Register to reply

QG five principles: superpos. locality diff-inv. cross-sym. Lorentz-inv.

Share this thread:
marcus
#73
Aug22-10, 10:38 AM
Astronomy
Sci Advisor
PF Gold
marcus's Avatar
P: 23,227
Quote Quote by atyy View Post
...
I think the interesting comparison is between Ashtekar et al's Eq 3.10 of http://arxiv.org/abs/1001.5147 (same as Eq 3.20 of http://arxiv.org/abs/1001.5147) and Oriti's unnumbered final equation on p5 of http://arxiv.org/abs/gr-qc/0607032, which is the same as Freidel's Eq 11 in http://arxiv.org/abs/hep-th/0505016 .
...
Again thanks! I think there was a typo in the post. You may have meant:
I think the interesting comparison is between Ashtekar et al's Eq 3.10 of http://arxiv.org/abs/0909.4221 (same as Eq 3.20 of http://arxiv.org/abs/1001.5147)
And in that case you are of course right---same equation.
=================

My main focus needs to stay on Rovelli's April paper, but I will keep intermittently chewing on the two Ashtekar papers and trying to understand them better. Ashtekar has a different perspective and has been a formative and greatly influential QG figure over the long haul. I have to pay attention especially to his overview of the field. Differences in formal detail can work themselves out---I can probably get along with just Marseille notation. But I have to try to assimilate Ashtekar's vision. Both the papers you pointed to have introduction and conclusion overview sections that I'm finding helpful that way.
marcus
#74
Aug22-10, 11:22 AM
Astronomy
Sci Advisor
PF Gold
marcus's Avatar
P: 23,227
Quote Quote by atyy View Post
... GR is not geometrical. It only is geometrical if you measure spacetime with test partcles and ideal clocks ('observers'). However, neither of those exist in GR,...
We may have a slight semantic difference here. When I think of a theory of geometry, I don't expect of it a "theory of everything" that would explain how life might evolve and how conscious beings able to make measurements and construct clocks might arise from the various matter fields.

All I ask from a classical theory of geometry is that it give me what GR gives----geometries.
A geometry is an equivalence class of metrics (with attendant matter) under diffeomorphism.

So for me GR is the paradigm theory of geometry---it more or less defines for me what geometry is. Granted the theory does not provide its own observers, but it is observer-ready in a kind of "plug-and-play" sense.

By itself a metric (with attendant matter distribution) gives the geometric relations among all material "events" (such as particle collisions). And it determines the world-lines of all "particles".

Admittedly the concept of a "particle" is either a bit ad hoc or a bit fuzzy---we must indulge the theory in small ways, allow it a few marbles. It does not explain or predict the existence of marbles. Or some people prefer clouds of dust---then the grains of dust are the marbles.

But that strikes me as a kind of comical quibbling. A theory of geometry does not have to explain how there could be a freely falling grain of dust. All it needs to be is ready for you to insert a marble or a cloud of dust into its picture of geometry---it will take charge from there on.

This may sound a pretty superficial and unphilosophical but that's how I think of classical geometry.

GR does what it needs to---explains what flat means and why geometry is usually nearly flat (because matter is sparse) and how distances to galaxies can expand and how you can get black holes and gravitational redshift and all that basic geometry stuff that we observe.
Anyway that is my simplistic attitude about geometry.

So your expressed reservation about GR seems like a non-reservation
atyy
#75
Aug22-10, 11:42 AM
Sci Advisor
P: 8,602
Quote Quote by marcus View Post
So your expressed reservation about GR seems like a non-reservation
It's not a reservation. I take the view that GR is not about geometry, except technically in the sense that all the fields of the standard model are geometrical because of the gauge symmetry. Thus in GR, observable geometry only emerges when one has matter. That, I believe, is the true lesson of GR. The plug and play view is not background independent, because you have test particles that move on a fixed background, without themselves affecting the background.
marcus
#76
Aug22-10, 12:08 PM
Astronomy
Sci Advisor
PF Gold
marcus's Avatar
P: 23,227
Atyy: "GR is not about geometry."
Marcus: "Geometry is precisely what GR is about. GR is the paradigm or model theory in that department."

No basis for discussion there---beyond sterile semantics. We had best get back to Rovelli's paper.
marcus
#77
Aug22-10, 12:15 PM
Astronomy
Sci Advisor
PF Gold
marcus's Avatar
P: 23,227
This will respond in part, as well, to Fra's concerns about the QG agenda.

