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Question for astrophocists. Does frame dragging originate at the event horizon or at the singularity/ringularity? If it's the latter, does it mean that there's

Question for astrophocists. Does frame dragging originate at the event horizon or at the singularity/ringularity? If it's the latter, does it mean that there's a limit in size for black holes above which this phenomenon becomes negligible? It feels similar to the counter intuitive notion that tidal forces are more extreme when closer to a small black hole.

marcyb5st 3y ago
Thinking about this more, it should be at the singularity level since the event horizon doesn't have mass to cause it. Unless my basic understanding is black holes is wrong.
- marcosdumay 3y ago
  This is a confusing question. We shouldn't be able to see tides caused by changes on the mass distribution inside the horizon. But yes, the mass is all there. That "originates" word is funny.
  - marcyb5st 3y ago
    Sorry, not a native speaker and typing from my phone. My understanding is that frame dragging is not about tidal forces, but more like a maelstrom of spacetime falling within the singularity. On the second part I was referring to this "fun fact": https://www.astronomy.com/science/ask-astro-why-are-small-black-holes-more-dangerous-than-big-ones/ https://www.astronomy.com/science/ask-astro-why-are-small-bl...
    - raattgift 3y ago
      1/2 Let's start with the most major feature of General Relativity: Freely falling objects (around which one will build inertial frames of reference, with the object always at the spatial origin) are in geodesic motion, with the choice of all possible geodesics determined by the whole spacetime, and the particular geodesic determined by "initial" position and velocity. The Schwarzschild metric for a black hole generates a set of available geodesics such that there is a large set of innermost stable circular orbits (ISCOs) permitted by the spherical symmetry and the absence of any outside matter to break the time-symmetry ("eternal, never growing or shrinking"). There are many initial <position, velocity> pairs that leave a test particle on an ISCO forever, and the ISCOs will generally be found at 3R_s, where R_s is the Schwarzschild radius. If we break the spherical symmetry and make the system cylindrically symmetrical or axisymmetric, there are fewer ISCOs available: just as there are many great circles around a sphere, there are fewer around a cylinder. Rotating black holes generally do not have spherical symmetry, even if they are fully time-symmetric ("never grows nor shrinks, never changes its angular momentum"). Very roughly, frame-dragging captures the difference in 'equatorial' orbits between a spherically symmetric black hole ("one cannot distinguish an equatorial orbit from a polar orbit") and an equatorial orbit around a black hole with reduced symmetry (and between equatorial and nonequatorial orbits). A rotating black hole with an axis of rotation lets one distinguish between an orbit that goes above both poles and an equatorial orbit (and orbital planes in between those extremes). It also lets one distinguish between a prograde and retrograde orbit. This affects available ISCOs: for an initial <position, velocity> where the velocity is prograde, the ISCO will be closer to the rotating black hole than if the velocity were retrograde. Notably, as the black hole rotation is increased to a maximum, the prograde ISCO "just touches" the event horizon. On every ISCO, the local physics of a test free-faller is that of Special Relativity. This is essentially the content of the weak and Einstein equivalence principles, which you can get a feel for in the two sections starting at <https://en.wikipedia.org/wiki/Equivalence_principle#The_weak_equivalence_principle https://en.wikipedia.org/wiki/Equivalence_principle#The_weak...>. Free-fallers behave like astronauts in the ISS or other orbiting or Earth-Moon spacecraft. Things just float about weightlessly, but the speed of light, Planck's constant, and so forth are unchanged compared to a lab on the surface of the Earth. Now, ignoring the further breaking of symmetry by introducing "fixed stars at great distances" to the spacetime, let's have a pair of counter-rotating equatorial ISCO-freefallers use a system of local East-North-Up coordinates (see <https://en.wikipedia.org/wiki/Local_tangent_plane_coordinates https://en.wikipedia.org/wiki/Local_tangent_plane_coordinate...>) where we choose an equatorial plane at random where zero black hole rotation does not pick one out. "Up" is normal to the black hole's surface. North follows the right hand rule convention. The observers can counter-rotate on the same orbit because they are magically transparent to each other ("test objects", which are mathematical tools used explore the geodesics of a spacetime). Let's have the initial <position, velocity> of our counter-rotating observers differ only by whether they are travelling East or West in these coordinates, and with "Up" pointing directly to a bright distant star. On a high-radius circular equatorial orbit above minimally-rotating black hole, each "year" our two observers will meet and look Up to see the bright star immediately overhead. As we harden the orbit towards ISCO, there is a growing identical