Prerequisits : Special Relativity For Laypeople, Tensors For Laypeople

Recommended : Quick And Dirty Tensor Calculus

In today’s article, we will be taking a closer, more detailed look at some further important concepts in Special Relativity ( henceforth abbreviated “SR” ). If you aren’t already familiar with SR, I would recommend you read the other articles linked to above first, as otherwise you may not be able to follow what I have to say here. Once again, this article is not intended as a substitute for a proper textbook, of which there are multitudes of very good ones. My goal is not to teach you how to pass your next exam on this subject matter by painstakingly running through pages of Lorentz transformations in different setups, but rather to provide an intuitive understanding of the concepts involved, mainly aimed at the interested lay person.

Let us just briefly summarize the main ideas behind SR again :

- An event is a point in space at a given instant in time; space-time is the collection of all events. It is
a physical substance or “fabric” with mechanical properties.**not** - The term “geometry of space-time” refers to how events are related to one another.
- The flat 4-dimensional space-time of SR is called Minkowski space-time; its geometry is not Euclidean, but Lorentzian in nature. This means that increments of space and time are not
, but**additive**within the space-time interval.**substractive** - An inertial frame is one wherein an accelerometer will read exactly zero at all times. All inertial frames experience the exact same laws of physics; in particular, this is true for the laws of electromagnetism, which is the why the speed of light must be the same in all inertial frames.
- Inertial frames cannot be distinguished from one another by any locally performed experiment. This means that
is not a universal, intrinsic property of an observer, but rather an expression of the relationship between two frames. The concept of inertial motion is meaningless, unless it refers to at least**motion**different frames.**two** - Inertial frames are related via Lorentz transformations of their coordinate systems, which is equivalent to a hyperbolic rotation in space-time. The concept of speed can therefore be described as a rotation angle.

The theory of Special Relativity describes events and their relationships in flat Minkowski space-time; this of course covers inertial frames and inertial motion, but it also includes accelerated observers. It is a common misconception that SR cannot handle acceleration, and that General Relativity is needed for that, but this is untrue – the only difference between accelerated and unaccelerated ( i.e. inertial ) frames is that the latter are related via Lorentz transformations, whereas the former are not. It is still possible to write transformations between frames when acceleration is involved, and this falls under the remit of SR; for example, the relationship between proper time in an accelerated frame, and coordinate distance as seen by a stationary outside observer, is

(1)

For those of you who are interested in the full treatment of arbitrary accelerated motion in SR, I recommend Dirk van de Moortel’s page on this topic.

The domain of applicability of SR does not end at acceleration, but rather at the presence of tidal gravity. SR can handle any situation where there are no sources of gravity involved, or the influence of such sources can be neglected. When we talk about General Relativity in a later article, we will learn that this is because the presence of sources of gravity changes the geometry of space-time in such a way that it is no longer adequately described by anything resembling the Minkowski metric. Hence, SR no longer applies. The presence of acceleration also changes the metric, but it does so in a way that the overall result remains * isometric* to the Minkowski metric. What this means is that there exists a coordinate transformation which takes the metric of an accelerated observer back into the familiar Minkowski metric, and this transformation is consistently defined for

*events in space-time. We will find later that this works fine for accelerated frames alone, since the metric describing such frames has the same structure as Minkowski space-time ( they are related via a simple coordinate transformation ), but it does*

**all***work if we add sources of gravity, since there is no isometry between gravitational metrics and the Minkowski metric.*

**not**In the special case of the acceleration being constant, the metric describing such frames is called the * Rindler metric*.

. Special Relativity can handle both purely inertial, as well as non-inertial ( accelerated ) frames of reference, so long as there are no sources of gravity present anywhere ( or the influence of such sources can be neglected ). SR therefore applies to all cases where space-time isDomain of Applicability, and that includes accelerated frames too. Special Relativity is therefore a model for space-times which are unaffected by tidal gravity.flat

We will, in the next article I am writing, learn what “flatness” actually means in this context, and how to mathematically describe it. For now I will just say that within such frames, an accelerometer will read a non-zero value, * but* it will not detect any tidal forces.