Several of Fra's posts responded to my couching the agenda in negative terms--a manifoldless QG+M.

To put what I see as the main direction is more positive terms, I'll propose this alternative---a more fully relational QG+M.

This notion of a goal to work towards has been around for decades (I don't know how long). The idea is that GR---the paradigm classical theory---only tells us about the web of geometric relations among events.

There is no substantive objective continuum, because of diff-invariance. One can morph the situation around. Points have no definable identity except where marked by some physical event, like an intersection of worldlines---or some identifiable feature of the gravitational field itself which can mark an event.

So if space is anything, it is an insubstantial web of relationships. To pass to a quantum picture basically means to construct a hilbertspace of webs of relationships, and define operators on it. Or? Do you have some more accurate and concise way to put it?

(looking back at Fra's post #69 I think I may have just now said some things that were contained in what Fra said---except that he went quite a bit further in certain directions---the importance of the observer and information-theoretical considerations.)

================
BTW re Atyy's "not about geometry" comment: Actually GR has matter. You can have dust or marbles adrift on the righthand-side of the main GR equation. In that sense it as plenty of observers already (assuming you do not require observers to be conscious and wear conventional timepieces on their wrists and so forth). If a grain of sand can serve as an observer (and I would argue that it can) then you can put in as many observers as you want---the main equation is set up for it. The effect of those observers will be taken account of in the gravitational field. Logically there is no need for "test particles".
Fra
#78
Aug22-10, 01:36 PM
Fra's Avatar
P: 2,799
I don't mean to just provide "negative terms", I actually wanted to drive the discussion in the constructive sense, by providing noting some provocative points with the picture and focus on some foundational issues that exists conflicting between a measurement theory.

It's nothing new as it's related to the problem of what is an observable in GR and QG, but for some reason the points doesn't seem to get the attention I think it deserves.

Quote Quote by marcus View Post
So if space is anything, it is an insubstantial web of relationships. To pass to a quantum picture basically means to construct a hilbertspace of webs of relationships, and define operators on it. Or? Do you have some more accurate and concise way to put it?
As far as I understand LQG, this sounds like a good summary of one of it's constructing principles.

But I have an objection to exactly this, but the objection is as much a critique against QM.

My clear conviction is that this is an inappropriate application of QM formalism taking out of context. I suggest that the hilbert space of states of the webs of relations are non-physical as they are not inferrable by an real inside observer. They make sense in the mathematical sense only - and if you accept is as a strucutral realism.

I'm not describing LQG here but I would want put it like something like this (to compromise with your phrasing):

Space, is an insubstantial web of relationships (ie. it's not "material") BUT the information needed to specify this web of relationships is physically coded in matter. Each material system encodes the subjective perspective (up to some horizon).

I further suggest that this picture means that each material observer (matter system) "sees" it's own "hilbert space" (I use quotes as I think this implies a modification of QM as we know it today), and moreoever this hilbert space is not timeless, it evolves with time (where time is just a parameterization of an the entropic flow; which is different to each observer).

Since different observers see different state spaces, that inconsistency is what forms the negotiated consensus and defines the local equivalence classes. So each observers, sees "equivalence classes" of nearby "material observers" whose definition genereally evolve. but one can certainly imagine equilibrium conditions where stable quasi-global classes emerge.

So as I see it the "quantum picture" doesn't involve applying the quantum formalism as is, to the equivalence classes of diff-generated observers, the quantum picture is there from the beginning if we consider the proper discrete measurement theory. What STARTS OUT as a classical measurement theory (ie probability theory, but discrete) gets mixed up by the set of different encoding structures.

The difference as I see it between classical and quantum logic, is that classical logic just uses as simple probability space, where quantum logic uses sets of relates spaces that are related by lossy compressions (such as truncated fourier transforms). This is why logical operators are different.

I agree this is radical and speculative, and maybe it's optimistic to expect anyone bot buy into this long train of though, but the simple point I have is that:

Quantum theory are we know it, are verified only for what smolin calls subsystems. Which means the cases where the statistics and hilbert spaces can be effectively constructed and encoded in some lab environment before the entire environment has completey evolved into something different.

