Exploring Space-Time: Is Einstein's Theory Overfitted?

[lukephysics]

TL;DR Summary

If you split space and time what would the equivalent functions look like?

I understand people collapse multi dimensional functions to make simpler visualisations, eg if you have a 500 dimension objective function in machine learning you can collapse it to 2D or 3D to get a visual idea of the objective-space.is this why Einstein did it as well? to make simpler visualisations or maths?

What would these same functions look like if you kept XYZ and time as separate dimensions, and just worked with the same concepts?

Also, I assume you could expand space time to N dimensions if you wanted, just by operating space-time on a hyperplane on that geometry. It wouldnt do anything, but just trying to point out that dimensions can be arbitrary, and understand why he combined space and time to spacetime.

I had one more q, would string theory be an example of expanding spacetime to say 11 dimensions arbitrarily in an effort to resolve singularities or divergences to whatever accuracy, on the spacetime hyperplane? and is this a form of over-fitting? you can explain anything if you add more dimensions.

 

[strangerep]

What would these same functions look like if you kept XYZ and time as separate dimensions, and just worked with the same concepts?

That's exactly what Einstein did, originally. One can work with the equations of Lorentz transformations quite successfully without ever drawing a Minkowskian spacetime diagram. However, people later found Minkowski diagrams to be helpful for visualizing what's going on.

 

[Dale]

TL;DR Summary: If you split space and time what would the equivalent functions look like?

What would these same functions look like if you kept XYZ and time as separate dimensions, and just worked with the same concepts?

The functions would all look the same as they do now. They would just be longer to write down. Spacetime doesn’t get rid of any dimensions or information. It just represents it nicely.

 

[FactChecker]

What would these same functions look like if you kept XYZ and time as separate dimensions, and just worked with the same concepts?

They are separate dimensions. They are inextricably related by the Lorentz transformations when two different inertial reference frames are considered.

 

[Ibix]

I understand people collapse multi dimensional functions to make simpler visualisations, eg if you have a 500 dimension objective function in machine learning you can collapse it to 2D or 3D to get a visual idea of the objective-space.

No, that's not what machine learning algorithms do. Some of them do find a lower-dimensional subspace of a high-dimensional feature space that still contains all of the data in order to process it more efficiently. This is not "collapsing" dimensions, just finding a clever representation. It has nothing to do with relativity's unification of space an time, which reduces nothing.

is this why Einstein did it as well?

He didn't reduce the number of dimensions. Relativity always uses four, the same three as Galilean relativity plus time. Sometimes you can find that all motion is in one spatial dimension and note that the behaviour is trivial in the other two, but this is a common feature of mathematical approaches across physics and is not specific to relativity.

What would these same functions look like if you kept XYZ and time as separate dimensions, and just worked with the same concepts?

They are separate dimensions, so all the maths is the same whether you treat them as forming spacetime or not. It just makes the interpretation messier. By the way, it was not Einstein who first presented relativity as a geometric theory but Minkowski, three years after Einstein's 1905 paper.

Also, I assume you could expand space time to N dimensions if you wanted, just by operating space-time on a hyperplane on that geometry. It wouldnt do anything, but just trying to point out that dimensions can be arbitrary, and understand why he combined space and time to spacetime.

You can add detailed dynamics for invisible non-interacting unicorns to the theory if you like, but since they are non-interacting and thus undetectable and change nothing, why would you do it? Dropping below four dimensions means you can't describe some real-world phenomena. Going above four dimensions is adding invisible unicorns.

I had one more q, would string theory be an example of expanding spacetime to say 11 dimensions arbitrarily in an effort to resolve singularities or divergences to whatever accuracy, on the spacetime hyperplane? and is this a form of over-fitting? you can explain anything if you add more dimensions.

String theory adds more dimensions in an attempt to explain particle physics and relativity together in one theory. It attempts to describe more phenomena than relativity alone so it needs more complexity.

 

[vanhees71]

That's exactly what Einstein did, originally. One can work with the equations of Lorentz transformations quite successfully without ever drawing a Minkowskian spacetime diagram. However, people later found Minkowski diagrams to be helpful for visualizing what's going on.

