Theory of encryption of power

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The theory of encryption of power (TEP) The theory of encryption of power is a cluster theory, mainly drawn up as a political philosophy that challenges the conventional understanding of power, beingness, and democracy proposing a novel way to relate those terms that, according to the proponents of the theory, will render a completely new ontological map of politics, law, and philosophy. According to TEP, power is not something that can be possessed or transferred, but something that is constantly created and transformed by human agency. However, this creative power has been systematically hidden, solidified and/or simulated by institutional structures that impose fixed and static models of identity, such as the state, the law, or the constitution. These structures, that define the backbone of power as domination, are called simulacra, and they act as forms of encryption that prevent or condition the expression of genuine political agency and crush democracy in its own name. One of the main tenets of the theory is that there is a unique shift in the forms of sovereignty and instituted power inaugurated by the imposition of constitutional law as the pinnacle of political societies in the last three centuries. Accordingly, the theory holds that this shift stands as the most sophisticated stage in the permanent thrive of western forms of power to neutralize and finally disarm democracy and political agency. As TEP sustains, coloniality exists because it encrypts power; hence the theory develops insights alongside and in parallel to decolonial theory and critical and subaltern studies. The central claim exposes how, with the synthesis of sovereignty and constitutional law and, in “the name of the people”, the people are made vulnerable to dispossession and exclusion, and how in the name of democracy, democracy is undermined and potentially destroyed.

TEP has been developed mainly by the Colombian legal and political philosopher Ricardo Sanín-Restrepo, but also by Marinella Machado-Araujo, James Martel, Gabriel Mendez, Sabelo J. Ndlovu-Gatsheni and Angus McDonald, among other scholars. Several panels regarding the concept of encryption have been held at the Critical Legal Conference in 2014 at the University of Sussex, in 2015 at the University of Wrocław, Poland, in 2017 at the University of Warwick and in 2021 at the University of Dundee. Lexington Books has recently launched a series of books devoted to the theory.

To further understand the implications of the theory as a cluster of political and theoretical possibilities it is paramount to begin with one of its basic constructions, according to Ricardo Sanin-Restrepo and Marinella Machado -Araujo what the encryption of power "inhibits is the possibility of communicating meanings that are not defined in advance by a transcendent model, where the political lexicon is hierarchized, and the possibility of its uses are reserved for a few. Where there is encryption of lexicons there is thus a hierarchical split of beings and objects in the world" (Sanin-Restrepo and Machado-Araujo, 2020). Hence, TEP understands that there is only a world where difference exists as the fundamental and sole order, but also that such a possibility is heavily obstructed by the concentration of power in forms of transcendent models that fashion oppression. It proposes a radical democracy that challenges the existing structures of power and law, and seeks to emancipate the hidden potentialities of difference.

The theory of encryption of power uses the concept of encryption to describe the process of hiding or protecting information from unauthorized access. In this case, the information that is encrypted is the political agency and creativity of the people, which is reduced to static and solid models of identity that pose as the only form of power. The forms of encryption are the simulacra, which are false representations or imitations of reality that conceal the absence or distortion of the original. For example, sovereignty is a simulacrum that encrypts the power of the people and imposes a transcendent and final authority over them.

TEP is a political and philosophical theory that analyzes and challenges the nature, origin, and effects of power and its encryption. It argues that power is not simply a relation of domination and resistance, but also a process of simulation and concealment that creates illusions of difference and democracy. The theory also claims that power encrypts itself by producing simulacra, which are false representations of reality that hide the true nature and source of power. Thus, the theory suggests that political agency can only be exercised by decrypting power, which means exposing and dismantling the simulacra and revealing the hidden structures and forces that shape reality. The theory draws on and integrates various sources and traditions of political and philosophical thought, such as Aristotle, Spinoza, Deleuze, Foucault, Derrida, and others. The theory of encryption of power is also relevant and applicable to different domains and contexts of power, such as law, economy, culture, or ecology; as it also inspires and informs different practices and movements of decrypting power, such as activism, art, education, or technology.

In understanding the theory it is paramount to grasp the relations between transcendent models, beingness, and the body politic. As Sanin-Restrepo puts it "Since Plato, politics is predefined through extenuating conditions of belonging to the body politic, where “to be” corresponds to an already existing qualification of life, the ‘idea’ as an inner split within forms of identity where some are welcomed into politics and some are excluded according to qualifications that are detached to beingness but to which beingness must conform to be. Hence the relationship between power and life is severed, qualified and utterly standardized to favour particular models of identity. The hidden origin is the malleable metaphysical center of Western discourse."