And some quite simple plausability arguments, and the quest for everything to be inferrable in the inductive rather than deductive sense suggest that the application of normal QM formalism to the equivalence class of GR observers in the suggested way may be the wrong way to approach the entire "QG" problem.

Note sure if that made sense? Because I have also deep concernts about QM foundations, it's not possible to comment on QG without getting into that as well.

/Fredrik
Fra
#79
Aug22-10, 01:45 PM
Fra's Avatar
P: 2,799
To try to make cleaner how we disagree.

"Since different observers see different state spaces, that inconsistency is what forms the negotiated consensus and defines the local equivalence classes."

LQG tries to make a "regular QM theory" to the STATES of the equivalence classes.

I think that we need to find the EVOLUTION of the SYSTEM of interacting observers.

So I guess what I say is that we need to make QM truly relational, like Einstein made SR into GR. Not, try to apply QM as we know it to the classical equivalence classeso GR. I think it's a mistake.

So I think we are seeking "Einsteins equation" for the relational QM. To apply non-relational QM formalism to Einsteins equation is not right.

So I'm suggesting that hte equivalence classes and their symmetries must be evovling, and that this pictures includes ALL interactions. Thus Strong, weak and EM as well. It's not something we can put "ontop" of the pure-gravity quantized. It makes no sense to me.

/Fredrik
atyy
#80
Aug22-10, 01:53 PM
Sci Advisor
P: 8,602
Quote Quote by Fra View Post
So I'm suggesting that hte equivalence classes and their symmetries must be evovling, and that this pictures includes ALL interactions. Thus Strong, weak and EM as well. It's not something we can put "ontop" of the pure-gravity quantized. It makes no sense to me.
So this would argue for unification, something like strings? In strings the graviton is sometimes a particle caused by an excitation of a string, but if you change your view it can become part of the background spacetime - and it can go the other way too, the background can become an excited string state about a different background.
Fra
#81
Aug22-10, 05:23 PM
Fra's Avatar
P: 2,799
Quote Quote by atyy View Post
So this would argue for unification, something like strings? In strings the graviton is sometimes a particle caused by an excitation of a string, but if you change your view it can become part of the background spacetime - and it can go the other way too, the background can become an excited string state about a different background.
Unification yes, and there are some ways for me to relate this construction to ST, but ST has many unsatisfactory traits. An certainly something is missing in the construction principles.

- ST makes use of the continuum, not only the manifolds, but maybe worse the string itself (which I view as a continuum index). This is highly unphysical and doesn't fit into the picture of a physical representation.

- ST have the same simple view of QM. So it does not solve the intrinsic measurement problem and coding of information problem of QM. ST is not the reconstruction of measurement and representation from the combinatorical perspective I think we need.

The second problem, is btw, what forces the higher background dimensions as it's the only way to "encode" all the variety ST wants to. But the problem is then that you do get this landscape that you don understand what it is. Is it real, is it an illusion? And why is there measure on the landscape?

From my point of view, some of the problems of ST might be gone if they replace the string with a more generic "set of sets" in the datacompression sense I mentioned before, that work from discrete indexes. But then, it just isn't string theory anymore.

Not to mention the action of the string, which is basically inherited from classical analogies.

In my view, all actions are generically related to probabilities or information divergences. The "action" is simply the generalized "entropy" in transition space, which is to be maximized. So all action forms should follow in this way (thus beeing inherently entropic).

There is a chance that "string like" structure, prove to be the simplest possible continuum structures in the large complexity limit, but that is still just a possible connection and the logic there is nothing like the logic of the string program.

Somehow, rovelli's reasoning as I've read it, although I object to it, is at least more clear and consistent that the string scheme which I find to be more of toyery.

/Fredrik
Fra
#82
Aug22-10, 05:30 PM
Fra's Avatar
P: 2,799
Quote Quote by atyy View Post
but if you change your view it can become part of the background spacetime - and it can go the other way too, the background can become an excited string state about a different background.
Generically this makes sense to me, and it would correspond to comparing two different observers. Just like any conditional assessment depends on perspective.

So such a general trait is I think sensible.

The Background should be part of the observer. The problem is that the way ST is constructed, the background complexity is not bounded. First of all because it's based on a continuum index, and it becomes highly ambigous IMHO at least how to COUNT and compare evidence in uncountable sets. The choice of limiting procedure becomes crucial. But no care is made about that in ST. The worst part is that the continuum itself is part of the baggage, and already there you have lost control before you've started as the counting procedure (from inference poitn of view) becomes more or less completely ambigous.

