Is the Universe Precisely Flat and Infinite?

[BurtJordaan]

In your very first post here, to the question 'Is the Universe Precisely Flat and Infinite?', you said 'The short answer to this question is, we do not know'.

Have you revised that? Sorry, I'm not trying to catch you out.

That was the short answer. I have since then also said many times, 'flat and infinite' is the most probable according to current observations, but since there are observational uncertainty of order 0.5% on spatial curvature, we don't know that for sure. Couple that to the fact that we use the 'flat equations' because they are simpler. This constitutes my 'long answer'.

[BurtJordaan]

Left: The observable universe. Right: With three dimensions of space compacted into any circumference (i.e. 3-ring), a curved-space, radial-time model accommodates a cosmos much greater than what we can ever hope to observe.

None of this rules out flat depictions, but I seek models with explanatory power for things like limit c and dark energy.

All this would have been more palatable to the mainstream if you would replace your 'BB' in the center by a horizontal line, i.e. a spatially spread out BB, which has happened everywhere simultaneously. Your preoccupation with the circular coordinate, where Epstein diagrams come to their right, is preventing you from making progress, IMO.

Also IMO, you have a better chance to get to grips with dark energy if you do that, because then you can start considering the 'branes' of M-Theory. But that must be the topic of another thread.

[Faradave]

more palatable to the mainstream if you would replace your 'BB' in the center by a horizontal line, i.e. a spatially spread out BB, which has happened everywhere simultaneously.

That would indeed work for a flat model, as your infinite lattice avatar reminds us. But that model and the mainstream remain mystified by many universal phenomena. That they are universal suggests that they relate to the underlying 4D structure of the universe. Einstein got us started. I think there's more to be gained from such an approach.

Keep in mind that placing the Big Bang (BB) as the center of a curved-space, radial-time model is consistent with the BB happening "everywhere simultaneously" as all space is contained in the central point (the cosmic simultaneity at t=0).

Your preoccupation with the circular coordinate, where Epstein diagrams come to their right, is preventing you from making progress, IMO.

I think it will confuse readers if you keep equating my interval-time coordinates with Epstein's space, proper-time coordinates. They both claim the intuitive simplicity of Euclidean geometry but Epstein hopelessly obscures the nature of light transmission. Interval-time reveals it with unsurpassed clarity. Epstein would slap his forehead and scold himself for not having abandoned his adherence to space as something real.


Left: Space-time coordinates depict a lightlike interval with indefinite span, though defined as "zero interval separation". Right: Interval-time correctly depicts a lightlike interval as zero ("pinhole contact"), bypassing equal spans of space and time.

Intervals are preferable as coordinates because they are invariant, thus agreed by all observers, including readers of this post (whether they know it or not). Yet intervals are highly distorted in space-time coordinates. Interval-time coordinates correct this in a way entirely consistent with recognized interval equations (shown). As noted earlier, interval-time coordinates are naturally available at all locations in a curved-space, radial-time model.

That is not to say curved-space, radial-time denies spatial coordinates. To be sure, 3-spherical coordinates would identify 4D events using radial time and three spatial arcs (or central angles) as coordinates. But designation as a coordinate is not the same as being a dimension.

Curved-space, radial-time simplifies to only one kind of dimension: unidirectional time, emanating radially in 4D from the BB. All lightlike phenomena (including all forces) then naturally propagate at universal speed limit c, tangent to space (as shown previously).

[BurtJordaan]

Your preoccupation with the circular coordinate, where Epstein diagrams come to their right, is preventing you from making progress, IMO.

I think it will confuse readers if you keep equating my interval-time coordinates with Epstein's space, proper-time coordinates. They both claim the intuitive simplicity of Euclidean geometry but Epstein hopelessly obscures the nature of light transmission. Interval-time reveals it with unsurpassed clarity. Epstein would slap his forehead and scold himself for not having abandoned his adherence to space as something real.

I was referring to these diagrams that you have previously offered as justification for your radial time presentation.



They have a decidedly Epsteinian 'look and feel'. ;)

I agree that a "time-interval" diagram is useful in explaining what a spacetime interval is, but IMO this is where its utility stops.


Left: Space-time coordinates depict a lightlike interval with indefinite span, though defined as "zero interval separation". Right: Interval-time correctly depicts a lightlike interval as zero ("pinhole contact"), bypassing equal spans of space and time.

