What is the Minkowski Metric and How Does it Relate to Special Relativity?

[Chris Hillman]

Books, books

Hi, again, aeroboyo,

Say i amend the list to this:

1) General Relativity from A to B by Geroch
2) Spacetime Physics by Wheeler
3) The Geometry of Physics: An Introduction by Frankel
4) A First Course in General Relativity by Schutz

Would i be better served by replacing 'The Geometry of Physics' with 'Flat and Curved Space-time'? I don't want to have more than 4 books just now... But because i learn quickly i would like to have two complex book in there, which is why i included the one by Frankel and Schutz... progressing will keep me motivated.

Clearly you will making your contribution to the economy this shopping season :-/

If you do purchase Spacetime Physics make sure you find a used copy of the FIRST edition (no longer in print) because the second edition drops a key topic ("rapidity", the hyperbolic analog of angle) which you will need to make the table I outlined.

Rob recommended a book which on second thought I also think would be much more appropriate for you right now than Frankel's Geometry of Physics, the book on mathematical methods by Boas. This really impressed me when I first saw it some years ago as covering a very good selection of topics in a well balanced way, plus it has some good problems. Could be a great way to teach yourself a whole lotta math in a hurry.

The Classical theory of Fields by Landau is about special/general relativity right? What do you think of their series in theoretical physics?

This series is sans doubt one of the great classics of the literature, although in places it might be a bit dated. However, I imagine that volumes 1 and 2 at least will never go out of style! Volume 2 (The Classical Theory of Fields) is remarkable for presenting both Maxwell's theory of EM and gtr in one volume. However, this is a graduate text--- on second thought, I'd tend to caution against biting off more than you can chew. Especially since money is tight, you might want to try one of the two popular books, say the one by Geroch, plus Boas, Mathematical Methods.

Chris Hillman

 

[aeroboyo]

hey Criss Hillman,

i have ordered the 1966 first edition of Spacetime physics from amazon, given that a few in this thread have said it's the best version. I'm curious about the significance of this table you've mentioned a few times now? I get the impression that it's a 'path' towards learning this stuff. I have already ordered the titles i listed, but i guess i could cancel the Geometry of Physics one in favour of Boas or Geroch's book. But what really is the difference? Don't all three cover the same things? I think the appeal of the Frankel one to me is that i read it expresses physical laws in terms of geometry (which apparently makes them more lucid and revealing). I'm intrigued by that.

 

[Daverz]

The Classical theory of Fields by Landau is about special/general relativity right? What do you think of their series in theoretical physics?

Mechanics was really useful in grad school. I have the next 3 volumes, but haven't used them much.

Do you have the Feynman Lectures yet? Maybe you could ask for the Definitive Edition set for Christmas.

 

[aeroboyo]

Feynman Lectures?

Are they a graduate kind of text? Because i think i should work through the books I've already ordered first... as someone pointed out, i'd best not bite off more than i can chew.

Daverz, I am wondering what is the 'geometric' approach to physics? The reviews of Spacetime Physics metioned that it's a 'geometric approach', and I've also seen that description used in other contexts too... what advantages is there to it? cheers.

 

[Daverz]

The Feynman Lectures were Richard Feynman's Freshman and Sophomore physics lectures at CalTech.

By geometric, they mean that they use spacetime diagrams and the invariant interval a lot instead of relying solely on algebraic methods.

 

[aeroboyo]

I'm just going to begin my self study by reading a book on linear alegbra, that should be a good first step. I actually didn't even know what linear algebra was until a few minutes ago. I assumed it was just like high school algebra and not something that was important! I'm using the online 'Linear Algebra' text by Jim Hefferon and it seems like a good one.

PS does 'Geometry of Physics' by Frankel cover linear alegbra or does it assume prior knowledge of it? I'm only asking because I've noticed that Boar's maths methods text does cover linear algebra (i haven't found a table of contents for Frankels text anywhere which is why I'm asking).

 

[Daverz]

Yes, Frankel assumes a knowledge of basic linear algebra and multi-variate calculus ("Calculus III" here in the States). Full table of contents is here:

http://assets.cambridge.org/052183/3302/sample/0521833302ws.pdf

 

[aeroboyo]

whoa, i can now see why the consensus was that Frankels text might not be suitable for a novice like myself... thanks Daverz you've been a great help.