/Fredrik
atyy
#83
Aug22-10, 05:48 PM
Sci Advisor
P: 8,602
Quote Quote by Fra View Post
The Background should be part of the observer. The problem is that the way ST is constructed, the background complexity is not bounded. First of all because it's based on a continuum index, and it becomes highly ambigous IMHO at least how to COUNT and compare evidence in uncountable sets.
Yes, I can never decide which I'd like better. On the one hand, it'd be nice if we only used integers in the formulation of the most basic theory. On the other hand, there are cases where discreteness emerges from the continuum - say eigenvalues in quantum mechanics - or non-relativistic quantum mechanics of atoms from relativistic quantum field theory.
Fra
#84
Aug23-10, 02:28 AM
Fra's Avatar
P: 2,799
Quote Quote by atyy View Post
Yes, I can never decide which I'd like better. On the one hand, it'd be nice if we only used integers in the formulation of the most basic theory. On the other hand, there are cases where discreteness emerges from the continuum - say eigenvalues in quantum mechanics - or non-relativistic quantum mechanics of atoms from relativistic quantum field theory.
I see what you mean. I think there is not really a conflict per see between continuum models and discrete ones, it's just that I think it's important to keep in mind from the point of view of inference and counting and rating evidence (inductive reasoning) what the physically distinguishable states are and what is "gauge".

By certain transformations (I'd like to call them datacompression) one can from limiting cases or continuum models compute key parameters that are independent from superficial embeddings or interpolated structures, that can be further used to "index" the continuum structures, maybe even in a countable way.

That's fine as long as we keep track of what the physically distinguishable states are, and what we should count. I prefer to start with the "backbone" and then picture this as indexing a continuum manifold if we need it for comparasion to old models, rather than start with a redundant description, get lost and try to figure out what's physical degrees of freedom and what's just continuum gauge.

For example when you start with a continuum structure, and try to apply inductive inference, construct various entropy or action measures, then it's crucial that we know how and what to count. In a continuum picture, by an ambigous choice of limiting procedure or measure one can pretty much get the results one wants.

This is even more important if one (like I want to) wants to construct also the expected action of this "observer complex", as they way I picture it, the prediction and computation of "probabilities" requires that the state spaces and transitions are countable. Actually finite, or if infinite, at minimum countable and have a well defined limiting procedure. Otherwise the physical measures are not computable.

/Fredrik
marcus
#85
Aug30-10, 12:10 AM
Astronomy
Sci Advisor
PF Gold
marcus's Avatar
P: 23,227
As we were talking about Rovelli's April paper in some other threads I was impressed by the level of misinformation/misunderstanding.

This is the paper that presents LQG in a manifoldless way giving it a "new look", as Rovelli's title indicates. Of course there is no distinction between canonical LQG and spinfoams here--those approaches were unified earlier. Network and foam are indeed inseparable but that is not what is new.

Someone in another thread stated with great confidence and authority that this version of Lqg had nothing to do with the Einstein-Hilbert action . (The Regge action is the relevant version of E-H, and is derived from the setup.)
Another person flatly stated his conclusion that the April paper merely presented a new spinfoam vertex. We need to get past a wall of ignorance/selective inattention. There is a kind of sea-change in progress---a general shift in the qg picture-- making it more important to be well informed.

In that other thread, Tom responded with a concise and helpful summary of what is happening in the April paper (1004.1780) the topic of this thread, so I'll copy here:

Quote Quote by tom.stoer View Post
So this new LQG is just a new SF model.
No!

It's about convergence of canonical approach and spin foams; it's about mapping of or identities between certain entities in both frameworks; it's about making LQG accessable for calculations; it's about long-distance limit / semiclassical approximations; it's about consistency of quantization, implementation of constraints, regularization of the Hamiltonian (which is notoriously difficult in old-fashioned LQG) ...

... the more you read the more you will find.
marcus
#86
Sep4-10, 08:28 PM
Astronomy
Sci Advisor
PF Gold
marcus's Avatar
P: 23,227
Here's some thematic material I want to develop here, taken from another thread about the April 2010 paper on "new look" LQG.

http://physicsforums.com/showthread....16#post2855316
marcus:...to better understand what underlies the relation of geometry to matter...