mismatch (the geodetic effect) between the locally maintained "Up" and the bright star that was initially "Up". The geodetic effect is not the subject of the paper press-released by northwestern, so I won't dwell on that here. If the black hole has significant rotation, as we harden the initially-soft orbit (with a long "year") there is a further but opposite "annual" mismatch between "Up" and the direction of the bright star. This is the Lense-Thirring effect. One can think of this in many ways, but it is related to prograde/retrograde geodesic motion. The <position, velocity> and <position, -velocity> starting points can only take one to a stable circular equatorial orbit. The other, with the opposite sign, must accelerate to be on the same circle but in the opposite direction. Or conversely, it will evolve into an orbit that is a combination of less-circular, less-equatorial, and different-height. We stop having the "annual" meetings below the distant star; each time the opposite-direction observers on approximately the same orbit meet, they will notice that the distant star is offset by an increasing angle. Or conversely, if they measure the year by being directly under the distant star, the prograde/eastbound observer will decide that the retrograde observer will arrive later and later every year, and the retrograde/westbound observer will decide that the prograde observer will arrive earlier and earlier every year. We might consider a simple accretion disc as a "dust" of many observers confined to the equatorial plane and circular orbits in the same direction. We have to make a perturbation like a dust-dust collision to accelerate a mote off its particular orbit and onto another one. This tends to cause an avalanche of interactions, so one then is inclined to treat the disc as a fluid. Additionally, one would draw some analogies between the orbital differences in the previous paragraphs and gyroscopic precessions or processions of electromagnetically-charged objects in a magnetic field. This is the root of "GRMHD", where the last three letters stand for magnetohydrodynamics, which is a technique used in the paper press-released by northwestern. In this approach one would not want to consider the local coordinates attached to each microscopic fluid element, but would work in some other coordinates suitable for studying the bulk fluid, such as spherical polar coordinates on the black hole itself. In those coordinates, the local "East-North-Up" of individual microscopic fluid elements along a <-ct,r,\theta=towards_our_bright_star,\phi=equator> will have to unalign, e.g., t_BH!=t_particle or \theta is at an angle to local Up/Down. The unaligning is stronger for fluid elements closer to the spinning black hole, and stronger as the black hole's spin increases. This unalignment is the 'general-relativistic "frame-dragging of the Kerr BH"' that the paper contemplates. We have to add something else: the fluid elements couple to each other non-gravitationally. The usual way of explaining this is to consider a figure skater who can extend and retract his arms, changing the velocity of his spin (while keeping angular momentum constant, since the latter is the former times the radius squared). If the figure skater were non-interacting dust, he would simply fly apart under rotation; it is the stickiness of the molecular forces in his body that keeps him intact, with his core preventing his fingertips from flying away by accelerating them back onto the core's preferred trajectory. In a far orbit around a spinning black hole, a freely falling spinning object that can extend and retract arms will notice nothing special compared to our figure skater: the arms' extremeties will be pulled towards the core by molecular forces and the like. The whole extended object will follow a free-falling trajectory. However, in a close orbit, the effects above manifest: the bits of the object closer to the black hole have to be accelerated more strongly back onto the freely-falling orbit of the centre of mass-momentum of the whole object compared to the bits further from the black hole. This relatively-higher drag on the closer-to-the-black-hole parts of the spinning generates a torque. If the orbiting object extends arms like a figure skater, but with a hand dipping closer to the black hole each rotation, the torque is increased. This is "paid for" by the whole spacetime, but for simplicity one usually points to the angular momentum of the black hole: the black hole's spin alters the spin of the (much much less massive) spinning orbiting object. In the accretion disc fluid, there are lots of small cells which are like the spinning object in the previous paragraph, and they bump into each other transferring angular momentum. The result is that inner regions of the disc experience a greater torque. The torques create differential misalignments of the black hole spin vector (in the spherical polar coordinates) and the spin vectors transported by each fluid element (in their local East-North-Up coordinates) that one an treat as an orbital precession. But an element initially travelling Eastbound in the equatorial plane will keep doing so no matter what sort of coordinates one puts on the bulk of all elements + the black hole, unless the element is acted on by some non-gravitational force. And fluid-element-to-fluid-element coupling does exactly that. The result is that the unalignment is forcibly realigned by bulk interactions, with the result that the disc warps in some combination of all four coordinates. The warp in \phi can be dramatic if the bulk is initially moving in a non-equatorial plane: the inner regions will be forced into the equatorial plane, leading to what the authors call a "torn disk". 1/2
