727 – James Bradley correctly identifies the peculiar behaviour of γ Draconis as stellar aberration. Bradley uses this fact to estimate the speed of light in air or vacuum, and his result is more accurate than Huygens’s: about 3.0×10⁸ m/s in modern units. For the first time, the measurement is correct to the first two significant figures.

19th century

Before 1880s

• 1810 – François Arago observes that the speed of light of stars – measured with stellar aberration – may be independent of the relative motion of stars and the Earth; or at least, no differences are observable with a naked eye.
• 1818 – Augustin-Jean Fresnel proposes his model of partial aether dragging to explain Arago’s finding.
• 1845 – George Gabriel Stokes creates his own model of complete aether dragging.
• 1851 – The Fizeau experiment with light in flowing water confirms Fresnel’s model.
• 1861 – James Clerk Maxwell publishes his equations of the electromagnetic field, which had a great impact on the later works on aether and special relativity.
• 1868 – Martinus Hoek modifies the experiment of Fizeau, with the same conclusions.
• 1871 – George Biddell Airy observes the stellar aberration in a telescope filled with water, confirming Fresnel’s model and contradicting Stokes’s.

1880s

Michelson and Morley's interferometric setup, mounted on a stone slab that floats in an annular trough of mercury

• 1881 – Albert Michelson performs his original interferometric experiment. It detects no aether wind, contradicting Fresnel’s model in favour of Stokes’s.
• 1885 – Ludwig Lange introduces the idea of inertial frame of reference. It is essential to relativity as an element of the modern formulation of the relativity principle.
• 1886 – Albert Michelson and Edward Morley repeat the Fizeau experiment with higher precision, confirming its result and contradicting the earlier conclusions of Michelson.
• 1887 – Woldemar Voigt publishes his coordinate transformations preserving the wave equation. They are very similar – but not equivalent – to the later Lorentz transformations.
• 1887 – the Michelson–Morley experiment fails to detect aether wind, disproving some aether theories and leading to new ones.
• 1889 – George FitzGerald conjectures the length contraction to explain the Michelson–Morley experiment.

1890s

• 1892 – Hendrik Lorentz – independently of FitzGerald – proposes the same explanation, with a formula only approximating the special-relativistic length contraction to the first order.
• 1893 – Oliver Lodge makes an interferometric experiment questioning the aether drag hypothesis.
• 1894 – Paul Drude introduces the symbol c for speed of light in vacuum.
• 1895 – Hendrik Lorentz corrects his 1892 model, proposing a contraction by the Lorentz factor (γ).
• 1895 – Albert Einstein probably makes his thought experiment about chasing a light beam, later relevant to his work on special relativity.
• 1897 – Oliver Lodge publishes another experimental result questioning aether drag.
• 1897 – Joseph Larmor publishes his coordinate transformations extending the length contraction formula. These transformations imply a form of time dilation and were an approximation of the full Lorentz transformations.
• 1898 – Henri Poincaré states that simultaneity is relative.
• 1899 – Hendrik Antoon Lorentz publishes an early version of his coordinate transformations, including the local time.

20th century

Hermann Minkowski, who introduced the spacetime formalism to special relativity in 1908.