The Loop enterprise is high risk. [But] it does seem to me ... philosophically sound. It gets away from dependence on the manifold. The labeled graph (spin network) is an economical representation of the experimenters' geometrical knowledge (a finite web of volume and area measurements which can also carry particle-detector readings and stuff like that). The program does seem at least to define a clear and reasonable direction.

Rovelli says that recent results provide some indication that they might get the Einstein equation for the simple matterless case. He explains why he thinks they might. That's all, he doesn't say they got it yet.
...
http://physicsforums.com/showthread....61#post2855561
sheaf:...Rightly or wrongly I'm impressed by the convergence of the various approaches. Also, being able to pull the Regge action out of that purely combinatorial framework sounds like good news to me. Even if all this, for the moment, only relates to the vacuum equations, that is an enormous achievement.

So yes, I'm watching all this with a great deal of interest.
http://physicsforums.com/showthread....64#post2858264
ensabah6:So this new LQG is just a new SF model.
http://physicsforums.com/showthread....14#post2858714
tom.stoer:No!

It's about convergence of canonical approach and spin foams; it's about mapping of or identities between certain entities in both frameworks; it's about making LQG accessable for calculations; it's about long-distance limit / semiclassical approximations; it's about consistency of quantization, implementation of constraints, regularization of the Hamiltonian (which is notoriously difficult in old-fashioned LQG) ...

... the more you read the more you will find.
marcus
#87
Sep4-10, 09:35 PM
Astronomy
Sci Advisor
PF Gold
marcus's Avatar
P: 23,227
One point to make is that you can look at the spin network graph as a truncation of geometry. Doing physics requires approximation and people habitually think in terms of a truncated series. Some will assume a perturbation series even where there is none(!) and expect to be presented with finite initial segment. But there are other ways to truncate.

So that's one thing: start seeing a graph as a finite truncation of geometry. I'll give an example using something that anyone reading this probably knows: the 3D hypersphere S3---the 3D analog of S2 the familiar 2D surface of a balloon.

For visual warmup I guess we could start with that simpler S2 case. Here's a primitive graph for it:
(|)
consisting of two nodes joined by 3 links, imagined dually as two equilateral triangles glued so as to make the S2 surface of a balloon. The two nodes pictured as the North and South poles.

But that's not what I want. I really want a graph used to approximate S3. It could be two nodes ("the point here and the point at infinity") joined by 4 links. Here is bad drawing:
([])
In a LQG graph the nodes can carry volume and the links represent adjacency and contact-area.
Links can represent area across which neighbor chunks of volume communicate.

So we can imagine this graph dually as two tetrahedra, each with 4 faces, and the faces glued so as to make it topologically the hypersphere.


As Rovelli mentions in the April paper, a LQG graph can carry other stuff as well. The nodes carry volume, but can also carry fermions. The links carry area, but can also be labeled with Y-M fields.

Still, their primary job is to carry the most rudimentary basic geometry information.

If you picture a more complicated graph, you can imagine how a surface in manifoldless LQG is defined. You define it as a collection of links (the links which the surface cuts, see equation (6) on page 2).

So an LQG graph is a finite truncation of geometric relationships which in "first order" cases can look like a crude simplification, but can also look naturalistic if you add more nodes and links.

Now let's look at how this graph ([]) is applied in COSMOLOGY. You see its picture on page 4 of the March paper http://arxiv.org/abs/1003.3483

A lot of cosmology involves considering the universe to be spatially the hypersphere S3 so we could expect this. There is section III "The Cosmological Approximation". And then Section III A is about "Graph expansion". (Here "expansion" means analogous to expansion in a power series, not expansion of the universe. But that's coming.)

Now they want to study the expansion of the universe and they want to calculate a transition amplitude between two labeled ([]) graphs, one bigger than the other.
So you look on page 5 and you see a spinfoam connecting two ([]) graphs. The simplest imaginable spinfoam doing that! (Because this is like "first order" truncation.)
And they calculate a spinfoam vertex amplitude because that is how you do dynamics in LQG.
That is section III B about "Vertex expansion".

Actually 1003.3483 is a good companion paper to 1004.1780 because it presents the same manifoldless development of LQG with concrete examples---and without the references, footnotes, and motivating discussion. The March paper gives essentially the same manifoldless treatment of LQG, self-contained, and in some respects easier to learn from.
One should read both.