After comparing the table of contents of Boas maths methods text and Frankels text, i can see that Frankels is a much more complex, higher level book. Boas on the other hand covers more generic things like linear alegbra and a lot of other topics. But does Boas text also cover the necessary mutli-variable calculus to start tackling Frankels text?

If it does, perhaps Boas text would give me a decent grounding in maths so i could confidently move onto something more difficult like Frankels text...

 

[Jimmy Snyder]

what is a metric?

aeroboyo, you asked this question in your original post. On the one hand, I don't see that it has been answered, and yet on the other hand, I see that you have progressed in your reading and may have already settled this question in your mind. I attempt to answer it for anyone reading this thread who wants to know. I invite real physicists to correct any errors I make.

As the name metric suggests, it is a way of measuring things. In this case, it is the interval that is being measured. In Euclidean space, we are familiar with the expression:

$$dL = \sqrt{dx^2 + dy^2 + dz^2}$$

valid for Euclidean spaces and it might seem natural to extend this to a Euclidean 3+1 space as:

$$dL = \sqrt{dt^2 + dx^2 + dy^2 + dz^2}$$

where t is the fourth dimension. However, through no fault of any physicist, this expression has no practical use, except perhaps to mathematicians. The world is not Euclidean. This expression:

$$dL = \sqrt{-dt^2 + dx^2 + dy^2 + dz^2}$$

on the other hand is useful in that it is invariant under Lorentz transformations. This is called the Minkowski metric.

This metric depends upon space being flat, which in turn depends upon there being no matter in it. In a space that is curved by the presence of matter, the metric is different, as you will see as you read further. Think of the way we measure large distances on the surface of the Earth. We use great circles rather than straight lines. In this sense, we use a non-Euclidean metric for 3 dimensional space. In GR, you will require a similar generalization of the Minkowski metric.

 

[aeroboyo]

I understand the Minkowski metric to be the space-time interval (distance) between two events in flat 4D space. Invarient under Lortenz transformation means that the distance between two events is the same no matter which frame of reference those events are observed from. So it is invarient under Lorentz transforms, but is it invarient under other transformations? Like the Poincare group? If i understand it right, a lorentz transformation is only a rotation in the xt plane, but there can also be rotations in the other plans and translations... I'm learning about matrix notation in linear algebra just now, it's starting to make sense.

Also, just because adding a negative to the time dimension makes the metric invarient under a Lorentz transformation, how can we be sure that that is representive of reality? I read something about how the Maxwell equations were initially found not to be invarient under galilean transformations, and so people thought the Maxwell equations were wrong. They tinkered with Maxwells equations until they were invarient under galilean transformations, but that introduced fields that there's no evidence of in reality.

PS this will sound ignorant, but why the 'd' infront of each term?

 

[robphy]

I understand the Minkowski metric to be the space-time interval (distance) between two events in flat 4D space. Invarient under Lortenz transformation means that the distance between two events is the same no matter which frame of reference those events are observed from. So it is invarient under Lorentz transforms, but is it invarient under other transformations? Like the Poincare group? If i understand it right, a lorentz transformation is only a rotation in the xt plane, but there can also be rotations in the other plans and translations... I'm learning about matrix notation in linear algebra just now, it's starting to make sense.

The metric at a point is actually a tensor that maps two input vectors at that point to an output scalar number... just like the dot-product. Here is some suggestive notation: for 4-vectors $$\vec V$$ and $$\vec W$$
$$\vec V \cdot \vec W = V^a g_{ab} W^b$$
(As written, this is not quite the same thing as matrix multiplication... a little more manipulation is needed.)

A rotation in the xt plane is called a Lorentz boost. A general Lorentz transformation is composed of boosts and "ordinary" rotations [and inversions].

Also, just because adding a negative to the time dimension makes the metric invarient under a Lorentz transformation, how can we be sure that that is representive of reality? I read something about how the Maxwell equations were initially found to be invarient under galilean transformations, and so people thought the Maxwell equations were wrong. They tinkered with Maxwells equations until they were invarient under galilean transformations, but that introduced fields that there's no evidence of in reality.

One way to answer your question about "reality" is to first emphasize the operational meaning of things in relativity with Radar measurements (as described in Geroch, Bondi, Ellis-Williams). Then, you'll have a better "physical" feeling as to how the signature (+---) relates to the principle of relativity and the speed-of-light principle.