1900s

• 1902 – Lord Rayleigh writes that Lorentz’s hypothesis of length contraction predicts a form of birefringence and tries to observe it. The null result questions Lorentz’s model, but it would be later explained by a combination of length contraction and time dilation.
• 1902 – Max Abraham develops his classical model of the electron. It anticipated some elements of special relativity like the non-linear dependence of momentum on velocity – or, in other, more debatable terms, the relativistic mass. However, Abraham’s formula was different than in SR or in Lorentz’s theory.
• 1902 – Walter Kaufmann publishes his measurements of how the electron’s momentum – or, using later terms, its relativistic mass – depends on its speed. The results seem to confirm Abraham’s model.
• 1903 – Olinto De Pretto presents his aether theory with some form of mass–energy equivalence. It was described by a formula looking like Einstein’s E = mc², but with different meanings of the terms.
• 1903 – Frederick Thomas Trouton and H.R. Noble publish the results of their experiment with capacitors, showing no aether drift.
• 1904 – DeWitt Bristol Brace conducts an improved version of Rayleigh’s 1902 experiment, again with null result.
• 1904 – Hendrik Lorentz explains the experimental results of Rayleigh, Brace, Trouton and Noble, using his refined coordinate transformations; he also proves that Maxwell’s equations are invariant under them. Lorentz also presents his own classical model of the electron, including the length contraction absent in the work of Abraham – but consistent with Kaufmann’s data so far.
• 1904 – Alfred Bucherer and Paul Langevin independently publish a model of the electron and its mass increasing with speed, in a way different both from Abraham’s and Lorentz’s theories. This hypothesis was also consistent with Kaufmann’s results at that stage.
• 1904 – Henri Poincaré presents the principle of relativity for electromagnetism.
• 1905 – Poincaré introduces the name Lorentz transformations and is the first to present them in their full form that would be later present in Einstein’s special relativity proper. Also, Poincaré is the first to describe the relativistic velocity-addition formula – implicitly in his publication and explicitly in his letter to Lorentz.
• 1905 – Albert Einstein publishes his special theory of relativity, including the mass–energy equivalence that would be later written as E = mc².
• 1906 – Alfred Bucherer introduces the name theory of relativity, based on Max Planck’s term relative theory.
• 1906 – Walter Kaufmann publishes his new measurements of the mass–velocity dependence, and claims to disprove the formula of Lorentz and Einstein. At the same time, he accepts that both the old model of Abraham (1902) and the later model of Bucherer & Langevin (1904) are consistent with the data.
• 1907 – Max Von Laue describes how the relativistic velocity-addition formula recreates the Fresnel drag coefficients.
• 1908 – Hermann Minkowski publishes his spacetime formalism of special relativity.
• 1908 – Frederick Thomas Trouton and Alexander Rankine conduct an experiment with electric circuit, proving that the length contraction is not the only relativistic effect and some form of time dilation is present – similarly to the previous experiments by Rayleigh (1902) and Brace (1904).
• 1908 – Walther Ritz publishes his ballistic theory of light as an alternative to special relativity and Maxwell’s electrodynamics.
• 1909 – Paul Ehrenfest publishes the Ehrenfest paradox about rigidity in special relativity.
• 1909 – Gilbert N. Lewis and Richard Tolman coin the disputed term relativistic mass.

1910s

Schematic representation of a Sagnac interferometer.

• 1910 – Vladimir Ignatowski makes the first derivations of Lorentz transformations that rely mostly – and almost entirely – on the relativity principle, without appealing to Maxwell’s equations; such derivations are sometimes called single-postulate.
• 1910 – Edmund Taylor Whittaker and Vladimir Varićak introduce the idea of rapidity, but without using this name.
• 1911 – Alfred Robb coins the term rapidity.
• 1911 – Paul Langevin presents the twin paradox implied by time dilation.
• 1911 – Max von Laue writes that special relativity and Lorentz aether theory predict the Sagnac effect, absent in Ritz's ballistic theory or in Stokes's theory of aether drag.
• 1913 – Georges Sagnac observes the effect named after him, disproving Ritz's ballistic theory or aether drag. However, he favours Lorentz's model and even claims – incorrectly – to contradict SR.
• 1913 – Willem de Sitter describes how the light of double stars contradicts Ritz’s ballistic theory of light.
• 1914 – Ludwik Silberstein gives the first description of Thomas–Wigner rotation, then underappreciated.
• 1914 – Günther Neumann measures the mass–velocity dependence for electrons. His result favours the formula of Lorentz & Einstein over the one by Abraham.
• 1915 – Charles-Eugène Guye and Charles Lavanchy make their own measurements of the inertia of cathode rays, much more exact than the earlier research by Kaufmann. Their conclusion is opposite to Kaufmann’s – again, the formula of Lorentz and Einstein is correct and Abraham’s model is disproved.

1920s and 1930s

• 1924 – Hans Thirring notices that ballistic theories of light contradict spectroscopic observations of the Sun.
• 1924 – Anton Lampa predicts a relativistic effect later known as Penrose–Terrell rotation.
• 1925 – the Michelson–Gale–Pearson experiment tests the Sagnac effect caused by the Earth’s rotation. The result disproves any aether drag; in combination with other experiments – disproving the stationary aether like the Michelson–Morley experiment – it proves the Lorentz transformations correct.
• 1925 – Llewelyn Thomas discovers Thomas precession, which can be explained by the effect described earlier by Silberstein and later by Wigner.
• 1928 – Paul Dirac describes the general energy–momentum relation, extending the equivalence of mass and energy.
• 1932 – Kennedy–Thorndike experiment confirms the Lorentz transformations in a new way, complementary to the Michelson–Morley experiment. These two results, if combined, prove some form of time dilation.
• 1932 – John Cockcroft and Ernest Walton prove the mass–energy equivalence via a nuclear reaction.
• 1933 – Dayton Miller conducts an improved form of the Michelson–Morley experiment, claiming to contradict special relativity. It would be later explained consistently with SR in the 1950s.
• 1935 – the Hammar experiment is another refutation of aether drag and evidence for special relativity.
• 1938 – Ives–Stilwell experiment measures time dilation via the relativistic Doppler effect. For the first time, the Lorentz transformations can be derived directly from empirical data, as would be noticed by Robertson in 1949.
• 1939 – Eugene Wigner rediscovers that SR predicts the Thomas–Wigner rotation.