Spacetime diagrams (including Epstein's) have high utility in depicting spacetime events and spacetime world-lines of objects in relative motion, not even to mention the all-powerful light-cone. I fail to see how time-interval coordinates can show any of these things. E.g. how would you show a light ray being reflected back from an object (with the point of reflection as an event) and (say) the spacetime path of the "away-twin" in the old "twin paradox"?

Maybe you can enlighten us.

[BurtJordaan]

more palatable to the mainstream if you would replace your 'BB' in the center by a horizontal line, i.e. a spatially spread out BB, which has happened everywhere simultaneously.

The more serious critique is your use of radial time instead of the radial cosmic scale factor. I can see why one can be tricked into thinking that would work: the scale factor and cosmic time (normalized to t/t_now is not too far from a straight line, especially around "now" i.e. a(t) = T/T(now) = 1

Cosmic scale factor vs. Normalized cosmic time T/T(now)

From Lightcone7

It is clear that it the relationship breaks down at early and late times.

[Faradave]

Appreciate the interesting questions. Busy time of year. My replies can sometimes take a day or two.

Am I the only one noticing a wonky color palette (shows just one row) and image upload screen (shows just part of a line) in the full edit screen? I've been using work arounds but they're time consuming.

[Faradave]

Sorry for the delay.

I was referring to these diagrams … for your radial time presentation. They have a decidedly Epsteinian 'look and feel'. ;) … Spacetime diagrams (including Epstein's) have high utility in depicting spacetime events and spacetime world-lines of objects in relative motion

That 'look and feel' is intuitive Euclidean geometry available to both Epstein diagrams and my interval-time diagrams. Epstein achieves this by rotating one coordinate frame relative to another. Interval-time coordinates don't require this because all frames have invariant interval coordinates in common, so one simply compares local worldlines.

Interval-time coordinates derive naturally from curved-space, radial-time (below) at any event (p). The rest frame of the cosmos is "special" as it is the oldest accessible reference frame and also exhibits perfect 3-spherical symmetry (equal recessional red shift in all directions).

From Left: A temporal field radiating outward in 4D from the Big Bang (BB) similar to electric charge. A 2D slice at rest with respect to the cosmos reveals interval-time coordinates corresponding to rest (v₀) and natural speed limit c (v_max) enforced by unidirectional time. A rest frame in relative motion (v₁) sees the cosmos shorter in the direction of motion. In moving "frame c" light emission & future absorption are a single interval contact event as simultaneity (t₂) collapses flatly on radial time.

A resulting finite, universal, constant and invariant speed limit is itself more than enough to justify a curved-space, radial-time model. c is the tangent limit because greater inclination violates unidirectional time. Interval coordinates correspond to this. There, c is an absolute speed limit because contact is an absolute proximity limit. Similarly, c is an absolute speed limit because nothing is slower than absolute rest (i.e. interval rest: ∆d/∆t = 0). Unlike spacetime, interval-time maintains "frame c", even as space and time collapse. That's utility! Explaining c in spacetime is futility.

[altered] I fail to see how [interval-time] coordinates can show… the all-powerful light-cone.

That's because for any event (p) in curved-space, radial-time there are an infinite number of incoming light cones and just one outgoing "light disk". This is best depicted from the perspective of a real (i.e. subluminal) observer, here conveniently in the rest frame of the cosmos. For incoming light cones, consider a paper cone resting tangent to a small balloon at its circle of contact. The cone keeps this relation, widening for expanded balloons. Those are like incoming light cones for a fixed vertex event p.

Like incoming light, outgoing light occurs tangent to the simultaneity of the emitter. From any event p, this describes a tangent disk.

Left: Incoming light cones (shown in cross section) widen with concentric, expanded, past cosmic simultaneities (3-spheres enclosing the Big Bang). Tracing their emission events yields a valentine shape. Right: Outgoing light occurs on a "light disk" tangent to the spatial simultaneity of the emitter (p) and aiming futureward but independent of aging (i.e. the emitter's timeline).

…how would you show a light ray being reflected back from an object (with the point of reflection as an event)…

As you know, that varies with reference frame. Since light transmission occurs by interval contact, emission is depicted with a mere dot p in frame c . Reflection is a future dot q. In both cases, future space (in the direction of transmission) is collapsed onto the emitter's timeline.