    - raattgift 3y ago
      2/2 Finally, whether or not it's tidal forces comes down to an interpretation of how one decomposes the Riemann curvature tensor. In some ways, the Weyl tidal tensor part is good for understanding the disc's warping and tearing. However, if one decomposes the fluid into a zillion particles of dust, this approach seems (to me) to be less intuitive (and certainly harder to compute) compared to the GRMHD approach in the paper. Additionally, for these effects the black hole's spin parameter is more relevant than the black hole's mass, and notably the paper is concerned about quasars and other active galactic nuclei, which implies supermassive black holes, so the tidal tensor will in general be quite gentle even at the ISCO. For none of the above do we need to know anything about the microscopic states within the horizon. This is entirely about solutions to the geodesic equations and equations of motion in the spacetime outside the horizon, and matter-matter interactions of disc material (or "figure-skater-like spacecraft" material).
  - raattgift 3y ago
    > We shouldn't be able to see tides caused by changes on the mass distribution inside the horizon. ... if we hold the horizon fixed. If we don't let the horizons move at all we get all sorts of useful results like Hawking radiation via linear analysis. But if we let apparent horizons evolve dynamically taking into account anisotropic radiation reaction, the backreaction on the metric of the accretion of mass and angular momentum, and so forth, the full nonlinear theory becomes so complicated that I think it is fair to say that it is not in general well understood. We do know (empirically) that merged black holes ring down, which involves values sloshing about in the Weyl tidal tensor (in the 3+1 formalism). I am not certain treating the apparent horizon as a substance that settles down (emitting gravitational radiation) is conceptually useful, as opposed to the dynamical apparent horizon being generated by the movement of whatever's inside the merged binary plus most of the outgoing waves falling back towards the merged binary. The paper press-released at the top is not about merger and ringdown, though, so the internal configuration of the Kerr BH is irrelevant. Also, Lense-Thirring torques, the topic of the paper, arise even in non-BH Kerr spacetimes. We even see them in Earth orbit, and in due course probably will have good data from solar orbits too. >>> Does frame dragging originate at the event horizon > That "originates" word is funny I think it would be enlightening if you were to expand on that.
Enginerrrd 3y ago
I think this question is a bad one. You can ask it only for an in-falling observer. To extend that to someone who remains outside the event horizon, it becomes meaningless. There are many hints that there is something special going on at the event horizon itself.
- raattgift 3y ago
  > I think this question is a bad one. With respect, I disagree. A pair of comments I made elsewhere in this thread bear that out. > > Does frame dragging originate at the event horizon or at the singularity/ringularity? > You can ask it only for an in-falling observer Both the geodetic effect and the Lense-Thirring (LT) effect, forms of "frame dragging", have been demonstrated in Earth orbit, for example by Gravity Probe B <https://en.wikipedia.org/wiki/Gravity_Probe_B https://en.wikipedia.org/wiki/Gravity_Probe_B>. I think we can agree that there is neither an event horizon nor a singularity associated with Earth, and that since GP-B continues to orbit Earth after almost twenty years, it was not doing any sort of in-falling during its mission. The tl;dr answer to the question you think is a bad one is that Kerr exterior spacetimes -- even those that are not black holes -- induce LT, which is roughly an altitude-dependent mismatch between a vector pointing to something far away that is parallel-transported around the spinning central mass and around the same mass if it were not spinning. On extended objects occupying a range of altitudes above a spinning mass, an LT torque is induced. No black hole needed. The paper press-released in TFA the spinning object is a black hole, which lets one get much closer to the centre of mass (because a BH is compact compared to a planet) and because can spin much faster than a planet (which would tend to fly apart), which lets the LT effect be stronger. It also contemplates a large accretion disc in which LT torques can create pronounced observable effects. The singularity is not relevant to the analysis (it is not even mentioned in the paper). The horizon is barely relevant: towards the horizon they reduce the resolution of the adaptive mesh they calculate with and justify doing so in a couple of ways -- most of the interesting physics are at least 2R and mostly above 4R, where R := r_g which is the Kerr equivalent of the Schwarzschild radius r_s.