One really has to spell out in detail what one means by "Maxwell's Equations" to properly identify what set of transformations they are invariant under. In one formulation, one can write them down in terms of differential forms without a metric. So, one really needs to explicitly specify the field equations, constitutive relations, and possibly transformation laws for the fields.

PS this will sound ignorant, but why the 'd' infront of each term?

The d describes an infinitesimal displacement... just like in Euclidean geometry.

 

[Jimmy Snyder]

I understand the Minkowski metric to be the space-time interval (distance) between two events in flat 4D space. Invarient under Lortenz transformation means that the distance between two events is the same no matter which frame of reference those events are observed from. So it is invarient under Lorentz transforms, but is it invarient under other transformations? Like the Poincare group? If i understand it right, a lorentz transformation is only a rotation in the xt plane, but there can also be rotations in the other plans and translations...

PS this will sound ignorant, but why the 'd' infront of each term?

Yes for the Poincare group, no for any transformation outside of this group. Indeed, I believe the definition of the Poincare group is "those transformations that leave the interval invariant".

The Poincare group includes translations along the three axes and rotations about the three axes and, by virtue of being a group, any combination of these. As stated above, the interval is invariant under all members of this group.

The d is for 'differential'. In the examples I gave, they could all be replaced by $\Delta$ and would correctly give distances. However, in curved spaces, they change from point to point and the metric has to be integrated along the path in order to get the distance. Therefore, you can think of them as deltas for now and don't worry about them being 'd's until you get to curved space.

Where this explanation overlaps that of robphy, you would do well to give his priority. As I had said, I am not a physicist.

 

[Daverz]

But does Boas text also cover the necessary mutli-variable calculus to start tackling Frankels text?

Looks like Boas's chapters 3-6 would give you what you need.

 

[aeroboyo]

mommy is buying me Boas for Xmas :) Only chapters 3 to 6? I guess if that's the case, Frankel doesn't require prior knowledge of infinite series, complex numbers etc

 

[Jimmy Snyder]

The d is for 'differential'. In the examples I gave, they could all be replaced by $\Delta$ and would correctly give distances. However, in curved spaces, they change from point to point and the metric has to be integrated along the path in order to get the distance.

Sorry, this is wrong.
It is the metric that changes from point to point. It is because the entries in the metric tensor are constants that you can replace d with delta. And it is because they are not constants in curved space that you can't.

 

[aeroboyo]

If it changes from point to point, i can see why the 'd' is required, and why 'd' is unecessary when the metric is the same for all points.

I'm currently learning about linear algebra from an excellent online pdf, and I'm at the point where it's explaining that the solution set of a linear system with n unkowns is a linear surface in R^n, the linear surface having k-dimensions, with k being the number of free variables when the linear system is expressed in echelon form. Well i do understand that, but i don't really understand what a 3-dimensional surface in 4D space would look like, for example.

(i'd like to thank every1 for explaining questions that i have while learning this stuff, it's helping me hugely)

 

[selfAdjoint]

If it changes from point to point, i can see why the 'd' is required, and why 'd' is unecessary when the metric is the same for all points.

I'm currently learning about linear algebra from an excellent online pdf, and I'm at the point where it's explaining that the solution set of a linear system with n unkowns is a linear surface in R^n, the linear surface having k-dimensions, with k being the number of free variables when the linear system is expressed in echelon form. Well i do understand that, but i don't really understand what a 3-dimensional surface in 4D space would look like, for example.

(i'd like to thank every1 for explaining questions that i have while learning this stuff, it's helping me hugely)

I deeply recommend that you lose the "look-like" meme when dealing with more than 3 dimensions and learn to satisfy yourself with "Well, it's analogous to a surface in three space". This is apparently harder for some people to do than for others, but you'll just be spinnng your wheels until you try.

 

[aeroboyo]

Ok. A system of linear equations with 3 unkowns, and 2 free variables would be a 2 dimensional linear surface in 3D space.

That's easy to picture: it's just a plane, like a piece of paper. I guess your right in that once one starts dealing with geometry in 4D and up then one has to be satisfied with not being able to picture it.

 

[Daverz]

mommy is buying me Boas for Xmas :) Only chapters 3 to 6? I guess if that's the case, Frankel doesn't require prior knowledge of infinite series, complex numbers etc

You should know complex numbers. You can probably leave the functions of a complex variable chapter for later. You'll need the infinite series chapter to understand many solutions to ODEs.

