Usually people emphasise that boosts are separate from rotations. however mathematically boosts are rotations rather than disaplacements or such.
I couldn’t read from the prose how the difference between lengths A and B get established.
I too have wondered about how solid the “no boosting to infinity” restrictions are. If one takes a unit de-sitter Space that should cover all the directions. From one point of intuition at first glance it would feel that all the directions should be “locally adjacent” ie one should be able to travel on all points of a sphere by sliding the angle the radius is off-set from the origin.
When drawn as a in a space-time diagram one gets an upwards opening parabola and a downwards opening parabola. But then it is puzzling on why they seem to be disconnected regions. I think I have something that speaks about the same are as your discussion 45 degree and 90 degree angles. The parabolas have asymptose at lightspeed ie 45 degree. However if one focuses on positive distance then the region from 45 to 135 doesn’t have anything, the elsewhere region where 90 degrees would lie is never visited. If one wants to include minus distances then there are going to be sidewyas parabola like strctures but the stark magnitude flip from positive to negative doesn’t seem like those would actually be in smooth contact. That is by having a constant radius and letting the angle be undetermined one gets the analog of the unit-sphere, the unit de-sitter space, and this does NOT include the “side-areas”. But it could be interesting if there was a smooth way to slide from the upregion to the downregion. Whether this is a cheating or allowed move is a bit beyond me.
I couldn’t read from the prose how the difference between lengths A and B get established.
I think I’ve corrected this; I failed to note that A is closer to the planet’s surface than B, which was more obvious in the original version where I had a picture. Or do you mean I don’t justify it in terms of the new abstraction I’ve established?
I too have wondered about how solid the “no boosting to infinity” restrictions are. If one takes a unit de-sitter Space that should cover all the directions. From one point of intuition at first glance it would feel that all the directions should be “locally adjacent” ie one should be able to travel on all points of a sphere by sliding the angle the radius is off-set from the origin.
I’m not 100% certain what you mean by “no boosting to infinity” here, so it’s hard to interpret the rest of the statement.
When drawn as a in a space-time diagram one gets an upwards opening parabola and a downwards opening parabola. But then it is puzzling on why they seem to be disconnected regions. I think I have something that speaks about the same are as your discussion 45 degree and 90 degree angles. The parabolas have asymptose at lightspeed ie 45 degree. However if one focuses on positive distance then the region from 45 to 135 doesn’t have anything, the elsewhere region where 90 degrees would lie is never visited. If one wants to include minus distances then there are going to be sidewyas parabola like strctures but the stark magnitude flip from positive to negative doesn’t seem like those would actually be in smooth contact. That is by having a constant radius and letting the angle be undetermined one gets the analog of the unit-sphere, the unit de-sitter space, and this does NOT include the “side-areas”. But it could be interesting if there was a smooth way to slide from the upregion to the downregion. Whether this is a cheating or allowed move is a bit beyond me.
I think we’re talking about the space “inside” a singularity; if I’m mistaken, let me know.
So the 45 and 90 degree explanation is built on an assumption of orthogonality between time and the three spacial dimensions that I’m not sure is entirely accurate. Consider a mass; now consider a point a meter away from that mass. I think the “natural” spacial dimensions here are: Distance (the line directly into/away from the mass), and then four cardinal directions relative to Distance. Is Distance actually meaningfully distinct from Time? If we take simultaneity seriously, a change in Distance is equivalent, in a specific sense, to a change in Time.
In your response to another post, you write “If you are thinking in euclidean terms it might seems that x,y,z,t can be relabeled into each other. However with relativity there is an “odd signature” going either (+---) or (-+++).”
So time is inverted, relative to the other dimensions. More, given that Time and Distance are in a specific sense equivalent, I’d suggest Time is not actually orthogonal to Distance, but parallel, and “pointed” in the opposite direction. So we have something like this:
<--------------------------------------->
+ Distance -
<--------------------------------------->
- Time +
So, if we are pointed in a positive direction in distance, we’re pointed in a negative direction in time; that is, moving away from an object is moving towards its history (because time is emitted, not kept). Considering an arrow pointed into a singularity, we expect it to have the following vector: (-+) - moving closer to the object, and also towards its future. Rotate it 90 degrees, so its orientation in distance becomes its orientation in time, and we get (+-). (Which okay, looks more like a 360 degree rotation, but I will insist it’s a 90 degree rotation for difficult-to-explain reasons.) That is, after the ninety degree motion, we’re now pointed away from the singularity, and also into its history.