marcyb5st 3y ago

Thinking about this more, it should be at the singularity level since the event horizon doesn't have mass to cause it. Unless my basic understanding is black holes is wrong.

marcosdumay 3y ago
This is a confusing question. We shouldn't be able to see tides caused by changes on the mass distribution inside the horizon. But yes, the mass is all there. That "originates" word is funny.
- marcyb5st 3y ago
  Sorry, not a native speaker and typing from my phone. My understanding is that frame dragging is not about tidal forces, but more like a maelstrom of spacetime falling within the singularity. On the second part I was referring to this "fun fact": https://www.astronomy.com/science/ask-astro-why-are-small-black-holes-more-dangerous-than-big-ones/ https://www.astronomy.com/science/ask-astro-why-are-small-bl...
  - raattgift 3y ago
    1/2 Let's start with the most major feature of General Relativity: Freely falling objects (around which one will build inertial frames of reference, with the object always at the spatial origin) are in geodesic motion, with the choice of all possible geodesics determined by the whole spacetime, and the particular geodesic determined by "initial" position and velocity. The Schwarzschild metric for a black hole generates a set of available geodesics such that there is a large set of innermost stable circular orbits (ISCOs) permitted by the spherical symmetry and the absence of any outside matter to break the time-symmetry ("eternal, never growing or shrinking"). There are many initial <position, velocity> pairs that leave a test particle on an ISCO forever, and the ISCOs will generally be found at 3R_s, where R_s is the Schwarzschild radius. If we break the spherical symmetry and make the system cylindrically symmetrical or axisymmetric, there are fewer ISCOs available: just as there are many great circles around a sphere, there are fewer around a cylinder. Rotating black holes generally do not have spherical symmetry, even if they are fully time-symmetric ("never grows nor shrinks, never changes its angular momentum"). Very roughly, frame-dragging captures the difference in 'equatorial' orbits between a spherically symmetric black hole ("one cannot distinguish an equatorial orbit from a polar orbit") and an equatorial orbit around a black hole with reduced symmetry (and between equatorial and nonequatorial orbits). A rotating black hole with an axis of rotation lets one distinguish between an orbit that goes above both poles and an equatorial orbit (and orbital planes in between those extremes). It also lets one distinguish between a prograde and retrograde orbit. This affects available ISCOs: for an initial <position, velocity> where the velocity is prograde, the ISCO will be closer to the rotating black hole than if the velocity were retrograde. Notably, as the black hole rotation is increased to a maximum, the prograde ISCO "just touches" the event horizon. On every ISCO, the local physics of a test free-faller is that of Special Relativity. This is essentially the content of the weak and Einstein equivalence principles, which you can get a feel for in the two sections starting at <https://en.wikipedia.org/wiki/Equivalence_principle#The_weak_equivalence_principle https://en.wikipedia.org/wiki/Equivalence_principle#The_weak...>. Free-fallers behave like astronauts in the ISS or other orbiting or Earth-Moon spacecraft. Things just float about weightlessly, but the speed of light, Planck's constant, and so forth are unchanged compared to a lab on the surface of the Earth. Now, ignoring the further breaking of symmetry by introducing "fixed stars at great distances" to the spacetime, let's have a pair of counter-rotating equatorial ISCO-freefallers use a system of local East-North-Up coordinates (see <https://en.wikipedia.org/wiki/Local_tangent_plane_coordinates https://en.wikipedia.org/wiki/Local_tangent_plane_coordinate...>) where we choose an equatorial plane at random where zero black hole rotation does not pick one out. "Up" is normal to the black hole's surface. North follows the right hand rule convention. The observers can counter-rotate on the same orbit because they are magically transparent to each other ("test objects", which are mathematical tools used explore the geodesics of a spacetime). Let's have the initial <position, velocity> of our counter-rotating observers differ only by whether they are travelling East or West in these coordinates, and with "Up" pointing directly to a bright distant star. On a high-radius circular equatorial orbit above minimally-rotating black hole, each "year" our two observers will meet and look Up to see the bright star immediately overhead. As we harden the orbit towards ISCO, there is a growing identical mismatch (the geodetic effect) between the locally maintained "Up" and the bright star that was initially "Up". The geodetic effect is not the subject of the paper press-released by northwestern, so I won't dwell on that here. If the black hole has significant rotation, as we harden the initially-soft orbit (with a long "year") there is a further but opposite "annual" mismatch between "Up" and the direction of the bright star. This is the Lense-Thirring effect. One can think of this in many ways, but it is related to prograde/retrograde geodesic motion. The <position, velocity> and <position, -velocity> starting points can only take one to a stable circular equatorial orbit. The other, with the opposite sign, must accelerate to be on the same circle but in the opposite direction. Or conversely, it will evolve into an orbit that is a combination of less-circular, less-equatorial, and different-height. We stop having the "annual" meetings below the distant star; each time the opposite-direction observers on approximately