 

[aeroboyo]

my first equation post... anyway this is what i was talking about in the previous post... Bellow would be the easy to picture, plane in 3 dimensions.

 

[aeroboyo]

ok so take a quick look at this solution set:

${ \left( {\begin{array}{*{20}c} 1 \\ 0 \\ 5 \\ \end{array}} \right) + t\left( {\begin{array}{*{20}c} 1 \\ 1 \\ { - 8} \\ \end{array}} \right) + s\left( {\begin{array}{*{20}c} { - 3} \\ 4 \\ { - 4.5} \\ \end{array}} \right)|t,s \in \Re \}$

The first column vector is one that is its canonical position right... so it goes from the origon to the point (1,0,5). Then the second column vector states that from this point you draw a vector with direction (1,1,-8) at the point (1,0,5) and the magnitude of this vector is the free variable t so it can be any length. Is that correct way to interpret the solution set? I'm trying to write this out because today is the 1st day of learning about the gemoetry of vectors so this helps me learn :)

 

[robphy]

aeroboyo,

...just to keep the focus of a given thread, it might be better to discuss that problem in a new thread in General Math, Linear and Abstract Algebra, or [even if it's not official homework] in the Homework section.

 

[aeroboyo]

Ok. The reason i started taking about geometry of vectors here is that i have an inkling that it is relevant to minkowski space... for example, four-velocity... which I'm assuming would be a line (1 dimensional linear surface) in 4 dimension space... something like this:

$${ \left( {\begin{array}{*{20}c} 1 \\ 0 \\ 5 \\ 2 \\ \end{array}} \right) + t\left( {\begin{array}{*{20}c} 1 \\ 1 \\ { - 8} \\ 3 \\ \end{array}} \right)|t \in \Re \}$$

could be a four-velocity vector i think... I'm just trying to apply what I'm learning from linear algebra to special relativity as i go along ;)

 

[Daverz]

That might be a representation in one particular coordinate system, yes.

 

[aeroboyo]

I hadn't thought about that, i guess that wouldn't be invariant when transformed into different coordinate systems... just the fact that the first 'canonical' vector in that set defines a point relative to the origon in that particular coordinate system, would mean that in a different coordinate system it would have to be different. I guess that's where expressing things as invariant tensors comes in, which is something i probably won't grasp until I've worked through boas.

I'm assuming that also, that represents the average four-velocity, because t isn't taken as a limmit. I'm guessing that instantaneous four-velocity might be expressed like this:
$${ \left( {\begin{array}{*{20}c} 1 \\ 0 \\ 5 \\ 2 \\ \end{array}} \right) + dt\left( {\begin{array}{*{20}c} 1 \\ 1 \\ { - 8} \\ 3 \\ \end{array}} \right)|t \in \Re \}$$
So now the free variable is infintismally small, and therefore the velocity vector would have to be as well. I haven't read this anywhere I'm just guessing.

 

[George Jones]

Ok. The reason i started taking about geometry of vectors here is that i have an inkling that it is relevant to minkowski space... for example, four-velocity... which I'm assuming would be a line (1 dimensional linear surface) in 4 dimension space... something like this:

$${ \left( {\begin{array}{*{20}c} 1 \\ 0 \\ 5 \\ 2 \\ \end{array}} \right) + t\left( {\begin{array}{*{20}c} 1 \\ 1 \\ { - 8} \\ 3 \\ \end{array}} \right)|t \in \Re \}$$

could be a four-velocity vector i think... I'm just trying to apply what I'm learning from linear algebra to special relativity as i go along ;)

The metric says that (in any inertial coodinate system in which the coodinates are written in either of the standard orders) the tangent vector to this line is not a 4-velocity, because this tangent vector is not timelike.

 

[George Jones]

Hi aeroboyo,

Sorry for being so brusque in my last post - let me give it another go.

Ok. The reason i started taking about geometry of vectors here is that i have an inkling that it is relevant to minkowski space... for example, four-velocity... which I'm assuming would be a line (1 dimensional linear surface) in 4 dimension space

Yes, Minkowski spacetime is usually modeled as a four-dimensional vector space. Minkowki spacetime is, within the context of special relativity, the space of all possible locations in space and time. Call these spacetime locations 4-positions, so Minkowski spacetime is the space of all possible 4-positions.