So I think the apparently unconnected parabolas of the upregion and downregion are actually the same region.
Yes, the added bit helps clarify what the difference is supposed to be.
However the scenarios don’t exactly work like that and I am having trouble parsing out whether it is just an unconventional framing or is there a signficant deviance. Yes, if we are far away and the bars are horizontal then a laser pulse will cover them in the same time. However the lower bar is subject to a distortion in time. If we place an analog clock with hands next to the lower bar that clock will appear to tick less than 1 time unit by the time the pulse has moved to the tips of the bar (assuming that local observer would think that it was covered in exactly 1 time unit). I am not sure you are using light as ruler in a correct way.
The “boost to infinity” is just that energy requirements for high speeds seem to grow without bound. One way to measure change is one measure changing per change in another measure for example x per t. If a draw a curly line that has an U-turn in it I could get into trouble that at some point in the U turn the “next instantenous moment in coordinate time” could get a bit ambigious. Another concept could be the “next dot of the curve” and for curves that go relatively straight that next blob is likely to be in the “global forward” direction. So thinking in terms of max displacement per global forward tick vs maximum curl between adjacent blobs that the swiggle can have don’t neccesarily meet.
Sorry, the signature is of the squares. The fuller equation is x^2+y^2+z^2-t^2=s^2. Increasing spatial separation gets you more of the measure, increasing time separation gets you less of the measure. If you take it in a certain way the underlaying number go complex but because people are allergic to the imaginary numbers people stay on the square level where it is just positive and negative reals.
The minus sign doesn’t get you “anti-directionality”, (which would be parallel but opposite). You took my prompt in a somewhat consistent direction but I was misleading in making you head that way. Riemanninan rotations are weird and I can’t tell whether you are working with euclidean rotations, whether you have yet to incorporate the riemannian weirdness or you have another take on the weirdness. I don’t example know whether it makes sense to use 360 degrees when talking about boosts. If you do 36 10 degree rotations and get back to where you started. But you can take a small/moderate boost and you can keep doing it for a very long time without it “wrapping around”.
If you are working from an euclidean standpoint then it will surprise you that taking all the events that are equidistant from a central event do not form a sphere. Your current language suggests that you use terminology with that kind of assumtion. We can talk about delta-Vs ie changes in velocity but that doesn’t lend to rotation language. To the extent that we use a rotation understanding numbers don’t make sense.
There is a thing where black holes make time and spcae reverse roles. I don’t know whether you diverge from that or have a unconvetional grab of it. For my part I was talking flat space and the implication of big curvature would require more explanation anyway (but I don’t want to hear about until rotation amounts are clear)
If we place an analog clock with hands next to the lower bar that clock will appear to tick less than 1 time unit by the time the pulse has moved to the tips of the bar (assuming that local observer would think that it was covered in exactly 1 time unit). I am not sure you are using light as ruler in a correct way.
That’s what I mean by the observers measuring different times for light crossing the different points; the major thing is that A is longer than B, because I’m trying (possibly failing) to suggest that the difference in distance arises from the fact that an observer in A is measuring some distance that an observer in B would call time, instead.
The “boost to infinity” is just that energy requirements for high speeds seem to grow without bound. One way to measure change is one measure changing per change in another measure for example x per t. If a draw a curly line that has an U-turn in it I could get into trouble that at some point in the U turn the “next instantenous moment in coordinate time” could get a bit ambigious. Another concept could be the “next dot of the curve” and for curves that go relatively straight that next blob is likely to be in the “global forward” direction. So thinking in terms of max displacement per global forward tick vs maximum curl between adjacent blobs that the swiggle can have don’t neccesarily meet.
I think I see, yes. I think it’s correct to say that energy requirements grow without bound—but I also think that that is a Newtonian framing of the question, in which “velocity” is a property of matter, which has an inexplicable maximum value.
...but I’m failing utterly to come up with an explanation that I think conveys that velocity isn’t just a scalar property of matter. I think it’s all going to sound like nonsense. I’ll think about this and return to it.
Sorry, the signature is of the squares. The fuller equation is x^2+y^2+z^2-t^2=s^2. Increasing spatial separation gets you more of the measure, increasing time separation gets you less of the measure. If you take it in a certain way the underlaying number go complex but because people are allergic to the imaginary numbers people stay on the square level where it is just positive and negative reals.