the same orbit meet, they will notice that the distant star is offset by an increasing angle. Or conversely, if they measure the year by being directly under the distant star, the prograde/eastbound observer will decide that the retrograde observer will arrive later and later every year, and the retrograde/westbound observer will decide that the prograde observer will arrive earlier and earlier every year. We might consider a simple accretion disc as a "dust" of many observers confined to the equatorial plane and circular orbits in the same direction. We have to make a perturbation like a dust-dust collision to accelerate a mote off its particular orbit and onto another one. This tends to cause an avalanche of interactions, so one then is inclined to treat the disc as a fluid. Additionally, one would draw some analogies between the orbital differences in the previous paragraphs and gyroscopic precessions or processions of electromagnetically-charged objects in a magnetic field. This is the root of "GRMHD", where the last three letters stand for magnetohydrodynamics, which is a technique used in the paper press-released by northwestern. In this approach one would not want to consider the local coordinates attached to each microscopic fluid element, but would work in some other coordinates suitable for studying the bulk fluid, such as spherical polar coordinates on the black hole itself. In those coordinates, the local "East-North-Up" of individual microscopic fluid elements along a <-ct,r,\theta=towards_our_bright_star,\phi=equator> will have to unalign, e.g., t_BH!=t_particle or \theta is at an angle to local Up/Down. The unaligning is stronger for fluid elements closer to the spinning black hole, and stronger as the black hole's spin increases. This unalignment is the 'general-relativistic "frame-dragging of the Kerr BH"' that the paper contemplates. We have to add something else: the fluid elements couple to each other non-gravitationally. The usual way of explaining this is to consider a figure skater who can extend and retract his arms, changing the velocity of his spin (while keeping angular momentum constant, since the latter is the former times the radius squared). If the figure skater were non-interacting dust, he would simply fly apart under rotation; it is the stickiness of the molecular forces in his body that keeps him intact, with his core preventing his fingertips from flying away by accelerating them back onto the core's preferred trajectory. In a far orbit around a spinning black hole, a freely falling spinning object that can extend and retract arms will notice nothing special compared to our figure skater: the arms' extremeties will be pulled towards the core by molecular forces and the like. The whole extended object will follow a free-falling trajectory. However, in a close orbit, the effects above manifest: the bits of the object closer to the black hole have to be accelerated more strongly back onto the freely-falling orbit of the centre of mass-momentum of the whole object compared to the bits further from the black hole. This relatively-higher drag on the closer-to-the-black-hole parts of the spinning generates a torque. If the orbiting object extends arms like a figure skater, but with a hand dipping closer to the black hole each rotation, the torque is increased. This is "paid for" by the whole spacetime, but for simplicity one usually points to the angular momentum of the black hole: the black hole's spin alters the spin of the (much much less massive) spinning orbiting object. In the accretion disc fluid, there are lots of small cells which are like the spinning object in the previous paragraph, and they bump into each other transferring angular momentum. The result is that inner regions of the disc experience a greater torque. The torques create differential misalignments of the black hole spin vector (in the spherical polar coordinates) and the spin vectors transported by each fluid element (in their local East-North-Up coordinates) that one an treat as an orbital precession. But an element initially travelling Eastbound in the equatorial plane will keep doing so no matter what sort of coordinates one puts on the bulk of all elements + the black hole, unless the element is acted on by some non-gravitational force. And fluid-element-to-fluid-element coupling does exactly that. The result is that the unalignment is forcibly realigned by bulk interactions, with the result that the disc warps in some combination of all four coordinates. The warp in \phi can be dramatic if the bulk is initially moving in a non-equatorial plane: the inner regions will be forced into the equatorial plane, leading to what the authors call a "torn disk". 1/2
  - raattgift 3y ago
    2/2 Finally, whether or not it's tidal forces comes down to an interpretation of how one decomposes the Riemann curvature tensor. In some ways, the Weyl tidal tensor part is good for understanding the disc's warping and tearing. However, if one decomposes the fluid into a zillion particles of dust, this approach seems (to me) to be less intuitive (and certainly harder to compute) compared to the GRMHD approach in the paper. Additionally, for these effects the black hole's spin parameter is more relevant than the black hole's mass, and notably the paper is concerned about quasars and other active galactic nuclei, which implies supermassive black holes, so the tidal tensor will in general be quite gentle even at the ISCO. For none of the above do we need to know anything about the microscopic states within the horizon. This is entirely about solutions to the geodesic equations and equations of motion in the spacetime outside the horizon, and matter-matter interactions of disc material (or "figure-skater-like spacecraft" material).