"…time [aging] would stop if you could travel at precisely the speed of light. You would abolish distance entirely, so that your point of departure and your destination seem to be at the same place." Calder p.96

In the cosmic rest frame, light maintains its direction tangent to the simultaneity of the emitter, but the path now appears to have length. This is consistent with the defining notion that a wormhole (here, a "pinhole") has different internal and external spans. To be consistently non-aging the light paths from emitter and reflector are each normal to their respective timelines.

Left: In frame c, incident light originates and terminates at the same event p despite different spacetime coordinates. Reflected light does the same from a future event q. Right: In the cosmic rest frame, light transmits without aging from event p and incident upon spatially-remote future event q. From there, light is reflected tangent to space at the reflector.

how would you show … [the worldline of] the "away-twin" in the old "twin paradox"?

The same as light in the cosmic frame above but with subluminal worldlines less inclined away from radial time. With interval-time coordinates, mutual age contraction is given by the simple Euclidean relation t' = tcosθ. Velocity as a fraction of c is given by v = csinθ, the degree to which a worldline is lightlike. This relates to the LFTs by the identity: cos² = 1 - sin², so cosθ = √(1-v²).

Inertial clocks (and aging) run mutually slow as depicted with interval-time coordinates. t' is the age each sees of the other at their own age t. In the rest frame of the "moving" object, its timeline would be vertical while the other tilts away at the same angle but opposite direction. Neither age is locked in relative to the other unless some force puts both clocks into the same rest frame.

A round trip combines two such paths. Let the trip be sufficiently local to neglect the curvature of space. Specify a forced instant path reversal at a point indicated on a vertical line. Now we know why F = ma! All force is lightlike, imparting a lightlike component to an object's worldline, thus changing its inclination. This is always a non-aging component in the rest frame of the object experiencing the force.

A round trip (green) of a massive object requires a lightlike force to accelerate its return. This locks in a non-aging component compared to a relatively stationary object (i.e. one not experiencing force) which ages t. The accelerated object ages t'.(Implied forces f to start and end travel have been neglected.)

Nothing ages faster than the cosmos, which is always at rest with respect to itself. All forces act within.

[Faradave]

An Einstein-Rosen bridge induced by extreme gravity in GR. Is there another way?

Most have seen some version of this wormhole diagram used in both science and fiction. Someone draws a long line, then bends the page to establish a shortcut between the line's ends. But why should space bend? Since gravity warps spacetime in General Relativity, some theorize a black hole doing this. I find Special Relativity sufficient in a curved-space, radial-time model.

Acceleration can be viewed as a continuous incremental sequence of velocities. In SR, pathlength decreases as a function of increasing velocity. The universe contracts in the direction of motion. This involves not just the current simultaneity (space now) but future space. In the limit as speed approaches c, tangent to the current simultaneity, the original pathlength shrinks to zero. That's a wormhole!

A non-aging, tangent light path from p sees real contact (zero interval separation) with future simultaneities.

The problem for real observers is rest mass. By definition, even the tiniest mass retains a rest component to its worldline, along time. Thus, all particles age to some degree at any achievable speed. Said another way, mass can never achieve a non-aging worldline, perpendicular to time (and tangent to a curved space).

This is no restriction to energy alone. Thus, I interpret light as the transmission of an energy quantum via direct zero-interval contact. That is, via pinhole (particle-interaction wormhole).

Does speed somehow cause space to bend? No. Just as particles can coexist at a wide continuous range of speeds, so too all pathlengths coexist, including potential contact. This means every spatial location is in contact with every other spatial location (offset by an equal temporal span), a condition described as its spacetime light cone (a.k.a. "null cone"). The future cone corresponds exactly to a particle's field (in the Gaussian sense, following the inverse square law). Of course, contact with an empty location is meaningless.

Space isn't bent by speed, preexisting curvature is revealed by speed.
The least of that curvature is seen in the rest frame of the cosmos, for a curved-space, radial time model.

[BurtJordaan]

A resulting finite, universal, constant and invariant speed limit is itself more than enough to justify a curved-space, radial-time model.

My original complaint was not against a radial time model, just that it needs a non-linear scale on the time axis. However, the "interval-space" diagram on the right of your graphic seems to be just a rotation of the radial time spacetime diagram through 90 degrees.



I have no problem with this step either, except that one must realize that the v₁ and v₂ arrows now sit on the black vertical line. Still no problem, until you get to this step as answer to my question on the 'twin=paradox'.