Transcendent models are fixed, final, solid and static structures that simulate power and prohibit or condition being while collapsing political agency into constituted power. Transcendent models are related to coloniality, sovereignty, simulation and mimesis. The implications and consequences of TEP and transcendent models for democracy, resistance and social change are profound and complex. On the one hand, TEP and transcendent models pose a serious threat to the possibility of genuine democracy, as they conceal and constrain the diversity and creativity of political agency and expression. On the other hand, TEP and transcendent models also open up spaces and opportunities for resistance and social change, as they reveal the cracks and contradictions of the dominant forms of power and identity. Some of the possible strategies and alternatives to decrypt power and create new forms of difference and political agency are: - Challenging the legitimacy and authority of transcendent models by exposing their historical and contingent nature, as well as their negative effects on human rights, social justice and ecological balance.

Main concepts

The theory of encryption of power uses several key concepts to explain and critique power and its encryption. Some of the main concepts are:

• The hidden people. The fundamental political and juridical totality of the of modernity is both a subject and an agency, “the people”. However, that said totality is divided in its intrinsic mechanism. "In modernity the key to encryption is the conversion of the concept of the people into a synecdoche. Accordingly, a false totality (the people of human rights and constitutions, the included) become to symbolize and falsely represent an impossible infinity (the excluded, the hidden people)" (Sanín-Restrepo and Araujo 2020). Accordingly, "the people as a totality is a pars pro toto synecdoche. An absolutely arbitrarily part (white people within a nation state) defines an unattainable infinity (the marginalized people, the forced migrant). The people as a synecdoche joins a part that is an excrement of the (simulated) totality and what the totality lacks in order to become a true totality. As the unrepresentable excess of liberal democracies, the hidden people escape all forms of representation and symbolize what exists beyond the representable" (Sanín-Restrepo 2016, 19; 40). However, the hidden people have to be falsely included to give consistency to the fantasy of the totality. The crucial point is that the people as a whole can only exist and exercise power, if and only if, it keeps that other zone of the people “hidden”. (Sanín-Restrepo 2016, 44). The constitutive ambivalence is as follows: We stand before the constitutive paradox of the legitimacy of liberal constitutionalism. On the one hand, we discover the rigid zone of codified law, of codified reality, that manifests itself in archetypal concepts such as the totality of the people as constituent power (We the People), or the totality of the human rights model (everyone) that announces an abrasive universality that holds together the fruit of reality. On the other hand, we have the excess that is compulsory in order to make such totality work as such, the all but one, the all minus one, as the exact mathematical formula of liberalism, the totality minus what it needs to exclude to keep itself immaculate. (Sanín-Restrepo 2016, 35). According to the theory, the hidden people are those who are excluded, marginalized or oppressed by the simulations that encrypt power, such as democracy, law, sovereignty, identity, representation, human rights and capitalism. The hidden people are those who do not have access to or influence over the institutions and agents that claim to administer justice and order in society. The hidden people are those who suffer the negative consequences of the encryption of power, such as violence, poverty, inequality and injustice However, the hidden people are not only passive victims of the encryption of power. They are also potential agents of resistance and transformation, as they are those who have the capacity to decrypt power, by exposing and challenging the simulations that conceal it. Consequently, the hidden people are those who have the solidarity to collaborate with others who share their vision and values, and to build networks and communities that foster empowerment and change. Therefore, the place of the hidden people within the theory of encryption of power is both a problem and a possibility. It is a problem because it reflects the injustice and oppression that the encryption of power produces. It is a possibility because it offers a source of hope and inspiration for decrypting power.

• Sovereignty.A s Sanin-Restrepo and Machado Araujo explain "Independently of geopolitical shifts, sovereignty continues to define the shape of the world from the definition of the exceptional. Power in coloniality depends on one thing alone, the creation of a hidden people as the exception, a feat that can only be achieved through the exercise of sovereignty (...). The supreme decision on the exception remains the core of the power machine as domination. There is a defining transformation in the modern concept of sovereignty that ties together the theory of encryption and the hidden people. This constitutive alteration requires us to create a new concept to explain and delimit the portentous and elusive realities that it creates, this concept is the “simulacrum” (Sanín-Restrepo 2016, 200). The galvanization between coloniality and liberalism creates the most sophisticated and impermeable machine of power in history. We can formulate it simply: “the people must be both the exception and the (simulated) sovereign!”. Coloniality achieves the most extraordinary exploit: it establishes the people as sovereign as it immediately seizes their sovereignty as absolute power (constituent power). All of this is done while maintaining the simulacrum of popular sovereignty as the political and legal axiom of the people. Therefore, it paradoxically merges the hidden people as sovereign and exception"