* Caution* : those of you who may have heard of the “equivalence principle” might interject at this point, and doubt what I am saying above. This is the reason why I am consistently writing

*, instead of just*

**tidal gravity***. We will discuss the relationship between the equivalence principle and the validity of SR in detail in a later article, and I will also give you the explicit proof that accelerated frames in the absence of gravitational sources are indeed completely flat.*

**gravity**So now we know that SR can handle acceleration just fine, so long as there is no tidal gravity involved; I urge you to file this new knowledge away for now ( but bear it in mind ! ), and briefly return to purely inertial frames, i.e. those without any acceleration. As we know already, inertial frames are related via Lorentz transformations, which are just hyperbolic rotations in space-time. Given some arbitrary vector , a Lorentz transformation is therefore just a simple matrix multiplication :

(2)

where the unprimed vector is the one we start off with ( the original frame ), and the primed one is the one we obtain after the transformation ( the new frame ). As expected, this looks just like any other rotation we are familiar with from classical physics. In index notation :

(3)

The inverse operation is the * same* one, only with the indices reversed :

(4)

In fact, the two Lorentz transformation matrices are just inverses of each other :

(5)

Physically, this means that inertial frames, as well as the Lorentz transformations between them, are perfectly symmetric – where frame A sees frame B to be in motion relative to itself, frame B also sees frame A to be in relative motion with respect to itself, and both frames see the same effects. You go from A to B in the same way as you go from B to A – a perfect symmetry, which is why the two frames are fundamentally indistinguishable. Since this is true for * all* inertial frames, there of course cannot be any one frame that is in any way “special” or “privileged”.

. All inertial frames are physically indistinguishable from one another, and all Lorentz transformations are invertible. There are therefore no privileged inertial frames. In addition, all inertial frames are perfectly symmetric.Relativity

The full Lorentz transformation matrix in 4 dimensions and in its most general form actually looks pretty intimidating, but most scenarios can be reduced to a simple transformation along just one axis, with some given relative speed *v* between frames; the Lorentz transformation and its inverse then becomes [1] :

In the first image on top, the frame F’ moves with constant velocity +*v* in the x-direction of the stationary frame F; in the bottom image, the opposite is the case, and F moves with velocity *-v* in the rest frame F’. In both cases, we have the same transformation, the only difference is the direction of *v,* which differs by a sign. While unfortunately not obvious, the vector *r* has been rotated into *r’*, and vice versa. To make the rotational aspect of this whole operation more visible, let us plot this scenario onto what is called a * Minkowski diagram* ( graphics and descriptions taken from [1], because I’m lazy and they are quite good ) :

This diagram shows the original configuration of *F* “stationary” while *F*′ is boosted away along the positive *x* direction, although it correctly gives the inverse transformation since the coordinates *ct*′, *x*′ of *F*′ are projected onto the coordinates *ct*, *x* of *F*. The event (*ct*′, *x*′) = (8, 6) in *F*′ corresponds to approximately (*ct*, *x*) ≈ (14.3, 13.28) in *F*, with rapidity *ζ* ≈ +0.66. Note that the difference in length and time scales is such that the speed of light is invariant; this is where the Lorentz geometry comes in. In essence, in this diagram, the *x* axis ( frame F ) is rotated counterclockwise into *x’* ( frame F’ ), whereas the time axis *ct* ( frame F ) is rotated clockwise into the time axis *ct’* ( frame F’ ). The direction of rotation is opposite between spatial and temporal coordinates, because they have different signs in the metric.

This diagram actually shows the inverse configuration of *F*′ “stationary” while *F* is boosted away along the negative *x*′ direction, although it correctly gives the original transformation since the coordinates *ct*, *x* of *F* are projected onto the coordinates *ct*′, *x*′ of *F’*. The event (*ct*, *x*) = (8, 6) in *F* corresponds to approximately (*ct*′, *x*′) ≈ (5.55, 1.67) in *F*′, with rapidity *ζ* ≈ −0.66. Again, the difference in length and time scales is such that the speed of light is invariant.

Take careful note of one important fact : because the red lines ( the rotated coordinate system ) must always intersect the blue curves somewhere, there is in fact no way to rotate your reference in such a way that the time and space axes coincide. In other words – you cannot bring yourself into the rest frame of a photon, and hence move at the speed of light; no Lorentz transformation exists which can achieve that.