This is exactly the Epstein space-propertime diagram, except that you have labeled the space axis: "interval (non-aging)" and the propertime axis: time (aging)". The green arrow is just the worldline of the green object in space-propertime, as Epstein defined it. Ditto for the round trip.

Can you see the rationale behind my misgivings about your interval-time diagrams? I can see confusion brewing.

[Faradave]

…all validated observational evidence on the large-scale spatial curvature sits so close to zero that the error bars straddle the zero. Hence, closed, flat and open are all on the cards.

With the best available measures ascribing a high degree of Euclidean flatness to space (e.g. Jorrie's infinite lattice), if the cosmos is curved, it is deemed too vast to reliably demonstrate.

The positive (i.e. "closed") curvature of earth's surface exhibits how interior angles of large triangles always sum to more than 180°. But relatively small triangles are insensitive to curvature, giving the impression of flat Euclidean geometry. - Wikipedia

Non-Euclidean surfaces undeniably exhibit distinct geometries (e.g. the fate of parallel lines, the value of pi and the sum of angles in a triangle). However, as a proponent of curved-space, radial-time (asserting spatial 3-spheres), I suggest a different explanation for observed flatness. It corrects the common misconception that light travels through space. No mass-energy ever travels through space, that would be instantaneous, clearly violating speed limit c!

Mass-energy can indeed change spatial coordinates but only accompanied by an obligatory change in time. That's why speed is defined as distance/time. Here's the problem. A triangle on earth's surface is drawn as if by walking along it. However, laser ranging between vertices will NOT do that. The light beams at best project off into space, tangent to the surface. Triangles constructed from those straight beams will always sum to 180°. That the beams incline so as to prevent a continuous triangle is of no consequence. Triangular stairs do the same thing and yet remain Euclidean.

Straight sides, even though inclined, form a Euclidean (180°) triangle.

In a curved-space, radial-time model, speed c has an interval path tangent to space. Thus, laser ranging between three widely separated satellites (away from strong gravitation – a different issue) will report geometry equally flat as for Jorrie's infinite lattice.

Temporally inclined triangle p(now)-q-r-p(future), constructed of straight interval sides reports a flat Euclidean geometry summing to 180°.

Extra Credit:

Triangles on negative curvature sum to less than 180°.

Negative curvature, exemplified by hyperbolic space, would have surface triangles summing to less than 180°. Such simultaneities would stack over time like Pringles potato chips. Thus, light departing from event p on one instance of space might then arrive simultaneously at another event q in the same space (as shown). Beyond that it would communicate with past spaces (at r)! Thus, negative spatial curvature should be ruled out.

Saddle-shaped hyperbolic space would allow light in some directions to be instantaneous or even retro causal!

[Faradave]

the "interval-space" diagram on the right of your graphic seems to be just a rotation of the radial time spacetime diagram through 90 degrees.

Rather the far right box envisions the flattening of the cosmos in the direction of motion taken to speed limit c, where future space makes interval contact with light emitter p. Relativity allows that increasing speed is like looking through increasingly powerful telescopes but instead of distant objects just appearing closer, they actually are closer. Speed c reveals contact with collapsed future spaces!

(And if you want to avoid confusion, try to get the name correct. Mine are "interval-time coordinates", simply replacing "space" in "spacetime" with "interval-". ;o)

Still no problem, until you get to this step as answer to my question on the 'twin=paradox'. This is exactly the Epstein space-propertime diagram, except that you have labeled the space axis: "interval (non-aging)" and the propertime axis: time (aging)".

What that means is Epstein got it wrong and I got it right! Neither he nor I take credit for Euclidean geometry. While Epstein deserves credit for intuiting a Euclidean relation, his pursuit is clumsy, making the same naïve mistake as Minkowski in adhering to nearly irrelevant spatial coordinates. Science progresses upon the mistakes of its predecessors.

Epstein completely fumbles photons and his coordinate rotations, though convenient, have no real basis. Minkowski correctly recognized the scissor-like collapse of space and time with increasing speed as his "rotations".

Left: Distortions always occur in mapping non-Euclidean relations on flatly. The yellow interval, like the south pole, actually has zero span. Center: Minkowski correctly recognized the collapse of space and time in the direction of motion. His frame is thus lost at speed c. Right: Interval-time coordinates correctly depict intervals, identifying light as contact, while deriving from a recognized interval formula.