• Beingness: The capacity or possibility of existing or acting in a certain way. Beingness is the source and expression of difference and political agency. Beingness is often encrypted or conditioned by the simulacra of power, which limit the possibilities of action and expression for most beings.
• Potentiality and actuality One of the key concepts that the TEP employs to decrypt the hidden mechanisms of power is the Aristotelian concept of entelecheia, which means being-at-an-end or completeness. Sanin Restrepo argues that entelecheia is the main principle that organizes the modern constitutional order, by imposing a predetermined end or purpose to the political community, which is usually identified with the common good, the general will, or the public interest. This end is presented as natural, universal, and rational, and therefore as superior to any other possible end that might be proposed by different groups or individuals within the society. Entelecheia thus functions as a normative criterion that legitimizes the exercise of power by those who claim to represent or embody it, and excludes or subordinates those who do not conform to it or challenge it. However, entelecheia is not the only concept that Aristotle used to describe a kind of action or actuality. He also used the concept of energeia, which means being-at-work or activity. Energeia refers to the dynamic and creative process of actualizing one’s potentialities, without being predetermined by a fixed end or purpose. Energeia is thus more open-ended, pluralistic, and contingent than entelecheia, and allows for more diversity and experimentation in the realization of one’s capacities.

Sanin-Restrepo claims that energeia is a more democratic concept than entelecheia, because it does not impose a single end or purpose to the political community, but rather allows for multiple and conflicting ends or purposes to coexist and interact in a dialogical and agonistic way. Energeia also recognizes the importance of difference and alterity as sources of creativity and innovation, rather than as threats or obstacles to be eliminated or assimilated. Energeia thus enables a more participatory and inclusive form of politics, where power is not monopolized or hidden by a privileged group or institution, but rather shared and distributed among diverse actors and agents. According to Sanin-Restrepo, one of the main tasks of critical constitutional theory is to revert the dominance of entelecheia over energeia in the modern constitutional order, and to promote a more democratic balance between them. This would entail challenging the naturalization and universalization of the dominant end or purpose that organizes the political community, and exposing its historical and contingent character. It would also entail fostering a more pluralistic and dynamic conception of politics, where different ends or purposes can be proposed, debated, negotiated, and revised in an ongoing process of collective deliberation and action. Finally, it would entail empowering the marginalized and oppressed groups and individuals who have been excluded or subordinated by the dominant end or purpose, and enabling them to express their own ends or purposes, as well as their own ways of actualizing them. In conclusion, Sanin Restrepo’s theory of encryption of power offers a novel and critical perspective on how power operates in the global context, by using the Aristotelian concepts of entelecheia and energeia as analytical tools. TEP reveals how entelecheia functions as a principle that encrypts power by imposing a predetermined end or purpose to the political community, which legitimizes the exclusion or subordination of difference. The TEP also suggests how energeia can function as a principle that decrypts power by allowing for multiple and conflicting ends or purposes to coexist and interact in a democratic way, which recognizes and values difference. TEP thus proposes a radical democratic alternative to the dominant forms of constitutionalism, which are based on liberal and republican principles that conceal the coloniality of power.

• Difference: The production or manifestation of diversity or multiplicity in beingness. Difference is the condition and possibility of political agency. Difference is often encrypted or reduced to static and solid models of identity that pose as the only form of power.

• Political agency: The exercise or fulfillment of beingness in producing difference. Political agency is the mode and effect of decrypting power. Political agency is often encrypted or neutralized by the simulacra of power, which simulate difference and democracy.

• Power: The capacity or possibility of beingness to produce difference and political agency. Power can be understood as potentia (potentiality) or actuality (energeia or entelecheia). Power as potentia is the capacity or possibility of beingness to produce difference and political agency.
• Encryption: The process or mechanism by which power conceals or simulates itself by producing simulacra. Encryption is a way of denying or limiting beingness, difference, and political agency. Encryption is also a way of imposing institutional simulations of difference that condition, neutralize or prohibit political agency, reducing difference to static and solid models of identity that pose as the only form of power.
• Simulacrum: A false representation or imitation of reality that hides or distorts the true nature or source of power. Simulacra are the products or effects of encryption.

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."