After 1930s

• 1940 – Bruno Rossi and D.B. Hall observe time dilation in cosmic rays, i.e. in the decay of muons.
• 1949 – Howard P. Robertson notices that the Lorentz transformations can be deduced (extracted) from three key experiments: Michelson–Morley, Kennedy–Thorndike and Ives–Stillwell.
• 1954 – Gerhart Lüders and Wolfgang Pauli prove that the Lorentz invariance in quantum field theories implies the CPT symmetry, allowing for new tests of special relativity.
• 1955 – Robert S. Shankland and others explain Miller’s experimental result from 1933 in a way consistent with special relativity.
• 1959 – Roger Penrose and James Terrell independently publish their rediscovery that SR predicts the Penrose–Terrell effect.
• 1959 – E. Dewan and M. Beran publish the thought experiment known as Bell's spaceship paradox.
• 1960 – Vernon W. Hughes et al. perform a spectroscopic experiment, later interpreted as evidence for the Lorentz invariance of particle interactions.
• 1961 – Ronald Drever independently conducts a similar experiment with the same conclusions.
• 1961 – Wolfgang Rindler presents and solves the ladder paradox.
• 1967 – Gerald Feinberg introduces the term tachyon for hypothetical particles with speeds higher than that of light in vacuum (c).
• 1971 – The Hafele–Keating experiment confirms time dilation predicted by special & general relativity.
• 1974 – Stefan Marinov claims to contradict special relativity by measuring a variation in c. His results are noted by the scientific community but rejected as incorrect.
• 1983 – the speed of light in vacuum (c) is used to define the metre in the SI system of units; the definition does not mention any frame of reference, assuming this speed is universal, and implicitly that special relativity is correct.

21st century

• 2011 – Faster-than-light neutrino anomaly is reported by CERN.
• 2012 – the anomaly in neutrino speed is explained by a failure of the equipment; this reason is officially reported.

Coherentism

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Coherentism

In philosophical epistemology, there are two types of coherentism: the coherence theory of truth; and the coherence theory of justification (also known as epistemic coherentism).

Coherent truth is divided between an anthropological approach, which applies only to localized networks ('true within a given sample of a population, given our understanding of the population'), and an approach that is judged on the basis of universals, such as categorical sets. The anthropological approach belongs more properly to the correspondence theory of truth, while the universal theories are a small development within analytic philosophy.

The coherentist theory of justification, which may be interpreted as relating to either theory of coherent truth, characterizes epistemic justification as a property of a belief only if that belief is a member of a coherent set. What distinguishes coherentism from other theories of justification is that the set is the primary bearer of justification.

As an epistemological theory, coherentism opposes dogmatic foundationalism and also infinitism through its insistence on definitions. It also attempts to offer a solution to the regress argument that plagues correspondence theory. In an epistemological sense, it is a theory about how belief can be proof-theoretically justified.

Coherentism is a view about the structure and system of knowledge, or else justified belief. The coherentist's thesis is normally formulated in terms of a denial of its contrary, such as dogmatic foundationalism, which lacks a proof-theoretical framework, or correspondence theory, which lacks universalism. Counterfactualism, through a vocabulary developed by David K. Lewis and his many worlds theory although popular with philosophers, has had the effect of creating wide disbelief of universals amongst academics. Many difficulties lie in between hypothetical coherence and its effective actualization. Coherentism claims, at a minimum, that not all knowledge and justified belief rest ultimately on a foundation of noninferential knowledge or justified belief. To defend this view, they may argue that conjunctions (and) are more specific, and thus in some way more defensible, than disjunctions (or).