The green arrow is just the worldline of the green object in space-propertime, as Epstein defined it. Ditto for the round trip.

Except that my diagram is not "space propertime". That would be wrong. Mine is actual Euclidean geometry, Epstein's is contrived. You're clinging to clever, well-intended garbage IMHO.

[BurtJordaan]

I suggest a different explanation for observed flatness. It corrects the common misconception that light travels through space. No mass-energy ever travels through space, that would be instantaneous, clearly violating speed limit c!

FD, I'm sorry to say, but your concept of doing/discussing science is bound to confuse the living daylights out of readers that are serious about learning modern science. I sincerely hope that they can 'smell the rat' here. I do not know of a scientist in the world that would agree with your statement.

Maybe you interpret what is being said/written differently and yes, science can be interpreted in different ways, but your insinuation that the scientific world has it wrong does not befit this section of the Forum.

The rest of your post does not provide anything to support your expressed views.

[BurtJordaan]

Except that my diagram is not "space propertime". That would be wrong. Mine is actual Euclidean geometry not Epstein's contrived one. You're clinging to clever, well-intended garbage IMHO.

Here is your applicable diagram:



Yes, sorry. It would have been a perfect Epstein diagram (minus the spatial axis) if you had the Euclidean trigonometry right, meaning had you shown the left-going dotted green arrow at 90 degrees to the solid green arrow! It looks suspiciously like "clever, well-disguised garbage"... ;-)

Your vertical axis is proportional to the proper time of the black and the green clocks respectively and then your horizontal axis must be proportional to the distance, as measured by the black clock's observer - by pure Euclidean trig. Please refrain from attacking well established science.

I seriously suggest that you continue to push your off-mainstream views on the Personal Theory side. Here it may attract sanctions.

[Faradave]

No mass-energy ever travels through space, that would be instantaneous, clearly violating speed limit c!

I do not know of a scientist in the world that would agree with your statement. …support your expressed views.

Spacelike trajectories are specifically prohibited by Relativity. Perhaps it would help to say light never travels purely through space. Allow me to explain for any that were confused. In a conventional spacetime diagram, space is horizontal, time is vertical, and any other path is "spacetime". Light's path is a characteristically inclined (typically 45°) spacetime interval. Thus, light never travels purely through space, nor purely through time. As speed c is invariant, this is agreed by all observers (including scientists), whether they know it or not!

Rather than traveling through space, light transits a characteristic spacetime interval.

Current presumptions about the overall flatness of space relate to "critical density" which ignores the possibility that space is inherently curved, as it would be in a curved-space, radial-time model. I read a proposal to place three satellites in space to check spatial curvature with light. I don't think it can work since light departs from space rather than adhering to the possible contours of any simultaneity.

Summing the angles of this straight edged triangle (180°) set on an expanding balloon can’t reveal the balloon's curvature. But a triangle drawn on the balloon's surface can (at >180°).

Nevertheless, I'm hopeful that the precession of an ultrasensitive gyroscope (e.g. those confirming gravitational frame dragging around earth) could, over a sufficient round trip, confirm underlying spatial curvature.

your applicable diagram…would have been a perfect…had you shown the left-going dotted green arrow at 90 degrees to the solid green arrow! It looks suspiciously like "clever, well-disguised garbage"

Then it must be perfect! I don't claim to be an artist, but the horizontal dotted black line makes a right angle with the vertical time coordinate (those I can draw). By side-angle-side (t, θ, t') both triangles with a dotted side are congruent, so the green angle in question must also be 90°. (I thought I drew it pretty close.)


The same angle was explicitly indicated (by the little blue squares) as 90°.

your horizontal axis must be proportional to the distance, as measured by the black clock's observer - by pure Euclidean trig.

Euclidean, yes! But please recall that spacetime is "pseudo-Euclidean" and my Euclidean interval-time coordinates put interval separation (not spatial "distance") on the horizontal. It does take some getting used to.

Please refrain from attacking well established science.

Unless there are further questions, I’ll take the hint.

[BurtJordaan]

your applicable diagram…would have been a perfect…had you shown the left-going dotted green arrow at 90 degrees to the solid green arrow! It looks suspiciously like "clever, well-disguised garbage"

....θ, t') both triangles with a dotted side are congruent, so the green angle in question must also be 90°.