You can even more successfully working with equations rather than drawing complicated diagrams. Indeed, some people find Minkowski diagrams a nice way to visualize things. For me it's always much more difficult to draw the diagrams than to do some matrix-vector multiplications :-).

 

[lukephysics]

The functions would all look the same as they do now. They would just be longer to write down. Spacetime doesn’t get rid of any dimensions or information. It just represents it nicely.

So why even talk about spacetime when explaining relativity if its just convenience for equation writing (Lorentz transformations?)? What if I am not a space time engineer, and don't care about equations, I just want to understand concepts. Where would I find the non-spacetime explanation of relativity?

In other areas I find as soon as you resort to equations to explain something, you dont really understand something deeply. so spacetime shouldnt really come in to it for most people not working on problems.

 

[PeterDonis]

why even talk about spacetime when explaining relativity if its just convenience for equation writing (Lorentz transformations?)?

It isn't just convenience. Spacetime is the invariant. "Space" and "time" are not. If you insist on viewing everything in terms of "space" and "time", you have to pick a particular reference frame. But if you view things in terms of spacetime, you don't. You can express everything in terms of invariants and you don't have to worry about frames at all. Which is good since reference frames are just human conventions.

 

[PeterDonis]

In other areas I find as soon as you resort to equations to explain something, you dont really understand something deeply.

This is backwards. If you don't resort to equations, you don't really understand something deeply, at least not in physics. Physics is done in math. If you don't understand the math, you don't really understand the physics.

 

[PeterDonis]

I just want to understand concepts

Spacetime is the central concept of relativity. You can't understand relativity correctly without it.

Special Relativity pedagogy sometimes tries to ignore this, but as soon as you deal with either curvilinear coordinates (non-inertial frames) or gravity (curved spacetime instead of flat), you will be hopelessly lost if you don't understand spacetime.

 

[FactChecker]

So why even talk about spacetime when explaining relativity if its just convenience for equation writing (Lorentz transformations?)? What if I am not a space time engineer, and don't care about equations, I just want to understand concepts.

It's not just convenience, it is the central concept. The relationship between space and time is the entire thing. If you studied a pendulum by considering the height and velocity separately then you are missing the central concept of the tradeoff between height and speed. The same is true of SR if you look at space and time separately.

Where would I find the non-spacetime explanation of relativity?

If you find such a thing, IMHO you should throw it in the trash.

 

[Dale]

So why even talk about spacetime when explaining relativity if its just convenience for equation writing (Lorentz transformations?)?

Two main reasons:

1) it is convenient (which we like anyway) but more than that it unifies and clarifies concepts that otherwise seem separate and ad hoc.

2) it is necessary for general relativity

Both immediate convenience and clarity plus future necessity are good motivations. However, the point

I just want to understand concepts

Are you seriously trying to say you want to understand concepts while complaining about the spacetime concept?

Where would I find the non-spacetime explanation of relativity?

You can find it in the “modern physics” section of the Serway textbook. I used that for 7 years to “learn” relativity until I accidentally stumbled on the spacetime concept. Then I made more progress in the next 7 days with it than I had made in 7 years without it. I don’t recommend it.

In other areas I find as soon as you resort to equations to explain something, you dont really understand something deeply

I completely disagree with this. However, spacetime involves both equations and geometry. The geometry, in particular, is what is most important for General Relativity.

 

[Ibix]

So why even talk about spacetime when explaining relativity if its just convenience for equation writing (Lorentz transformations?)?

I would say the only way of understanding relativity without maths is spacetime. Although I'd note that I think you probably need the maths to understand it properly - even tools like Minkowski diagrams are hard to manipulate (and certainly totally unjustifiable) without maths.

What if I am not a space time engineer, and don't care about equations, I just want to understand concepts. [Snip] In other areas I find as soon as you resort to equations to explain something, you dont really understand something deeply.