After responding to foundationalism, coherentists normally characterize their view positively by replacing the foundationalism metaphor of a building as a model for the structure of knowledge with different metaphors, such as the metaphor that models our knowledge on a ship at sea whose seaworthiness must be ensured by repairs to any part in need of it. This metaphor fulfills the purpose of explaining the problem of incoherence, which was first raised in mathematics. Coherentists typically hold that justification is solely a function of some relationship between beliefs, none of which are privileged beliefs in the way maintained by dogmatic foundationalists. In this way universal truths are in closer reach. Different varieties of coherentism are individuated by the specific relationship between a system of knowledge and justified belief, which can be interpreted in terms of predicate logic, or ideally, proof theory.

Definition

As a theory of truth, coherentism restricts true sentences to those that cohere with some specified set of sentences. Someone's belief is true if and only if it is coherent with all or most of his or her other (true) beliefs. The terminology of coherence is then said to correlate with truth via some concept of what qualifies all truth, such as absoluteness or universalism. These further terms become the qualifiers of what is meant by a truth statement, and the truth-statements then decide what is meant by a true belief. Usually, coherence is taken to imply something stronger than mere consistency. Statements that are comprehensive and meet the requirements of Occam's razor are usually to be preferred.

As an illustration of the principle, if people lived in a virtual reality universe, they could see birds in the trees that aren't really there. Not only are the birds not really there, but the trees aren't really there either. The people may or may not know that the bird and the tree are there, but in either case there is a coherence between the virtual world and the real one, expressed in terms of true beliefs within available experience. Coherence is a way of explicating truth values while circumventing beliefs that might be false in any way. More traditional critics from the correspondence theory of truth have said that it cannot have contents and proofs at the same time, unless the contents are infinite, or unless the contents somehow exist in the form of proof. Such a form of 'existing proof' might seem ridiculous, but coherentists tend to think it is non-problematic. It therefore falls into a group of theories that are sometimes deemed excessively generalistic, what Gábor Forrai calls 'blob realism'.

Perhaps the best-known objection to a coherence theory of truth is Bertrand Russell's argument concerning contradiction. Russell maintained that a belief and its negation will each separately cohere with one complete set of all beliefs, thus making it internally inconsistent. For example, if someone holds a belief that is false, how might we determine whether the belief refers to something real although it is false, or whether instead the right belief is true although it is not believed? Coherence must thus rely on a theory that is either non-contradictory or accepts some limited degree of incoherence, such as relativism or paradox. Additional necessary criteria for coherence may include universalism or absoluteness, suggesting that the theory remains anthropological or incoherent when it does not use the concept of infinity. A coherentist might argue that this scenario applies regardless of the theories being considered, and so, that coherentism must be the preferred truth-theoretical framework in avoiding relativism.

History

In modern philosophy, the coherence theory of truth was defended by Baruch Spinoza, Immanuel Kant, Johann Gottlieb Fichte, Karl Wilhelm Friedrich Schlegel, and Georg Wilhelm Friedrich Hegel and Harold Henry Joachim (who is credited with the definitive formulation of the theory). However, Spinoza and Kant have also been interpreted as defenders of the correspondence theory of truth.

In late modern philosophy, epistemic coherentist views were held by Schlegel and Hegel, but the definitive formulation of the coherence theory of justification was provided by F. H. Bradley in his book The Principles of Logic (1883).

In contemporary philosophy, epistemologists who have significantly contributed to epistemic coherentism include: A. C. Ewing, Brand Blanshard, C. I. Lewis, Nicholas Rescher, Laurence BonJour, Keith Lehrer, and Paul Thagard. Otto Neurath is also sometimes thought to be an epistemic coherentist.

The regress argument

Both coherence and foundationalist theories of justification attempt to answer the regress argument, a fundamental problem in epistemology that goes as follows. Given some statement P, it appears reasonable to ask for a justification for P. If that justification takes the form of another statement, P', one can again reasonably ask for a justification for P', and so forth. There are three possible outcomes to this questioning process:

1. the series is infinitely long, with every statement justified by some other statement.
2. the series forms a loop, so that each statement is ultimately involved in its own justification.
3. the series terminates with certain statements having to be self-justifying.

An infinite series appears to offer little help, unless a way is found to model infinite sets. This might entail additional assumptions. Otherwise, it is impossible to check that each justification is satisfactory without making broad generalizations.