Yup says Epstein (and I), but what is the length of the black dotted line? It is t sin(theta) = vt/c, which is the spatial distance that the green clock has traveled in time t, as observed in the black inertial frame. So it does not represent a relevant spacetime interval, but rather a pure spatial distance.

So rather label your horizontal axis "space", using a black line. Having "interval" there is confusing in the extreme!

You are welcome to continue the crusade in the Personal Theory section.

[charon]

Well, looky here... damn! The comments are quite fun too.

https://www.dailymail.co.uk/sciencetech ... ieved.html

[TheVat]

https://www.nature.com/articles/s41550- ... =affiliate

This is the abstract from Nature Astronomy.

So rather label your horizontal axis "space", using a black line. Having "interval" there is confusing in the extreme!

- Jorrie

Thank you!

[Positor]

So rather label your horizontal axis "space", using a black line. Having "interval" there is confusing in the extreme!

I think it would be helpful if Faradave could give a concise restatement of the difference between "space" and his "(invariant) interval".

[BurtJordaan]

A much better article on the findings is available on LiveScience.

Melchiorri pointed out that his team's interpretation isn't conclusive. According to the group's calculations, the Planck data point to a closed universe with a standard deviation of 3.5 sigma (a statistical measurement that means about 99.8% confidence that the result isn't due to random chance). That's well short of the 5 sigma standard physicists usually look for before calling an idea confirmed.

[Faradave]

When first prescribed glasses as a teen, I was highly distracted by the clarity they provided, preferring the focus and isolation of near vision. To this day, I use them primarily for safety and legality while driving. I say this because even in science, introducing simple clarity can at first seem strange and intrusive. Indeed, Einstein's Relativity met with substantial initial resistance.

your horizontal axis... Having "interval" there is confusing in the extreme!

Thank you!

As this is a legitimate concern, answerable with "mainstream science", I give readers the convenience of a reply here.

what is the length of the black dotted line?

I've indicated the right angles with small boxes and the dotted lines with d and d'. d = d'

[d] is t sin(theta)

I (and Epstein) agree that d = t sinθ, which obviously has units of time.

[d = t sinθ] = vt/c, which is the spatial distance that the green clock has traveled in time t, as observed in the black inertial frame.

Now that is confusing (to me). vt/c is still a time. vt is a spatial distance r but so what?
Velocity is indicated by slope (or angle) such that v/c = sinθ.
Thus, θ = 0° for rest. θ = 90° for speed limit c.
Who says d is not an interval?

So [d = t sinθ] …does not represent a relevant spacetime interval, but rather a pure spatial distance.

This is the heart of the problem! Ask yourself, in what units is interval given?
A foundation of Relativity is that space, time and interval are all merely separations.
Units don't matter!
They can be seconds, meters or natural units as convenient.

I think it would be helpful if Faradave could give a concise restatement of the difference between "space" and his "(invariant) interval".

Thanks! But it's not my invariant interval. An interval is any separation in our 4D continuum. It conventionally has both temporal and spatial components. Einstein considered interval separation to be the most "real" because it is invariant (agreed by all inertial observers). That's what makes interval a superb coordinate choice!

Interval-time coordinates are readily obtained from the mainstream interval equation: interval² = space² – time² rearranged to the Pythagorean form: space² = interval² + time². The Pythagorean relation applies exclusively to Euclidean geometry. Here, interval-time coordinates are clearly implied, corresponding to the sides of a right triangle.



Confirmation is found in the correct representation of a lightlike interval, which is defined as zero (yellow dot) in the limit as spatial and temporal separation become equal. Less distortion means a better match between the map and the region it represents. In this, interval-time coordinates are unsurpassed!

Like it or not, a corrective Euclidean lens is required for an undistorted view.

[charon]

https://www.dailymail.co.uk/sciencetech ... ieved.html

They've ignored it!

[BurtJordaan]

FD, discussing the units issue is superfluous, because as long as the units are consistent throughout the interval definition, it's fine. BTW, the original interval definition used x,y,z,ct for convenience, i.e. it had spatial units, but that's irrelevant.

I think it would be helpful if Faradave could give a concise restatement of the difference between "space" and his "(invariant) interval".

An interval is any separation in our 4D continuum. It conventionally has both temporal and spatial components. Einstein considered interval separation to be the most "real" because it is invariant (agreed by all inertial observers). That's what makes interval a superb coordinate choice!