- raattgift 3y ago
  > We shouldn't be able to see tides caused by changes on the mass distribution inside the horizon. ... if we hold the horizon fixed. If we don't let the horizons move at all we get all sorts of useful results like Hawking radiation via linear analysis. But if we let apparent horizons evolve dynamically taking into account anisotropic radiation reaction, the backreaction on the metric of the accretion of mass and angular momentum, and so forth, the full nonlinear theory becomes so complicated that I think it is fair to say that it is not in general well understood. We do know (empirically) that merged black holes ring down, which involves values sloshing about in the Weyl tidal tensor (in the 3+1 formalism). I am not certain treating the apparent horizon as a substance that settles down (emitting gravitational radiation) is conceptually useful, as opposed to the dynamical apparent horizon being generated by the movement of whatever's inside the merged binary plus most of the outgoing waves falling back towards the merged binary. The paper press-released at the top is not about merger and ringdown, though, so the internal configuration of the Kerr BH is irrelevant. Also, Lense-Thirring torques, the topic of the paper, arise even in non-BH Kerr spacetimes. We even see them in Earth orbit, and in due course probably will have good data from solar orbits too. >>> Does frame dragging originate at the event horizon > That "originates" word is funny I think it would be enlightening if you were to expand on that.

Enginerrrd 3y ago

I think this question is a bad one. You can ask it only for an in-falling observer. To extend that to someone who remains outside the event horizon, it becomes meaningless. There are many hints that there is something special going on at the event horizon itself.

raattgift 3y ago
> I think this question is a bad one. With respect, I disagree. A pair of comments I made elsewhere in this thread bear that out. > > Does frame dragging originate at the event horizon or at the singularity/ringularity? > You can ask it only for an in-falling observer Both the geodetic effect and the Lense-Thirring (LT) effect, forms of "frame dragging", have been demonstrated in Earth orbit, for example by Gravity Probe B <https://en.wikipedia.org/wiki/Gravity_Probe_B https://en.wikipedia.org/wiki/Gravity_Probe_B>. I think we can agree that there is neither an event horizon nor a singularity associated with Earth, and that since GP-B continues to orbit Earth after almost twenty years, it was not doing any sort of in-falling during its mission. The tl;dr answer to the question you think is a bad one is that Kerr exterior spacetimes -- even those that are not black holes -- induce LT, which is roughly an altitude-dependent mismatch between a vector pointing to something far away that is parallel-transported around the spinning central mass and around the same mass if it were not spinning. On extended objects occupying a range of altitudes above a spinning mass, an LT torque is induced. No black hole needed. The paper press-released in TFA the spinning object is a black hole, which lets one get much closer to the centre of mass (because a BH is compact compared to a planet) and because can spin much faster than a planet (which would tend to fly apart), which lets the LT effect be stronger. It also contemplates a large accretion disc in which LT torques can create pronounced observable effects. The singularity is not relevant to the analysis (it is not even mentioned in the paper). The horizon is barely relevant: towards the horizon they reduce the resolution of the adaptive mesh they calculate with and justify doing so in a couple of ways -- most of the interesting physics are at least 2R and mostly above 4R, where R := r_g which is the Kerr equivalent of the Schwarzschild radius r_s.