There is no Lorentz transformation which could bring you into the “rest frame” of light; photons are thereforePhotons.a valid frame of reference, and nothing starting off at subluminal speeds can ever be accelerated to the speed of light.not

So what does all of this physically mean ? To understand that, consider what happens when an observer watches a clock that is in relative motion with respect to himself [2] :

Even in this very simple graphic, we immediately see the three main physical consequences of Lorentz transformations :

- Length are
along the direction of motion – the moving clock appears “compressed” across the centre, as seen from the stationary observer**contracted** - Times are
on the moving clock – it reads less time as seen from the stationary observer**dilated** - Times displayed on the clock do not coincide – the moving clock and the stationary clock do not share the same notion of
**simultaneity**

All three of these effects – length contraction, time dilation, and relativity of simultaneity -, are * coordinate effects*, meaning they depend on which observer measures them. Specifically, they depend on the relative speed between the observer and the distant frame which is being observed. This is to say, that different observers may measure different numbers for the same effects and the same object – these effects are different in each frame of reference. In addition, they always vanish in the

*of the observer – if you hold a clock and a ruler in your own hand, you will never see anything special happening on them ( since their relative speed with respect to yourself is zero ); the ruler is always 1m long, and the clock always ticks at “one second per second”. Relativistic effects such as these appear only in distant frames in relative motion with respect to the observer. And again, the situation is exactly symmetric – if you go into the rest frame of the “moving clock” above, then it is the stationary clock which is now moving, and hence time dilated and length contracted.*

**rest frame**

The main physical effects of the relationship between inertial frames are time dilation, length contraction, and the relativity of simultaneity. These are coordinate effects, and hence observer-dependent.Lorentz transformations.

However, it is important to understand that these effects of relativistic motion are * not* merely optical illusions of some kind, but physically quite real phenomena, which have real physical consequences. In my previous article Special Relativity for Laypeople, we have already encountered one such scenario – the curious case of the atmospheric muon, which, despite not living “long enough”, nevertheless reaches the detector at the surface of the Earth. This is because, with respect to the stationary Earth detector, the life time of the muon particle “really is” dilated. This is

*some kind of mechanical action on the particle, on a mere optical illusion, but a manifestation of how events in space-time are related to one another. The particle really*

**not***arrive on Earth, so it really*

**does***“live” longer than it should have, based on old Newtonian mechanics. And this is just one of many, many experimental test of ( kinematic ) time dilation; such tests can now be directly performed under controlled circumstances, e.g. in particle accelerators. The same is true for length contraction – this is a real, physical effect, not just an optical illusion. This can also be directly demonstrated in particle accelerators, for example in an experiment called the “Relativistic Heavy Ion Collider” ( RHIC ). In this setup, heavy gold ions ( which are approximately spherical when at rest ) are accelerated to very high speeds in opposite directions, close to the speed of light; seen from the frame of the laboratory observer, length contraction means that these ions are now “flattened” into thin disks, and collided head-on. The trajectories of the resulting collision by-products ( a state of matter called a “quark-gluon plasma” ) are very sensitive to the initial shapes of the ions prior to collision, and as expected they are consistent with the collision of two thin disks of matter –*

**did***spherical ions. These particles*

**not***contracted, or else their collision wouldn’t produce the results it does. Watch physicist Peter Steinberg explain it in this short video [3].*

**really are**The interesting question in this particular experiment is of course how to reverse the situation, and ask what the ion itself would see. In its own rest frame, the ion’s shape will remain spherical, * but* both the internal dynamics of the second ion ( the one that is on-coming on a head-on collision course ) as well as its shape will be distorted, which can be shown to yield the same results. The mathematics of this are not at all trivial though, so I will say no more on this, except that the experimental results are conclusive, and fully support SR.

I am not aware of a simple experiment that directly tests the relativity of simultaneity, but do note that this follows directly from length contraction and time dilation. If there are no universally applicable concepts of time and space, there cannot be a universal concept of simultaneity either, as the simple graphic of the two clocks above already demonstrates. The most direct confirmation of this is probably the * