[ Last updated: October 2022. ]
Here are notes on the theory of relativity — special, general, and their intersection — from my undisciplined learning of the subject. The focus is on concepts.
A paradoxical warm-up.
___ Let us begin with what I dub the double contraction paradox to make sense of what is meant by length. Isaac and Albert slide past each other, and Isaac holds up a metre-stick. Albert indicates its length by holding up his hands — at a contracted length of 1/$\gamma$ metres. That's understandable, for that is how length contraction works. The trouble is, to Isaac, who wishes to record the length of the stick he is holding by looking at Albert, the distance between Albert's hands is further contracted to 1/$\gamma^2$ metres. What's going on?
___ Isaac's is no way to measure lengths. The absurdity of this method is at once apparent if Isaac were to drop the stick and keep his hands apart at the distance he obtained from Albert, a distance that Albert in his turn would now register as 1/$\gamma^3$ metres — and so on until the observers clap their hands. So what must Isaac do to obtain the proper length of the stick from Albert? First he must remember that in his worldview not only are Albert's lengths contracted, but that Albert's clocks are slowed down. And not just that — Isaac will record events happening in a different order from Albert. For instance, if Albert were to raise both hands at the same time to indicate his measurement, Isaac would see one hand going up before another. This is the relativity of simultaneity by which events that are simultaneous for Isaac need not be so for Albert, and vice versa. If Isaac accounted for this time lag, he would find that the distance moved by Albert in this duration supplies the rest of the stick's proper length.
___ And that is no surprise if you're conscious that in relativity, times and distances are measured in some system of co-ordinates, and the only invariant is the spacetime interval. Here is a simple, Euclidean analogy of this paradox. You wish to learn the unknown location of a fountain that happens to be 1 km east of you. You have a friend with a telescope, but she uses a system of directions that is tilted by 45 degrees with respect to the one everyone else uses. After a quick search, she tells you that the tree is $1/\sqrt{2}$ km along her positive x-axis. Should you then conclude that, since your axes are at 45 degrees to hers, the tree is $1/\sqrt{2} \times 1/\sqrt{2}$ = 1/2 km in the easterly direction? No, you should also ask her for her y-coordinate, before mapping to your co-ordinates correctly. Just as Isaac should have asked Albert to report on his reading of Isaac's wristwatch.
___ The barn-pole paradox is perhaps a more dramatic illustration of these concepts. Albert has a ten-foot pole but his barn is only three feet wide, so has got no place to store it. While lamenting about this to his brother Isaac he has a bright idea that he wishes to test. He would run very fast with the pole at a speed that shrinks it to within the length of the barn. Isaac must try to shut the barn doors at either end (say, using remote control) the moment he sees that the entire pole has made it in. Of course, he should reopen the doors immediately to avoid a fratricidal collision, but this would at least demonstrate that the pole can fit in the barn. Isaac argues that such a scheme would never work, for in Albert's reference frame it is the barn that would shrink to smaller than three feet while the pole remains at ten feet. Who is correct?
___ Both are. Albert will see the front door close when the front end of the pole is near it, and the back door close when the back end of the pole comes up to it. But he will not see these events occur simultaneously, even though "simultaneously" is just how Isaac shut the doors in his frame of reference.
___ The twin paradox teaches one just about everything essential to relativity.
___ Tara the astronaut flies to outer space in a relativistic rocket while her twin Bhumi the accountant stays behind on Earth. Finding nowhere to land after coasting for 10 years (per Bhumi's watch), Tara turns around and coasts back at the same speed as her forward journey. When Tara returns, we expect Bhumi to have aged 20 years, and Tara to have aged less — say, 12 years — per relativistic time dilation. But then, in Tara's reference frame, it is Bhumi who made the journey at relativistic speeds while she stayed at rest in the rocket. Shouldn't Tara then have aged 20 years and Bhumi 12?
___ The resolution of the paradox is straightforward if you compute the proper time elapsed in a round trip. A trivial path integral shows it to be always smaller than the proper time in the lab frame. So it is indeed Tara who is younger than Bhumi, just as short-lived particles are seen to go around accelerators for much longer than their stationary counterparts that decay quicker. In the case of the twins, it is clear that Tara was not always in an inertial frame, for she turned back after all. In more detail, let us say it took her two minutes to apply brakes, switch direction, and speed back up. In those two minutes she occupies a non-inertial frame, and the ticking of Bhumi's clock in Tara's frame depends entirely on Bhumi's exact position, in this case "up above" Tara at a great distance. It took about 10 years each in Bhumi's frame for Tara's inertial motion on either side of those two minutes. In those near-twenty years, Bhumi maintains that Tara has aged near-twelve years. This is just the part of the journey when Tara maintains that Bhumi has aged but 7.2 years. But her acceleration during turnaround is so enormous, and Bhumi so far above her, that owing to extreme gravitational time dilation, during these two minutes in Tara's rocket Bhumi ages 12.8 years.
___ Bell's spaceship paradox throws more light on acceleration in special relativity. You see two spaceships, one behind the other, at rest and connected by a delicate string of length equal to the distance between them. They now start moving at the same time with accelerations that you measure as equal. Does the string break? You may be tempted to say no, because at every instant the velocities of the spaceships are equal, hence at every instant the entire system is Lorentz-contracted by the same amount (see "The clock postulate" below).
___ Yet it breaks. In your frame the string shrinks, the spaceships shrink, but by design the inter-spaceship distance doesn't. And in the moving frame? It turns out that in the momentarily co-moving inertial frame of either spaceship, the inter-ship distance is greater than the one you measure. As that exceeds the proper length of the string, it duly snaps. (Ehrenfest's paradox below clarifies how rigidity and acceleration mix in relativity.) Again at the heart of this resolution is the relativity of simultaneity: unlike you, neither spaceship observes the other one start at the same time, or match its velocity at every instant.
Superluminal life.
Kosher faster-than-light.
___ If Nature imposed no speed limit on signals, information may be transmitted instantaneously. For instance, should the Sun explode in a supernova, an observer in outer space may see the Earth vapourize at the same moment. Say there are three astronauts Alpha, Beta and Gamma in indestructible pods near Venus, and Alpha witnesses this event. Beta is moving toward the Sun relative to Alpha, and thanks to the relativity of simultaneity she would report that the Sun went off first, and the Earth broke up next. Gamma, who is moving toward Earth relative to Alpha, would report this sequence in reverse! That cannot stand, since an effect has preceded a cause — time has flown backward — to an inertial observer. Such violations of the cherished principle of causality can rapidly get us into unstraightenable temporal paradoxes.
___ Yet it is incorrect to state that "nothing can go faster than light". To avoid causal paradoxes it suffices to say that no signal or information that carries energy can exceed the natural speed of relativity, $c$, in every inertial frame. Or as I like to think of it, "no thing can go faster than light". Below are situations where faster-than-light motion is perfectly legitimate.
- receding galaxies (space can expand at a rate "faster than light" as it carries no information from Point A to Point B; no galaxy passes another superluminally)
- closing speeds (two photons can collide head on and a lab frame observer would measure the separation between them reduce at a rate greater than $c$. So what?)
- proper speeds (is the unfortunate term given to speeds computed by dividing the distance in the lab frame by the time in the rocket frame; a relativistic astronaut will reach the stars in a few years by their calendar, but that doesn't mean they travelled faster than light: their spatial coverage was compressed)
- movement of shadows and laser beam spots (see the lighthouse paradox; nothing physical moves here, for instance a lightbeam shifting sideways faster than lightspeed does not carry photons likewise; a similar effect is produced by spots of light in a row activated by a lightbeam passing through them with successively shrinking delays in the activation)
- movement of objects in the sky around you (it is their local speed that counts)
- phase velocities of waves (carrying no information of signal, unlike group velocities)
- quantum entanglement (quantum mechanics is inherently non-local, so what is really spooky is not "action at a distance" but that the laws of physics allow us to effectively describe particles as though their behaviour is local)
- virtual particles (no information moves with them)
- the Cerenkov effect (Nature's speed limit happens to be the speed of light in vacuum, which may be seen as a consequence of the Lorentz invariance of Maxwell's equations; you can always outrun light in a medium)
- tachyons (their superluminal existence is permanent; see more below)
- wormholes (by connecting two faraway regions of space they appear to a global observer to provide a means of superluminal travel, however in a race inside the wormhole don't bet against the photon; in any case an exotic negative energy density source is required to keep the wormhole stable through transmission)
- Alcubierre drive (again, by warping space ahead of and behind a spacecraft, swifter-than-light travel can be demonstrated to a global observer, but nobody inside the unwarped bubble of space containing the ship is fooled; again an unheard-of source of negative energy is required to obtain the Alcubierre metric from Einstein's equations)
For more discussion see Baez's excellent page.
Tachyons.
___ Due to their negative squared mass, they slow down if they gain energy, so they too cannot cross the lightspeed barrier, where the energy is infinite. Conversely, they speed up if they lose energy. So a charged tachyon would induce a runaway reaction of speeding up to infinity by emitting infinite Cerenkov radiation. This tallies with our intuition that tachyonic vacua are unstable, because the vacuum can spontaneously create tachyon-antitachyon pairs. Thus interacting tachyons are hard to contrive. However, energy-carriers always interact gravitationally, casting doubt on the existence of tachyons as this means causal paradoxes seem inevitable, as we will see in the next paragraph. Further weakening their case is the fact that photons can generically decay to tachyon-antitachyon pairs and some prima facie model-building element must be introduced to safeguard against it (see "Extra time dimensions" below).
___ Detection has nevertheless been unsuccessfully attempted seeking the Cerenkov radiation in presumably exotic nuclear reactions, and in scattering processes. You cannot send messages faster than light, though. Localized tachyons disturbances are subluminal and superluminal disturbances are non-local. A simpler way to see that is that a faster-than-light signal would travel backwards in time in some inertial frame, which can be immediately gleaned from a spacetime diagram. That would stir the pot of causes and effects — would lead to the tachyonic anti-telephone paradox. Alisha could tell Babu she'd call him at lunchtime if he failed to call her at breakfast. Sure enough, Babu sleeps in and gets her call at lunchtime. On a lark, he then calls back Alisha, and her phone rings during breakfast.
What is special?
___ General relativity pertains to physics on the curved spacetime of true gravity (causing tidal forces) sourced by masses and energies. Special relativity is then on the stage of flat spacetime. In particular, special relativity can have accelerating frames where pseudo-gravity is felt. (As an aside, flat spacetime is sufficient for explaining the twin paradox.) These senses of the terms are modern: Einstein meant "special" to denote inertial frames and "general" to all frames, but the line of demarcation on these overlapping domains has since moved.
___ Let us take this opportunity to recall some fundamentals of general relativity. A uniform gravitational field will not cause tidal forces. The equivalence principle states that the feeling of uniform-field gravity is indistinguishable from the pseudo-force experienced by an observer in an accelerating frame. The implication is that frames in free-fall are inertial. Thus objects in free-fall follow geodesics, whether the spacetime is flat or curved. On the Earth's surface we are not on a geodesic of spacetime, because free-fall is arrested by the presence of other forces, rather our spacetime trajectory is a longer path. It helps to think of Bill Unruh's picture of a one-dimensional Earth embedded on the surface of a balloon along with an orthogonal axis for time. The Earth's surface may be represented by a small (not Great) circle at some non-zero latitude. A dot jumping off the surface can be seen to follow the geodesic until landing.
___ There is one other analogy that is a great grokking aid, one that set Einstein himself on the path to formulating general relativity. Consider the co-rotating occupants of a uniformly spinning disk. Were they to measure its circumference using tiny rigid metre-sticks, they would find it to exceed 2$\pi R$ . This is because an external observer would expect Lorentz contraction in the tangential direction but not in the radial one, yet would insist that the circumference is 2$\pi R$ as they view the setup in Euclidean space. To accommodate this the disk observer will see a longer periphery (while agreeing on the length of the radius). The inescapable conclusion is that geometry is non-Euclidean to the disk observer, which can only be due to their being in a non-inertial frame experiencing a pseudo-force, here the centrifugal acceleration. Analogously, rotating clocks cannot all tick at the same rate along the radius. Thus spacetime curvature must be generic to accelerating frames of reference. Note that the centrifugal force is like a tidal force in that its "field" is not uniform, so the analogy is as clean as they come. I'm not even sure it's an analogy.
___ This argument is a pivotal spin-off (well, yeah) from Ehrenfest's paradox that we will encounter below in the context of rigidity.
The freakiness of time.
Extra time dimensions.
___ That we inhabit a single dimension of time and multiple dimensions of space is intimately linked to the observation that massless particles like the photon do not spontaneously decay. If there is only one dimension of time, a time-like vector can only bent in space-like directions (the perpendiculars connecting the original and bent vectors orient in space-like angles), which shortens their interval. Thus time-like vectors in our universe are already maximal in length. Au contraire, a space-like vector can be bent in directions in both time-like (shortening the interval) and space-like (in a new dimension, lengthening the interval).
___ This insight applied to the three-body decay of a particle gives the adjacent diagram depicting four-momentum conservation. The length of each vector denotes the length of the interval. The daughters' four-momenta are seen to "bend" the mother's four-momentum, and hence from the argument above, $m_a \geq m_b + m_c + m_d$ . If you have other dimensions of time this condition is not guaranteed, and no particle is stable against decay.
___ This also implies that particles with space-like four-momenta — tachyons — may be unstable to decay to fellow tachyonic species. How about luxons, particles with light-like four-momenta, e.g. photons? Decays to sub-luminal tardyons are still protected by the above bend-in-timelike-direction argument, but decays to super-luminal tachyons are unguarded against a bend-in-spacelike-direction argument. Thus a theory of tachyons must contain a mechanism of the photon's stability.
___ Incidentally, tardyons, those massive slower-than-light species, have two other names, bradyons and ittyons. It is almost as if physicists discovered them independently over and over again.
Synchronizing clocks.
___ If you want to measure the speed of light "one way", i.e., without bouncing the light off something, you will run into an insurmountable problem of synchronizing the clocks at the ends of emission and detection. If the clocks are apart, you cannot synchronize them without knowing the speed of light beforehand. If you synchronize them close to each other, taking them apart implies relative acceleration, and you introduce time dilation effects. Thus you must follow some convention for synchrony. The Einstein convention fixes the time of the second clock by bouncing a light beam off a mirror close to it, detecting it at the place of emission, and noting the time lag. The slow clock-transport convention transports the second clock slowly to the desired point, with the error introduced by time dilation reduced for slower transports. In either convention one cannot disentangle a possible anisotropy in the velocity of light. In the Einstein convention this is obvious: the forward speed may differ from the backward speed. In the slow clock-transport convention one cannot remove an anisotropy in the direction of transport, which gets into the gamma factor of the time dilation.
Time on a rotating platform.
___ Your phone shows time ticking differently from your stopwatch by up to 30 nanoseconds each day. In special relativity there is no such thing as a rotating inertial frame: on a rotating platform observers cannot agree on what events are simultaneous (they cannot synchronize clocks by the Einstein convention): so, e.g., they cannot define a $t=0$ axis. Another way to see this is you can tell if you are in an inertial frame by releasing an object you're holding — if it doesn't move, you are. On a rotating platform the object will move away due to centrifugal and Coriolis forces. So GPS clocks cannot be synchronized this way, and people in India have internal clocks ticking differently from people in Canada. What UTC (shifted from GPS time by $O(10)$ leap seconds) measures is time in an Earth-centred inertial frame (ECI) containing the Earth's spin, whereas we live on an Earth-centred Earth-fixed frame (ECEF), a rotating platform.
___ The difference in ECEF and ECI ticking rates can also be understood in terms of the proper time of a round trip (encountered in the twin paradox) on the ECEF, which is different from the proper time measured in the ECI. The non-inertial nature of rotational frames also gives rise to the Sagnac effect by which the arrival time of a light beam on a round trip measured on a rotating setup differs for either direction of departure. One can compare this to the difference in arrival times of light that has been gravitationally lensed; there the cause is the non-local nature of the measurement due to spacetime curvature, whereas here it is the violation of inertiality of the co-ordinate system.
Event horizon.
___ If you go from an inertial to an accelerated frame, due to shift in simultaneity the clocks "above" you jump ahead, the clocks "below" you jump behind. In particular, there is a plane below you at a critical distance $x = c^2/a$ where clocks stop ticking and light is frozen, the Rindler horizon. (Of course, you will keep moving away from this horizon, i.e. will increase: think of the spacetime diagram with hyperbolic motion.) This is another way of saying that as you speed up, you will be able to stay ahead of light beams chasing you (which you couldn't have without a handicap). Beyond the horizon time runs backward, and is a place from which no signal can reach you. Einstein had had the means to guess the existence of horizons (and could have later identified their physical sources, e.g., black holes), but didn't.