So a graph (the spin-network with nodes and links) can be a truncation of spatial geometry, but also a spinfoam (the 2-complex analog of a graph, with vertices, edges and faces) can define a truncation as well--of the dynamical evolution. And the authors calculate with it.

They get standard cosmology in the limit. The usual Friedman-Robertson-Walker model that cosmologists use.
Fra
#88
Sep5-10, 02:06 AM
Fra's Avatar
P: 2,799
Marcus, I have a question that perhaps you can answer, since you are well informed about the LQG program and it's neighbourhoods.

Your last post makes me again associate to the way I hoped LQG was before I learned it was not. But maybe there are some published versions or speculative connections to LQG that isn't standard-LQG?

Has anyone considered the following idea: To try to infer matter and matter interactions by considering two different INTERACTING spin-networks? What I mean is to associate the "truncated geometry" with the natural truncation that any observer has due to horizon and information capacity constraints? Then what would the rules be for interacting spin network? And would they possibly reveal non-gravitational interactions? This would be a possible natural link to be put ontop of matter, and there would be two view of it: one view is that somehow matter would be some additional stuff living in the spinnetworks (some additional complexity of some sort) but the other dual view would be simpler: simply that each material particle ENCODES a spinnetwork or a complex of them?

If there is anything like that I would be interested in that. So my this is what I have been "missing". I'm not sure if it exists but maybe you know?

Edit: a good thing with that idea is that LQG would not really be a "pure QG" theory anymore where you have to add manully the other interactions ontop, without matter Encoding the spinnetworks there would be no pure gravity eiter. It's just that if you don't acknowledge that the observer, encoding the relations of the geometry is in fact material and needs somewhere to encode it, it looks like a pure graivty scenario. But the non-gravitational character may possible we encoded in two such views interact. That would be great and it would laso be much close to my own visions. At least someone must have thought of this and aleast tried it and say ran into problems? I'd be interested to review that.

/Fredrk
atyy
#89
Sep5-10, 10:46 AM
Sci Advisor
P: 8,602
If the boundary state is conceived as the boundary of 4D spacetime, is this still manifoldless?
marcus
#90
Sep5-10, 11:43 AM
Astronomy
Sci Advisor
PF Gold
marcus's Avatar
P: 23,227
Quote Quote by atyy View Post
If the boundary state is conceived as the boundary of 4D spacetime, is this still manifoldless?
The short answer is yes. The smooth manifold of diff. geom. is a set with complicated specialized structure. It's just one possible way to think of 4D spacetime--not the only one.

Atyy, you realize that Rovelli and the others do not say "manifoldless". The technically correct term for this presentation of LQG is "combinatorial"---that's the word used in the April paper.

I decided to say manifoldless because it gets across the salient point that, when presented this way, the theory has no set which you can identify with the spacetime continuum.

Labeled graphs (dubbed spin-networks) merely represent disembodied finite information.

In this mathematical presentation there is no set corresponding to the points of spacetime, or of space, or of the boundary of any region of space or spacetime. No continua, or continuums, however you say it. Only finite webs of information, which in a rather vague sense one can imagine resulting from a series of measurements (including particle detections) or from the preparation of an "experiment" involving geometry and matter.

The idea of the labeled graph is not to BE spacetime (perhaps with some particles in it) but to represent in a very concise way the state of knowledge---what we might be able to SAY.
Able to say, that is, about the initial and final conditions, or about the boundary conditions, on the basis of some finite bunch of data-taking.

So in this presentation of QG the continuum does not exist. I mean it is not presented as a mathematical object (a set with some structure described by other sets--the usual way math objects, such as for instance smooth manifolds, are described).

I call it a "manifoldless" presentation to emphasize that feature. If it weren't such an awkward mouthful I would say "smoothmanifold-less" because technically it's a smooth manifold that people usually mean when they say manifold and that's the element which has been eliminated from the picture.


Register to reply

Related Discussions
Finding the shear centre of a non-homogeneous cross section cross Engineering, Comp Sci, & Technology Homework 2
Volume by cross-section: ellipse and equilateral triangle cross sections? Calculus & Beyond Homework 0
Difference between cross section and cross sectional area? Classical Physics 0
Locality and non-locality in period measurement Special & General Relativity 2
Difference between lorentz invariant and lorentz covariant Introductory Physics Homework 4