I fail to spot any spacetime events indicated in either of your last two diagrams. I'm asking becasue the invariant spacetime interval is defined as the separation between two spacetime events, irrespective of from which inertial reference frame the events are observed. To make your depictions more understandable, please indicate the two events that you are referring to and in which inertial frames their coordinates are measured.

[charon]

Sorry, wrong link on previous post. Apologies, don't know how that happened. Here's the right one again.

I don't think it should be ignored. Looks suspicious :-)

https://www.dailymail.co.uk/sciencetech ... ieved.html

Incidentally, the video on that site is awful. Here's a really good one. You'll need Adobe Flash.

http://htwins.net/scale2/

[Positor]

Interval-time coordinates are readily obtained from the mainstream interval equation: interval² = space² – time² rearranged to the Pythagorean form: space² = interval² + time². The Pythagorean relation applies exclusively to Euclidean geometry. Here, interval-time coordinates are clearly implied, corresponding to the sides of a right triangle.



Confirmation is found in the correct representation of a lightlike interval, which is defined as zero (yellow dot) in the limit as spatial and temporal separation become equal.

Let me see if I have this right:

Your diagram above shows a lightlike interval, where x = t and d = 0.
For a timelike interval, d > 0, x > t, and x forms a hypotenuse.
Timelike intervals can be shown in invariant (4D) form along the horizontal (d) axis.
Spacelike (faster-than-light) intervals do not really exist, so there is no need for your diagram to show them.

[Faradave]

as long as the units are consistent throughout the interval definition, it's fine… the invariant spacetime interval is defined as the separation between two spacetime events, irrespective of from which inertial reference frame the events are observed.

Agreed! So, if d happens to have units of space, that doesn't make it "purely spatial". d is the interval separating the resting clock (black) and moving clock (green) at proper time t'. As d is determined by θ, it is also the degree to which a moving worldline is lightlike (horizontal, thus non-aging), so t’ < t.

d' = d is invariance. d' is horizontal in the rest frame of the moving clock.

I fail to spot any spacetime events indicated in either of your last two diagrams. …To make your depictions more understandable, please indicate the two events that you are referring to and in which inertial frames their coordinates are measured.

The above diagram, in the rest frame of the black (unprimed) clock, intervals correspond to one spatial direction (say x), ignoring the other two. At the origin both clock paths conveniently cross. Their interval separation is zero, corresponding to classical contact and both clocks are given arbitrary x,t coordinates (0,0). While at rest, the x coordinate of the black clock is unchanged, giving us the red dot event at (0,t') and purple dot event above at (0,t).

d leads us horizontally to the orange dot event on the moving clock's worldline (green), which must also have the t' coordinate in the black rest frame shown. It is the lightlike view of the rest clock as seen by the moving clock at its own proper time t. As v is subluminal, the timelike interval formula: d² = t'² – x² gives its spatial coordinate. Thus, the orange dot event has coordinates (√(t'² – d²), t').

The events in interval-time coordinates: origin (0,0); red dot (0,t'); purple dot (0,t); orange dot (d,t').

Hope that's enough. (I'm running out of time and colors ;o)

[Faradave]

Your diagram above shows a lightlike interval, where x = t and d = 0.

Yes. Lightlike intervals form a boarder between timelike intervals (where Δt > Δx) and spacelike intervals (where Δx > Δt).

Our continuum is divided into regions as seen in spacetime (left) and interval-time (right).

Spacelike (faster-than-light) intervals do not really exist, so there is no need for your diagram to show them.

I wouldn’t say they don't exist. It's spacelike velocities which are expressly forbidden by Relativity. However, each type of interval has its uses. Timelike intervals correspond to translation (of mass-energy), lightlike intervals do communication (of energy with its information) and spacelike intervals can represent non-traversable entanglement connections which may serve as a common correlation reference.

A drawn spatial coordinate is spacelike: all locations at a given time (designated t = 0), also referred to as a "spatial simultaneity" or "spatial foliation". "Relativity of simultaneity", can be represented by spatial coordinates at various angles to each other. Each is spacelike in its own rest frame.