___ This horizon plays an interesting part in the paradox of a charged particle in a gravitational field. The paradox is this. By the equivalence principle, acceleration in an inertial frame is equivalent to a gravitational force. And per Maxwell's equations, accelerating charges radiate. Putting these facts together, one must expect a falling charged particle to radiate and slow down, and in particular reach the ground later than a neutral particle released at the same time from the same height. The difference in speeds also gives the means for an observer to tell whether they are in an accelerating frame or in a gravitating field, violating the equivalence principle. The violation can be stated in a more vexing way. If the equivalence principle is true, why don't charges at rest on the Earth's surface radiate? Are they not equivalent to charges undergoing acceleration? The resolution to these statements is that they are made with the viewpoint placed in the wrong frame. Maxwell's equations hold only in inertial frames. In this case it is the free-fall frame. To a free-faller falling alongside the charge, there is no radiation, and no slowdown with respect to a neutral particle. One can transform co-ordinates to the non-inertial frame that is the Earth's surface and find that the charge does radiate here, but this is not the usual radiation observed from an inertial frame — the radiation rate is not Lorentz-invariant. Analogously, to a free-faller the charge resting on the ground does radiate at a Lorentz-invariant rate, and it can be shown that to the co-accelerator on the ground it doesn't radiate due to the lack of a magnetic field. Thus there is no paradox. At this point one may then ask: if the free-falling inertial observer sees real radiation, where does it then go for the grounded observer? The answer is that it goes beyond the horizon.
The clock postulate.
___ There is a third postulate of relativity to be added to the first two. (The first two are (i) the laws of physics are the same for all inertial observers, (ii) the speed of light is the same in all inertial frames. These postulates, in combination, evict the concept of absolute simultaneity.) It is the clock postulate. It states that the ticking rate of an accelerating clock does not depend directly on the acceleration, but only parametrically via the changing velocity. There is no direct dependence on higher derivatives of the velocity either. That is, the time dilation factor is always the usual gamma factor in a momentarily co-moving inertial frame. This is akin to the wind-chill factor of a biker, which depends directly only on their speed.
___ The clock postulate truly is independent of the other postulates and cannot be derived; it has been empirically verified. It forms the basis of the covariant transformations one has to make between non-inertial frames in general relativity. As such, it is seen to hold for clocks moving in the curved spacetime of general relativity as well.
Did you know?
- There is a satellite altitude at which velocity time dilation due to special relativity and gravitational time dilation due to general relativity cancel. ISS clocks tick slower, GPS clocks tick faster.
- Gravitational time dilation renders the Earth's core 2.5 years younger than the surface.
Further delights.
Gold.
___ Gold has a golden colour because of relativistic effects in orbitals. If you don't account for them, gold will be silvery.
Aberration.
___ What is seen, not measured, by a relativistic traveller is aberration. When you tilt your umbrella while walking in a vertical rain, that's aberration. Stellar aberration was observed four centuries ago by astronomers who were only looking for parallax. James Bradley even measured lightspeed with it in 1727. Per aberration objects "behind" may appear in front. If one is going fast enough, the entire radiation of the universe may appear in a small patch in front, faint due to Doppler shifting.
A double boost is a tilted boost.
___ Consider the paradox of the metre-stick and hole. A metre-stick is sliding east along its length toward a pit. At the same time coming up through the pit is a thin plate with a metre-long slit in the east-west direction. The stick and the plate arrive at the top of the pit at the same time. Will the stick fall through the slit in the plate? To a lab observer, the stick has shrunk to sub-metre length and will of course fall through the slit. But to an observer on the stick, isn't the slit shrunk length-wise due to the component of the plate's velocity approaching them, and so shouldn't the stick clatter on the plate?
___ Unlike in the barn-pole paradox, where the pole had a way to get out of the barn in the runner's frame, the relativity of simultaneity by itself will not save us here. We will need, in addition, to invoke Thomas-Wigner rotation. When a system is observed from a frame that is reached after two consecutive Lorentz boosts (that are not necessarily parallel), the system appears not only boosted in a new direction but also rotated. In the case under study, the stick observer can be thought of as having got there by boosting first from the plate frame to the lab frame, then from there to the stick frame. To this observer the plate will not only appear to fly in a west-and-up direction, but it will also be tilted with an easterly upper end (see adjoining figure). The stick will then first pass through the slit's upper end, then exit the slit just as its lower end catches up.