I've certainly come across this attitude. It often comes with a rude awakening. In the machine learning space (which you mentioned) there are instances of people making grandiose claims for the superiority of their simple neural network over "traditional" statistical methods, only for someone to come along and point out that the simple network is a traditional statistical method when stripped of its go-faster stripes. Understanding the maths of tools you are using and their limitations is vital to understanding the tools properly.

 

[haushofer]

Where do I find a nice book on Russian language usage in Dostojevski's novels without emphasis on the Russian language or grammar? I'm not a student in Slavic languages.

 

[pervect]

So why even talk about spacetime when explaining relativity if its just convenience for equation writing (Lorentz transformations?)? What if I am not a space time engineer, and don't care about equations, I just want to understand concepts. Where would I find the non-spacetime explanation of relativity?

In other areas I find as soon as you resort to equations to explain something, you dont really understand something deeply. so spacetime shouldnt really come in to it for most people not working on problems.

Let me ask you a related question. Suppose you have a plane (the 2d surface sort of plane, not an airplane), and you have directions on the plane, which we will call north-south and east-west. What would happen if you regarded these directions as fundamentally different? Why do we regard east-west and north-south as being unified, rather than separate.

To give a hint at my answer, I'd point to "the parable of the surveyor", a somewhat long example. The version I am most familiar with is in a textbook, but there are various sources online. Google finds for instance http://spiff.rit.edu/classes/phys150/lectures/intro/parable.html which has a reference to the originating texts. I think that there might even be online versions of some of the texts, but I'd have to track them down. Which I'd be willing to do if there was interest and I didn't get yanked away, but I digress.

There are various levels of sophistication involved in thinking about this parable, but perhaps one of the most basic is to consider how one would handle rotation if one regarded north-south as different between east west. What sort of mysterious process converts north-south distances on a ruler to east-west distances if you rotate the ruler, since we now trying to think about them as being different? Is the idea of the existence of rotation a good and sufficient justification to regard the two sorts of distances as being the same?

To skip ahead even more, the space-time equivalent of rotation, which has very similar math with a few sign changes, is called a "boost". If the term is not familiar, a "boost" is just the process of changing one's velocity. So the analogue of rotational symmetry, that links north-south and east-west differences, is the "boost" symmetry of physics, that says that physics itself does not change if you change your state of motion.

To expound on this more, rotation can "turn" (pun intended) north-south distances into east-west distances. Similarly, in relativity, "boosts" can turn simultaneous events that are separated only by space, into non-simultaneous events that are both separated in space and occur at different times.

This feature of boosts (velocity changes) is called "The Relativity of Simultaneity", and has historically been hard for people to grasp from short posts. Actually, it seems to be hard even if they read any of the numerous, longer literature on the topic. I don't know if you are familiar with the idea of the relativity of simultaneity, but if you are not already familiar, I hope I've written enough of a teaser where you can see how it might be relevant to your question. FOr that matter, I hope you can see the relevance if you are already familiar with the concept - that it sheds some light on why the treatment of special relativity that unifies space and time is so powerful.

This is not the only way of thinking about the parable of the surveyor, it is at least intended to be one of the easiest and most accessible ways to think about the issue.

 

[lukephysics]

So I read the surveyor parable. It talked about a Spacetime interval which seems to just describe a scalar length equivalent to distance. And that space and time are convertible. Maybe I misunderstand that space time is not combining space and time it’s just defining a relationship between these dimensions. D = t / c

 

[FactChecker]

Two inertial reference frames moving relative to each other disagree on how clocks are synchronized in the direction of relative motion (aka the relativity of simultaneity). The amount of disagreement is a function of the spatial separation of clocks. So space and time are intimately linked.
With very natural assumptions, there are only two possible geometries of space and time that are consistent with relativity: A paper by Polash Pal, "Nothing but Relativity" shows that the assumption of relativity will only allow Galilean relativity (all inertial frames share a universal time) or Einseteinian relativity (constant speed of light in a vacuum). (See https://arxiv.org/abs/physics/0302045)
These geometries were studied before Einstein developed his theory of Special Relativity.