If it helps—it probably won’t—I think time is distance projected onto (desuspending into?) an imaginary plane. Specifically I think time is probably a spiral, in which distance is the arc length, and my unified field theory is one of the two complex dimensions.
The minus sign doesn’t get you “anti-directionality”, (which would be parallel but opposite). You took my prompt in a somewhat consistent direction but I was misleading in making you head that way. Riemanninan rotations are weird and I can’t tell whether you are working with euclidean rotations, whether you have yet to incorporate the riemannian weirdness or you have another take on the weirdness. I don’t example know whether it makes sense to use 360 degrees when talking about boosts. If you do 36 10 degree rotations and get back to where you started. But you can take a small/moderate boost and you can keep doing it for a very long time without it “wrapping around”.
So this is where the “We need to consider rotation as hyperbolic” approach comes in. A ten degree rotation from one frame of reference is not a ten degree rotation from another frame of reference; frames of reference won’t agree on how far a given rotation is. From the perspective of an object inside space-time, I think you have to think of rotation as hyperbolic. When I think about these things, however, I’m usually thinking about them from a perspective I can only describe as outside all frames of reference.
That said, the space-time relationship I am attempting to describe is definitely not Euclidean.
I mention Matryoshka dolls; my view of time is something like nesting Matryoshka dolls. Consider the event horizon of a black hole, notice that it is the surface of a 3-sphere (two dimension making the obvious surface of a sphere, plus a distance dimension); now “unroll” the event horizon’s distance dimension along the dimension that is directional with the distance dimension of the event horizon. The other two dimensions remain closed—you have three dimensions, made up of one “linear” dimension and two closed dimensions. However, we didn’t actually unroll that third dimension, we more … mapped it onto a linear dimension, so it still has curvature (also it wasn’t precisely a closed dimension like the other two in the first place, it was more like the desuspension of a closed dimension). As we move away from the black hole, our mapping moves in a loop. Relative to our linear direction we have mapped onto, curvature is positive for part of the loop, and negative for part of the loop; this curvature is “impressed” upon space-time itself. So the “structure” of the black hole is, through this mapping, infinitely superimposing itself on spacetime as we move away. (But I think that maybe thinking of the black hole as the “real” structure, which is duplicated/imprinted outward from itself, is wrong; the black hole is this repeated structure in its entirety).
Time, then, is sometimes anti-directional, and sometimes directional, with respect to distance, as a function of distance. When it’s anti-directional, time is bent towards the object; with it’s directional, time is bent away from the object.
Or we could express all that as rotation-of-rotation; the rotation of space-time itself is rotating, as we move away from the black hole.
If any of that makes sense, which I kind of doubt.
If you are working from an euclidean standpoint then it will surprise you that taking all the events that are equidistant from a central event do not form a sphere. Your current language suggests that you use terminology with that kind of assumtion.
...huh. No, I don’t mean a Euclidean sphere. Or, above, a Euclidean 3-sphere, although maybe the surface of a 3-sphere remains at least approximately correct, I’d have to think about that.
We can talk about delta-Vs ie changes in velocity but that doesn’t lend to rotation language. To the extent that we use a rotation understanding numbers don’t make sense.
True, I think. Well, they can make sense, but it’d probably require the hyperbolic version of rotation in order to make sensible use of numbers.
There is a thing where black holes make time and spcae reverse roles. I don’t know whether you diverge from that or have a unconvetional grab of it.
Maybe unconventional? Strictly speaking it doesn’t matter if you start with two space dimensions and one time dimension, or three space dimensions, but I think singularities bend time into space, or space into time, “creating” the fourth dimension, which is why it doesn’t act like a properly independent fourth dimension in some respects; it’s just a superimposition. (Also all particles in this are expected to be singularities, or possibly quasi-singularities in the case of neutrinos)
Usually people emphasise that boosts are separate from rotations. however mathematically boosts are rotations rather than disaplacements or such.
I couldn’t read from the prose how the difference between lengths A and B get established.
I too have wondered about how solid the “no boosting to infinity” restrictions are. If one takes a unit de-sitter Space that should cover all the directions. From one point of intuition at first glance it would feel that all the directions should be “locally adjacent” ie one should be able to travel on all points of a sphere by sliding the angle the radius is off-set from the origin.