Rigidity.
___There is no satisfactory notion of rigidity in special relativity — think of Lorentz contraction. Think especially of Rindler's length contraction paradox, which conceptually rhymes with the barn-pole paradox. A metre-stick sliding lengthwise approaches a metre-long pit on the pavement. Will it fall in? Yes, says the pavement dweller, the metre-stick is shrunk. No, says the stick rider, 'tis the pit that's shrunk. By now we know that the stick must fall in, and that the process will look different to either observer. The figure shows how the stick rider sees it: the stick takes on a rubbery quality, "flowing" along a parabola into the pit, as the rear compresses. (We will re-encounter such parabolic curving in the context of Supplee's paradox; it is generic to inertial motion with perpendicular acceleration.)
___ Max Born, he that struck at the epicentre of all vexation regarding quantum mechanics by linking the wavefunction to observational probability, set about defining what rigid motion might mean in relativity. He defined it as a lack of change with proper time the proper distances between the particles of a moving body: $ds/d\tau = 0$. Born rigidity demands a conspiracy of internal forces and can be trivially violated such as in Bell's spaceship paradox.
___ More illuminating is Ehrenfest's paradox, which upends the very notion of Born rigidity by considering a disk that is being sped up. For the particles of such a disk will measure non-uniform distances from their neighbours in their co-rotating frame whilst undergoing tangential acceleration. If due to unspecified forces they don't — in order to keep Born rigidity — then in the lab frame they Lorentz-contract and we have the circumference smaller than 2$\pi$ times the radius, a radius that doesn't contract as there is no motion along it. As the disk is in Euclidean space, this cannot be: Born rigidity must be violated. When the disk is spinning uniformly, however, one has non-Euclidean geometry in the co-rotating frame, as discussed in the section "What is special?"
Torque.
___ The Trouton-Noble paradox a.k.a. the right-angled lever paradox appears to produce torques out of nowhere. Take the L-shaped lever in the figure, moving to the right, with forces on the arms adjusted — in the lever frame — to keep the setup from spinning. In the lab frame not only is the horizontal bar shrunk, but also the vertical force is reduced, both effects seeming to put a spin on the lever. But oh, it is the accelerations you must consider, not the forces. Transform the accelerations correctly between frames, and the torque, hence the paradox, goes away.
Dragged light.
___ Hippolyte Fizeau was puzzled in 1851. With cunning interferometry he had just measured the speed of light through moving water. He had expected the speed to come out as the sum of the speeds of his water flow and of light in water. But it came out way less.
___ Annus Mirabilis was still 54 years away, so Fizeau couldn't have known that never should he have expected his desired outcome. For sufficiently rapid-moving water his expectation would have produced a superluminal signal. What he got instead was an empirical pre-verification of Einstein's velocity-addition formula.
Sink or swim?
___ Buoyancy is tricky in relativity, as evidenced in Supplee's paradox. Picture a submarine at rest just about floating with neutral buoyancy — the acceleration due to buoyancy is $g$. If it starts moving, Isaac on the shore would expect it to sink as its increased mass and lengthwise contraction make it denser, whereas Albert on the vessel would expect to float as to him it is the lake that gets denser by becoming heavier and shortened. Who is right?
___ The sailor would sink. Isaac notices two things: the force of buoyancy is reduced by $\gamma$ because the volume of water displaced is, and the acceleration due to buoyancy is taxed another factor of 1/$\gamma$ because Newton's second law now features the relativistic mass. Thus the upward acceleration of the vessel is smaller than the upward acceleration of the lake by 1/$\gamma^2$, and the submarine sinks. Albert notices that the acceleration due to buoyancy is unchanged from $g$, and that the acceleration due to gravity at one corner of the lake is smaller because he is in a non-inertial frame. Yet he finds himself sinking. That is because the lake also lives in a non-inertial frame. In particular, Albert sees the lake floor curve upward, a floor he will strike "horizontally".
___As an aside, this paradox helped me see why a person hired to (un)load ships at a dock calls himself a "longshoreman". He ain't the "shortseaman" he sights on moving vessels, where, as the poet noted,
_ Water, water everywhere,
_ And all the boards did shrink.