Regarding the use of mathematical equations in Special Relativity: While you should be aware that there are equations describing the spacetime geometry of Special Relativity, that does not mean you need to use them unless you want to calculate specific, accurate values. As an interested amateur, I practically never do any calculations. I only want to understand the concepts. Books like L. Epstein, Relativity Visualized do a good job with very little mathematics and practically no actual calculations.

 

[Dale]

seems to just describe a scalar length equivalent to distance

The existence of such a scalar implies that it is useful to consider a combined spacetime.

 

[Janus]

Let me ask you a related question. Suppose you have a plane (the 2d surface sort of plane, not an airplane), and you have directions on the plane, which we will call north-south and east-west. What would happen if you regarded these directions as fundamentally different? Why do we regard east-west and north-south as being unified, rather than separate.

To give a hint at my answer, I'd point to "the parable of the surveyor", a somewhat long example. The version I am most familiar with is in a textbook, but there are various sources online. Google finds for instance http://spiff.rit.edu/classes/phys150/lectures/intro/parable.html which has a reference to the originating texts. I think that there might even be online versions of some of the texts, but I'd have to track them down. Which I'd be willing to do if there was interest and I didn't get yanked away, but I digress.

There are various levels of sophistication involved in thinking about this parable, but perhaps one of the most basic is to consider how one would handle rotation if one regarded north-south as different between east west. What sort of mysterious process converts north-south distances on a ruler to east-west distances if you rotate the ruler, since we now trying to think about them as being different? Is the idea of the existence of rotation a good and sufficient justification to regard the two sorts of distances as being the same?

To skip ahead even more, the space-time equivalent of rotation, which has very similar math with a few sign changes, is called a "boost". If the term is not familiar, a "boost" is just the process of changing one's velocity. So the analogue of rotational symmetry, that links north-south and east-west differences, is the "boost" symmetry of physics, that says that physics itself does not change if you change your state of motion.

To expound on this more, rotation can "turn" (pun intended) north-south distances into east-west distances. Similarly, in relativity, "boosts" can turn simultaneous events that are separated only by space, into non-simultaneous events that are both separated in space and occur at different times.

This feature of boosts (velocity changes) is called "The Relativity of Simultaneity", and has historically been hard for people to grasp from short posts. Actually, it seems to be hard even if they read any of the numerous, longer literature on the topic. I don't know if you are familiar with the idea of the relativity of simultaneity, but if you are not already familiar, I hope I've written enough of a teaser where you can see how it might be relevant to your question. FOr that matter, I hope you can see the relevance if you are already familiar with the concept - that it sheds some light on why the treatment of special relativity that unifies space and time is so powerful.

This is not the only way of thinking about the parable of the surveyor, it is at least intended to be one of the easiest and most accessible ways to think about the issue.

The way I've tried to explain it it that before Relativity it was assumed that Time and Space behaved like North-South and East-West; Something everyone would agree on no matter what direction they themselves were facing. Relativity instead treats them like Forward-Backward and Left-Right; Something that depends on each individual's own orientation.

 

[pervect]

So I read the surveyor parable. It talked about a Spacetime interval which seems to just describe a scalar length equivalent to distance. And that space and time are convertible. Maybe I misunderstand that space time is not combining space and time it’s just defining a relationship between these dimensions. D = t / c

That's basically right. Modulo some sign and scaling issues (the later being due to the factor c, which is nowadays regarded as basically a conversion constant - look for instance at the SI definition of the meter, https://physics.nist.gov/cuu/Units/current.html), the space-time interval in an inertial frame is equal to the distance for events separated only in space, and equal to the time interval for events only separated by time. For events separated by both time and space, you need the complete formula.

This is very similar, except for the sign and scaling issues, to the pythagorean theorem. If we have two spatial dimensions on an Euclidean plane, which we will call x and y, then distance^2 = dx^2 + dy^2. The result distance does not change (i.e. is invariant) under rotations.