When drawn as a in a space-time diagram one gets an upwards opening parabola and a downwards opening parabola. But then it is puzzling on why they seem to be disconnected regions. I think I have something that speaks about the same are as your discussion 45 degree and 90 degree angles. The parabolas have asymptose at lightspeed ie 45 degree. However if one focuses on positive distance then the region from 45 to 135 doesn’t have anything, the elsewhere region where 90 degrees would lie is never visited. If one wants to include minus distances then there are going to be sidewyas parabola like strctures but the stark magnitude flip from positive to negative doesn’t seem like those would actually be in smooth contact. That is by having a constant radius and letting the angle be undetermined one gets the analog of the unit-sphere, the unit de-sitter space, and this does NOT include the “side-areas”. But it could be interesting if there was a smooth way to slide from the upregion to the downregion. Whether this is a cheating or allowed move is a bit beyond me.
I think I’ve corrected this; I failed to note that A is closer to the planet’s surface than B, which was more obvious in the original version where I had a picture. Or do you mean I don’t justify it in terms of the new abstraction I’ve established?
I’m not 100% certain what you mean by “no boosting to infinity” here, so it’s hard to interpret the rest of the statement.
I think we’re talking about the space “inside” a singularity; if I’m mistaken, let me know.
So the 45 and 90 degree explanation is built on an assumption of orthogonality between time and the three spacial dimensions that I’m not sure is entirely accurate. Consider a mass; now consider a point a meter away from that mass. I think the “natural” spacial dimensions here are: Distance (the line directly into/away from the mass), and then four cardinal directions relative to Distance. Is Distance actually meaningfully distinct from Time? If we take simultaneity seriously, a change in Distance is equivalent, in a specific sense, to a change in Time.
In your response to another post, you write “If you are thinking in euclidean terms it might seems that x,y,z,t can be relabeled into each other. However with relativity there is an “odd signature” going either (+---) or (-+++).”
So time is inverted, relative to the other dimensions. More, given that Time and Distance are in a specific sense equivalent, I’d suggest Time is not actually orthogonal to Distance, but parallel, and “pointed” in the opposite direction. So we have something like this:
<--------------------------------------->
+ Distance -
<--------------------------------------->
- Time +
So, if we are pointed in a positive direction in distance, we’re pointed in a negative direction in time; that is, moving away from an object is moving towards its history (because time is emitted, not kept). Considering an arrow pointed into a singularity, we expect it to have the following vector: (-+) - moving closer to the object, and also towards its future. Rotate it 90 degrees, so its orientation in distance becomes its orientation in time, and we get (+-). (Which okay, looks more like a 360 degree rotation, but I will insist it’s a 90 degree rotation for difficult-to-explain reasons.) That is, after the ninety degree motion, we’re now pointed away from the singularity, and also into its history.
So I think the apparently unconnected parabolas of the upregion and downregion are actually the same region.
Yes, the added bit helps clarify what the difference is supposed to be.
However the scenarios don’t exactly work like that and I am having trouble parsing out whether it is just an unconventional framing or is there a signficant deviance. Yes, if we are far away and the bars are horizontal then a laser pulse will cover them in the same time. However the lower bar is subject to a distortion in time. If we place an analog clock with hands next to the lower bar that clock will appear to tick less than 1 time unit by the time the pulse has moved to the tips of the bar (assuming that local observer would think that it was covered in exactly 1 time unit). I am not sure you are using light as ruler in a correct way.
The “boost to infinity” is just that energy requirements for high speeds seem to grow without bound. One way to measure change is one measure changing per change in another measure for example x per t. If a draw a curly line that has an U-turn in it I could get into trouble that at some point in the U turn the “next instantenous moment in coordinate time” could get a bit ambigious. Another concept could be the “next dot of the curve” and for curves that go relatively straight that next blob is likely to be in the “global forward” direction. So thinking in terms of max displacement per global forward tick vs maximum curl between adjacent blobs that the swiggle can have don’t neccesarily meet.
Sorry, the signature is of the squares. The fuller equation is x^2+y^2+z^2-t^2=s^2. Increasing spatial separation gets you more of the measure, increasing time separation gets you less of the measure. If you take it in a certain way the underlaying number go complex but because people are allergic to the imaginary numbers people stay on the square level where it is just positive and negative reals.