Now, consider an observer, say A. A doesn't experience all possible locations in spacetime - only a one-dimensional subset of 4-positions, i.e., a (possibly curving) line. This line, the set of events that A experiences, is called the worldline of A. At any event on A's worldline, A's 4-velocity is a vector tangent to the worldline.

$${ \left( {\begin{array}{*{20}c} 1 \\ 0 \\ 5 \\ 2 \\ \end{array}} \right) + t\left( {\begin{array}{*{20}c} 1 \\ 1 \\ { - 8} \\ 3 \\ \end{array}} \right)|t \in \Re \}$$

This looks more like a worldline (a set of 4-positions), than a 4-velocity. Since this is a straight line, it might represent the woldline of an inertial velocity. if $t$ is the proper time (usually denoted by $\tau$) of the observer, then differentiating with respect to [tex]t[/itex] gives the observer's 4-velocity.

However, 4-velocity is always a timelike 4-vector. In fact, if units are chosen such that the speed of light is one, a 4-velocity is always a unit length timelike vector.

My advice - dive into Spacetime Physics without, at first, worrying too much about mathematics. After you've mastered the concepts in this book, then have a look at the mathematics.

 

[aeroboyo]

I've decided i'd rather focus on the basic areas of maths in a concurrent way before diving into SR or GR. After all, the maths methods would be applicable to other areas of physics. I'll be learning linear algebra, vector and tensor analysis once my Boas Maths Methods book arrives... I'm wondering, how deeply does one have to study tensor analysis, linear algebra etc in order to be able to master general relativity? I'm guessing one would have to take the study of tensor analysis a bit further than a maths methods book would. I see on amazon there are a number of texts which focus soley on those topics... after I've grasped the basics of these areas of study should i try to find a text which takes each further?

 

[Daverz]

Well, you don't need a lot of math for Spacetime Physics, nothing much beyond what you've indicated you know, but it should give you a good math workout by working the problems and checking your solutions in the back. I guess you'll see when it arrives.

Introductory GR books usually develop a lot of the math, so a good place to pick up tensor analysis when you're ready is... a good intro GR book.

Sure, you can learn some of the math itself in more depth, and we've already discussed several books in this regard. I'd wait until you have some more mathematical experience under your belt. Also, mastering GR requires a good knowledge of mechanics and electrodynamics, so don't neglect your physics education.

 

[aeroboyo]

i have no idea what electrodynamics is... is it another word for electromagnetism? Would Landau's The Classical Theory of Fields be a good intro to electrodynamics? I wonder if Landau's course in theoretical physics would be a decent thing to work through, after I've got some basic maths down... although someone on these forums said that they're antiquated.


In regard to the maths, a theoretical physicist should understand maths as well as someone who studies pure mathematics right? Because how can you understand and develop theories unless you have a mastery of maths... Thats why i was guessing that althouth Boas will give me an intro to many different areas of maths, each of those areas probably isn't developed in enough detail in that particular text. I have orderd 'A first course in GR' so i guess that will develop some tensor calculus in that case.

 

[Chris Hillman]

Tensor analysis prereqs?

Hi, aeroboyo,

I've decided i'd rather focus on the basic areas of maths in a concurrent way before diving into SR or GR. After all, the maths methods would be applicable to other areas of physics. I'll be learning linear algebra, vector and tensor analysis once my Boas Maths Methods book arrives... I'm wondering, how deeply does one have to study tensor analysis, linear algebra etc in order to be able to master general relativity? I'm guessing one would have to take the study of tensor analysis a bit further than a maths methods book would. I see on amazon there are a number of texts which focus soley on those topics... after I've grasped the basics of these areas of study should i try to find a text which takes each further?

I am glad to hear you plan to study from the Boas Math. Methods textbook---- I think you will that a lot of fun, and invaluable for all kinds of applications.