In the analogous case in special relativity, the invariant space-time interval is either dx^2 - (c dt)^2, or (c dt)^2 - dx^2, depending on one's choice of sign convention. And the resulting interval is invariant under boosts (changes in velocity), just as the distance was invariant under rotations. It's convenient to chose units that make c=1, called geometric units, in which case the analogy becomes even clearer - the space time interval then becomes dx^2 - dt^2, or possibly dt^2 - dx^2, depending on sign convention.

So the rationale for why we unify space and time in special relativity is very similar to the rational as to why we regard north-south distances the same as east-west distances. In the case of the plane, I'd point to rotational symmetry, in the case of space-time, I'd point to the boost symmetry, as being the motivation for the unification. It'd be possible to do surveying with different notions of distance for north-south and east-west difference as in the parable, but it'd be very clunky. And things get even clunkier when you have people with different definitions of "north".

This is not to say that space and time are exactly the same thing - but clearly in special relativity there is a very intimate relation between them. Distance intervals are not invariant under boosts in SR due to Lorentz contraction, and time intervals are not invariant either, due to "time dilation" - but the space time interval notably IS invariant.

 

[vanhees71]

I don't understand, what you want to say with this.

In Newtonian physics time is absolute. Mathematically Newtonian spacetime is most naturally described as a socalled fiber bundle, i.e., you consider a directed 1D real affine manifold for time with the usual "metric" and put at each point of this manifold a 3D Euclidean manifold for space.

This structure is related to the corresponding spacetime symmetry, i.e., the Galilei group, which maps one inertial frame of reference to another via
$$t'=t-t_0, \quad \vec{x}'=\hat{D} \vec{x}-\vec{v} t-\vec{x}_0,$$
where $t_0 \in \mathbb{R}$, $\vec{x}_0 \in \mathbb{R}^3$, and $\hat{D} \in \text{SO}(3)$.

It should be obvious that the most natural description of the physical laws in Newtonian physics is to work with Euclidean three-vectors/tensors and time as a real parameter, and indeed that's how we express both point-particle and continuum mechanics in mathematical terms without much questioning why, because is looks "pretty natural" and "intuitive" from our everyday experience.

In special-relativistic physics the natural description of the spacetime manifold is a pseudo-Euclidean affine manifold with signature (1,3) (or (3,1) if you prefer the east-coast convention of signs). Correspondingly you introduce four-vectors $\underline{x}=(x^{\mu})=(c t,\vec{x})$, and then the spacetime symmetry is described by the (proper orthochronous) Poincare group,
$$\underline{x}' =\hat{\Lambda} \underline{x}-\underline{a}$$
with $\hat{\Lambda} \in \mathrm{SO}(1,3)^{\uparrow}$ and $\underline{a} \in \mathbb{R}^4$.

It should be obvious that the most natural description of the physical laws in special-relativistic physics is a field theory expressed in terms of tensor fields. Point particles are strangers in SR, but also they can be treated manifestly covariant, at least as long as you consider massive particles. Then there's a natural invariant measure for time, the proper time, and everything can be expressed in a manifest covariant way by equations of motion of the form
$$m \mathrm{d}_{\tau}^2 \underline{x}=\underline{K}(\underline{x},\mathrm{d}_{\tau} \underline x)$$
such that
$$(\mathrm{d}_{\tau} \underline{x}) \cdot \underline{K}=0,$$
because by definition of proper time
$$(\mathrm{d}_{\tau} x) \cdot (\mathrm{d}_{\tau} x)=c^2=\text{const}.$$

 

[vela]

Maybe I misunderstand that space time is not combining space and time it’s just defining a relationship between these dimensions. D = t / c

When you say "combining space and time," what exactly are you thinking? It seems you may perhaps have had something different in mind than how the rest of us interpret that phrase.

 

[PeterDonis]

Maybe I misunderstand that space time is not combining space and time it’s just defining a relationship between these dimensions. D = t / c

It's both. It is recognizing that the physical invariant is spacetime, not space or time separately, and it is recognizing that the most natural relationship between the units used to measure space and time is units where $c = 1$.