Coherentism is sometimes characterized as accepting that the series forms a loop, but although this would produce a form of coherentism, this is not what is generally meant by the term. Those who do accept the loop theory sometimes argue that the body of assumptions used to prove the theory is not what is at question in considering a loop of premises. This would serve the typical purpose of circumventing the reliance on a regression, but might be considered a form of logical foundationalism. But otherwise, it must be assumed that a loop begs the question, meaning that it does not provide sufficient logic to constitute proof.

Foundationalism's response

One might conclude that there must be some statements that, for some reason, do not need justification. This view is called foundationalism. For instance, rationalists such as Descartes and Spinoza developed axiomatic systems that relied on statements that were taken to be self-evident: "I think therefore I am" is the most famous example. Similarly, empiricists take observations as providing the foundation for the series.

Foundationalism relies on the claim that it is not necessary to ask for justification of certain propositions, or that they are self-justifying. Coherentists argue that this position is overly dogmatic. In other words, it does not provide real criteria for determining what is true and what is not. The Coherentist analytic project then involves a process of justifying what is meant by adequate criteria for non-dogmatic truth. As an offshoot of this, the theory insists that it is always reasonable to ask for a justification for any statement. For example, if someone makes an observational statement, such as "it is raining", the coherentist contends that it is reasonable to ask for example whether this mere statement refers to anything real. What is real about the statement, it turns out, is the extended pattern of relations that we call justifications. But, unlike the relativist, the coherentist argues that these associations may be objectively real. Coherentism contends that dogmatic foundationalism does not provide the whole set of pure relations that might result in actually understanding the objective context of phenomena, because dogmatic assumptions are not proof-theoretic, and therefore remain incoherent or relativistic. Coherentists therefore argue that the only way to reach proof-theoretic truth that is not relativistic is through coherency.

Coherentism's response

Coherentism denies the soundness of the regression argument. The regression argument makes the assumption that the justification for a proposition takes the form of another proposition: P" justifies P', which in turn justifies P. For coherentism, justification is a holistic process. Inferential justification for the belief that P is nonlinear. This means that P" and P' are not epistemically prior to P. Rather, the beliefs that P", P', and P work together to achieve epistemic justification. Catherine Elgin has expressed the same point differently, arguing that beliefs must be "mutually consistent, cotenable, and supportive. That is, the components must be reasonable in light of one another. Since both cotenability and supportiveness are matters of degree, coherence is too." Usually the system of belief is taken to be the complete set of beliefs of the individual or group, that is, their theory of the world.

It is necessary for coherentism to explain in some detail what it means for a system to be coherent. At the least, coherence must include logical consistency. It also usually requires some degree of integration of the various components of the system. A system that contains more than one unrelated explanation of the same phenomenon is not as coherent as one that uses only one explanation, all other things being equal. Conversely, a theory that explains divergent phenomena using unrelated explanations is not as coherent as one that uses only one explanation for those divergent phenomena. These requirements are variations on Occam's razor. The same points can be made more formally using Bayesian statistics. Finally, the greater the number of phenomena explained by the system, the greater its coherence.

Problems for coherentism

A problem coherentism has to face is the plurality objection. There is nothing within the definition of coherence that makes it impossible for two entirely different sets of beliefs to be internally coherent. Thus there might be several such sets. But if one supposes—in line with the principle of non-contradiction—that there can only be one complete set of truths, coherentism must therefore resolve internally that these systems are not contradictory, by establishing what is meant by truth. At this point, Coherence could be faulted for adopting its own variation of dogmatic foundationalism by arbitrarily selecting truth values. Coherentists must argue that their truth-values are not arbitrary for provable reasons.

A second objection also emerges, the finite problem: that arbitrary, ad hoc relativism could reduce statements of relatively insignificant value to non-entities during the process of establishing universalism or absoluteness. This might result in a totally flat truth-theoretic framework, or even arbitrary truth values. Coherentists generally solve this by adopting a metaphysical condition of universalism, sometimes leading to materialism, or by arguing that relativism is trivial.

However, metaphysics poses another problem, the problem of the stowaway argument that might carry epistemological implications. However, a coherentist might say that if the truth conditions of the logic hold, then there will be no problem regardless of any additional conditions that happen to be true. Thus, the stress is on making the theory valid within the set, and also verifiable.

A number of philosophers have raised concerns over the link between intuitive notions of coherence that form the foundation of epistemic forms of coherentism and some formal results in Bayesian probability. This is an issue raised by Luc Bovens and Stephen Hartmann in the form of 'impossibility' results, and by Erik J. Olsson. Attempts have been made to construct a theoretical account of the coherentist intuition.