The minus sign doesn’t get you “anti-directionality”, (which would be parallel but opposite). You took my prompt in a somewhat consistent direction but I was misleading in making you head that way. Riemanninan rotations are weird and I can’t tell whether you are working with euclidean rotations, whether you have yet to incorporate the riemannian weirdness or you have another take on the weirdness. I don’t example know whether it makes sense to use 360 degrees when talking about boosts. If you do 36 10 degree rotations and get back to where you started. But you can take a small/moderate boost and you can keep doing it for a very long time without it “wrapping around”.
If you are working from an euclidean standpoint then it will surprise you that taking all the events that are equidistant from a central event do not form a sphere. Your current language suggests that you use terminology with that kind of assumtion. We can talk about delta-Vs ie changes in velocity but that doesn’t lend to rotation language. To the extent that we use a rotation understanding numbers don’t make sense.
There is a thing where black holes make time and spcae reverse roles. I don’t know whether you diverge from that or have a unconvetional grab of it. For my part I was talking flat space and the implication of big curvature would require more explanation anyway (but I don’t want to hear about until rotation amounts are clear)
That’s what I mean by the observers measuring different times for light crossing the different points; the major thing is that A is longer than B, because I’m trying (possibly failing) to suggest that the difference in distance arises from the fact that an observer in A is measuring some distance that an observer in B would call time, instead.
I think I see, yes. I think it’s correct to say that energy requirements grow without bound—but I also think that that is a Newtonian framing of the question, in which “velocity” is a property of matter, which has an inexplicable maximum value.
...but I’m failing utterly to come up with an explanation that I think conveys that velocity isn’t just a scalar property of matter. I think it’s all going to sound like nonsense. I’ll think about this and return to it.
If it helps—it probably won’t—I think time is distance projected onto (desuspending into?) an imaginary plane. Specifically I think time is probably a spiral, in which distance is the arc length, and my unified field theory is one of the two complex dimensions.
So this is where the “We need to consider rotation as hyperbolic” approach comes in. A ten degree rotation from one frame of reference is not a ten degree rotation from another frame of reference; frames of reference won’t agree on how far a given rotation is. From the perspective of an object inside space-time, I think you have to think of rotation as hyperbolic. When I think about these things, however, I’m usually thinking about them from a perspective I can only describe as outside all frames of reference.
That said, the space-time relationship I am attempting to describe is definitely not Euclidean.
I mention Matryoshka dolls; my view of time is something like nesting Matryoshka dolls. Consider the event horizon of a black hole, notice that it is the surface of a 3-sphere (two dimension making the obvious surface of a sphere, plus a distance dimension); now “unroll” the event horizon’s distance dimension along the dimension that is directional with the distance dimension of the event horizon. The other two dimensions remain closed—you have three dimensions, made up of one “linear” dimension and two closed dimensions. However, we didn’t actually unroll that third dimension, we more … mapped it onto a linear dimension, so it still has curvature (also it wasn’t precisely a closed dimension like the other two in the first place, it was more like the desuspension of a closed dimension). As we move away from the black hole, our mapping moves in a loop. Relative to our linear direction we have mapped onto, curvature is positive for part of the loop, and negative for part of the loop; this curvature is “impressed” upon space-time itself. So the “structure” of the black hole is, through this mapping, infinitely superimposing itself on spacetime as we move away. (But I think that maybe thinking of the black hole as the “real” structure, which is duplicated/imprinted outward from itself, is wrong; the black hole is this repeated structure in its entirety).
Time, then, is sometimes anti-directional, and sometimes directional, with respect to distance, as a function of distance. When it’s anti-directional, time is bent towards the object; with it’s directional, time is bent away from the object.
Or we could express all that as rotation-of-rotation; the rotation of space-time itself is rotating, as we move away from the black hole.
If any of that makes sense, which I kind of doubt.
...huh. No, I don’t mean a Euclidean sphere. Or, above, a Euclidean 3-sphere, although maybe the surface of a 3-sphere remains at least approximately correct, I’d have to think about that.
True, I think. Well, they can make sense, but it’d probably require the hyperbolic version of rotation in order to make sensible use of numbers.
Maybe unconventional? Strictly speaking it doesn’t matter if you start with two space dimensions and one time dimension, or three space dimensions, but I think singularities bend time into space, or space into time, “creating” the fourth dimension, which is why it doesn’t act like a properly independent fourth dimension in some respects; it’s just a superimposition. (Also all particles in this are expected to be singularities, or possibly quasi-singularities in the case of neutrinos)