 

[Mister T]

So I read the surveyor parable. It talked about a Spacetime interval which seems to just describe a scalar length equivalent to distance.

Note that this "scalar length" is calculated using both space and time coordinates. In other words, by combining them.

 

[lukephysics]

Ok so I think my confusion comes from the name. Spacetime is confused with for example kilowatthours. This is concatenating two dimensions and measuring the integral. Spacetime ain’t doing that it’s just a unit of measure. Why arent we calling it a 4D system with units called Einsteins? Would be less confusing. i guess its not 4D there is some sphereical or rotational geometry here. anyway. just annoying makes it hard to learn in 5 mins that i have to study it!

 

[Ibix]

Spacetime ain’t doing that it’s just a unit of measure.

No it's not. Space isn't a unit of measure - it's the "background" on which things happen. Spacetime is similar, but includes time as part of the structure rather than some unrelated parameter.

 

[Dale]

makes it hard to learn in 5 mins that i have to study it

Renaming spacetime isn’t going to fix that. These are fundamentally challenging concepts. You will not be able to do any challenging task in five minutes at a time. That isn’t specific to relativity.

In any case, however long you spend, you will learn relativity faster with spacetime than without it.

 

[Nugatory]

Ok so I think my confusion comes from the name. Spacetime is confused with for example kilowatthours.
This is concatenating two dimensions and measuring the integral. Spacetime ain’t doing that...

So far so good.

...it’s just a unit of measure.

Better to say that it is a continuous set of points such that there is a measurable distance between them. The mathematical terminology (invaluable because it avoids the sort of misunderstanding you describe above, which happens when we rely too much on imprecise natural language) would say something like "It is a manifold that is equipped with a metric".

Why arent we calling it a 4D system with units called Einsteins? Would be less confusing.

We already have perfectly good units called "meters" and "seconds", any distance between two points in spacetime can be measured in either unit, and the conversion between them is just arithmetic, as when I convert feet to inches by multiplying by 12. So there's no clear advantage to introducing a new unit - we still have to do the hard part, which is recognizing that one unit is all we need.

i guess its not 4D there is some sphereical or rotational geometry here. anyway.

No, it's four-dimensional, just not Euclidean. The history here is somewhat interesting - long before relativity was discovered, mathematicians had invented non-Euclidean geometry, basically by considering what would happen if one of Euclid's postulates were wrong and parallel lines could intersect. To their surprise they found that the resulting unrealistic geometry was internally consistent and made as much sense as "real" geometry, it just didn't seem to have anything to do with the real world. Because it had nothing to do with the real world it remained an academic curiosity for decades, until the physicists came up with General Relativity and suddenly this academic toy was a crucial part of our understanding of the universe.

just annoying makes it hard to learn in 5 mins that i have to study it!

I'll give you a different slant here: Relativity solved a problem that tormented some of the the smartest people in the world for almost a half-century (roughly 1863 to 1905) and then it took another generation of equally smart people almost another century to work out all the implications and hammer out the modern concept of spacetime. And now you can cover all of that ground in a semester or so of college... I'd call that deliriously good news.

 

[Mister T]

just annoying makes it hard to learn in 5 mins that i have to study it!

Then you will be often-annoyed. Most things worth learning take much much longer than 5 minutes.

 

[pervect]

Ok so I think my confusion comes from the name. Spacetime is confused with for example kilowatthours. This is concatenating two dimensions and measuring the integral. Spacetime ain’t doing that it’s just a unit of measure. Why arent we calling it a 4D system with units called Einsteins? Would be less confusing. i guess its not 4D there is some sphereical or rotational geometry here. anyway. just annoying makes it hard to learn in 5 mins that i have to study it!

We call the units that are the same for time and space geometric units. They can be (and are, see for instance MTW's "Gravitation" which, however, is a graduate level text) used for charge and mass as well. It's somewhat conventional to use a length scale for the base unit of geoemtric units. MTW, for example, uses centimeters as the base geometric unit for distance, time, mass, and charge.

Wiki's webpage on the topic is https://en.wikipedia.org/wiki/Geometrized_unit_system