About tensor analysis--- I don't think Boas covers that, but don't worry, there's really nothing much to tensor analysis that Boas+Schutz won't prepare you for. Ideally, you'd study linear algebra and be comfortable with vectors, matrices, linear operators, binlinear forms, vector spaces and vector bases, before starting in on gtr, but this is inessential compared to trig, a strong visual imagination, and good mathematical "situational awareness" generally (those last two probably can't be taught, so you'll just have to see if you have them by trying to learn gtr). While you are waiting for Boas to arrive, you might try some on-line tutorials to try to start developing your geometric intuition for Minkowski spacetime:

http://www.astro.ucla.edu/~wright/relatvty.htm
http://casa.colorado.edu/~ajsh/sr/sr.shtml
http://physics.syr.edu/courses/modules/LIGHTCONE/

If you can get a used copy of the FIRST edition of Spacetime Physics by Taylor & Wheeler (e.g. via amazon or another such website) that would be ideal. Apparently the second edition dropped one of the topics you would most need, "rapidity" (analog of angle).

Chris Hillman

 

[Daverz]

i have no idea what electrodynamics is... is it another word for electromagnetism?

Yes. Electrodynamics emphasizes the dynamic aspects of electromagnetism, like radiation from moving charges. Roughly, it means "E&M beyond electrostatics and magnetostatics."

Would Landau's The Classical Theory of Fields be a good intro to electrodynamics?

No. That's a book for those who've already mastered E&M.

Feynman Lectures Volume 2 is a good informal introduction (and the parts of volume 1 dealing with EM waves).

In regard to the maths, a theoretical physicist should understand maths as well as someone who studies pure mathematics right?

Ideally, but there's only so much time in the day, and physics comes first. Which is one reason there are so many of these math methods books rehashing the math for physicists.

Thats why i was guessing that althouth Boas will give me an intro to many different areas of maths, each of those areas probably isn't developed in enough detail in that particular text.

True, but hopefully you'll have an opportunity to fill out your knowledge at university. Some of this will be required coursework anyway.

I have orderd 'A first course in GR' so i guess that will develop some tensor calculus in that case.

Sounds good.

 

[aeroboyo]

yes I've ordered the 1st edition of spacetime physics used from amazon... from 1966! I hope it doesn't fall to pieces in the mail. I believe Boas does cover tensor analysis, according to the TOC:

Chapter 10 Tensor Analysis:
1. Introduction
2. Cartesian Tensors
3. Tensor Notation and Operations
4. Inertia Tensor
5. Kronecker Delta and Levi-Civita Symbol
6. Pseudovectors and Pseudotensors
7. More About Applications
8. Curvilinear Coordinates
9. Vector Operations in Orthogonal Curvilinear Coordinates
10. Non-Cartesian Tensors
11. Miscellaneous Problems

I'm not sure how 'extensive' a review that is of tensor analysis... but it'll be a start for sure. I read a little about tensors today from a NASA tutorial online... it was quite interesting. Did einstein invent tensors as a way to describe GR or did he just realize that it was the best language to describe it in? I can see how useful tensors would be in describing natural laws, since they're invarient in different spaces and coordinate systems. I now understand why this:
$${ \left( {\begin{array}{*{20}c} 1 \\ 0 \\ 5 \\ 2 \\\end{array}} \right) + t\left( {\begin{array}{*{20}c} 1 \\ 1 \\ { - 8} \\ 3 \\\end{array}} \right)|t \in \Re \}$$
would only describe a line in one particular coordinate system, the point vector is coordinate dependent and so it's not a rank 1 tensor (but I've learned that point vectors can be tensors is you differentiate them, or in the case that you have a difference of two of them). :)

Another thing I've learned from an online book on linear algebra is that you can prove that in any R^n space, a line is always straight and a plane is always flat... the proof involves a trig identity that says the shortest distance between two points is always a straight line. Preety fascinating stuff. I was wondering if a 2 dimensional linear surface (a plane) was actually 'flat' in 4D spacetime... i guess it must be.

 

[Daverz]

I believe Boas does cover tensor analysis, according to the TOC:

Chapter 10 Tensor Analysis:
[snip]

That should be a good intro. Schutz should get you the rest of the way.

Did einstein invent tensors as a way to describe GR or did he just realize that it was the best language to describe it in?

The tensor analysis Einstein used was worked out by Ricci and Levi-Cevita. Einstein was made aware of their work by a mathematician friend. Levi-Cevita was still working out details even after Einstein figured out the GR field equations, so it was cutting edge math at the time.

http://www-history.mcs.st-andrews.ac.uk/HistTopics/General_relativity.html

 

[aeroboyo]

i was enrolled in physics at St Andrews Uni back in 2003... long story.