[Faradave]

Sorry, I had to step away.

there is no need for your diagram to show [space]

In the specific diagram you refer to, it's important to show space collapsed onto time as it does so in both spacetime and interval-time coordinates at speed limit c. The reason is that speed is a ratio of spatial and temporal separations ∆t/∆x. At limit c, in natural units, that ratio is exactly 1 and the corresponding interval span is zero, as you acknowledge. As far as I know, only interval-time coordinates show this as it is, interval contact, with space and time intrinsically equal.


Interval contact satisfies the definition of a lightlike interval (∆d = 0) whenever ∆t = ∆x regardless of their spans. Light (and fields) appear to bypass space and time to achieve remote contact, explaining action-at-a-distance.

In 4D a more general definition of contact is required than what Newton used. All contact is interval contact (i.e. ∆d = 0). Newton considered only the minor case, where ∆t = ∆x = 0. The interval formulas now require us to extend that all other cases where ∆t = ∆x ≠ 0. With Euclidean clarity, light appears to bypass both space and time in a single emission-absorption event (though it has two sets of coordinates in spacetime).

[charon]

...

Everything we know about the universe may be WRONG after new evidence suggests it is curved and not flat as previously believed.

For years scientists have believed our universe was as flat as a piece of paper, but new evidence has suggested it is curved like a giant inflated balloon.

A recent study analyzed data from the cosmic microwave background, the faint echo of the Big Bang, and discovered gravity seems to bend the microwaves.

These findings point towards a closed universe – the idea that if you travel far enough into space, you will loop back around to where you started.

In the paper, published in Nature Astronomy

https://phys.org/news/2019-11-planck-sp ... phere.html

researchers noted the 2018 Plant Legacy release that confirmed the presence of 'gravitational lensing' in cosmic microwave background (CMB), which suggests its microwaves bend.

The CMB is the oldest thing found in the universe and it is made up of ambient microwave light.

'A closed Universe can provide a physical explanation for this effect, with the Planck cosmic microwave background spectra now preferring a positive curvature at more than the 99% confidence level,' reads the study.

These findings contradict years of 'conventional wisdom and other studies based on the same CMB data set', Live Science reported.

https://www.livescience.com/universe-may-be-curved.html

And the theory of a flat universe may actually be 'masking a cosmological crisis where disparate observed properties of the Universe appear to be mutually inconsistent', the authors write.

Sapienza University of Rome cosmologist Alessandro Melchiorri, who is involved in the recent study, explained to Live Science that the closed-universe model would raise a range of problems for the field of physics.

'I don't want to say that I believe in a closed universe,' he told Live Science. 'I'm a little bit more neutral. I'd say, let's wait on the data and what the new data will say. What I believe is that there's a discrepancy now, that we have to be careful and try to find what is producing this discrepancy.'

Although the 2018 Plant Legacy release is confirming the closed universe with 99.8 percent accuracy, the researchers still noted that 'future measurements are needed to clarify whether the observed discordances are due to undetected systematics, or to new physics or simply are a statistical fluctuation'.

These findings come just a month after new calculations had suggested that the universe could be a couple billion years younger than scientists now estimate, and even younger than suggested by two other calculations published this year that trimmed hundreds of millions of years from the age of the cosmos.

The huge swings in scientists' estimates - even this new calculation could be off by billions of years - reflect different approaches to the tricky problem of figuring the universe's real age.

The generally accepted age of the universe is 13.7 billion years, based on a Hubble Constant of 70.

However, Inh Jee, of the Max Plank Institute in Germany, lead author of the study in Thursday's journal Science, and his team came up with a Hubble Constant of 82.4, which would put the age of the universe at around 11.4 billion years.

Jee used a concept called gravitational lensing - where gravity warps light and makes far away objects look closer.

https://www.dailymail.co.uk/sciencetech ... ieved.html

[Faradave]

Thanks charon!

That (and Jorrie's reply) hadn't escaped my notice. It's made the rounds in science news but you had it up front.

Though I prefer a closed curved-space, radial-time model, I don't rely on critical mass-energy density vs. dark energy as the current model does. In fact, I suspect mass-energy actually accelerates cosmic expansion (and may obviate dark energy) in contrast to conventional models.

The findings are interesting nonetheless and I look forward to additional scrutiny in this regard. There's still a lot to be learned!

[charon]

Faradave -

I won't pretend to understand it all but I think 99.8% accuracy, if true, needs some attention. Mind you, they cast a lot of doubt on it as well so I definitely don't understand it. However, better doubtful than cocksure.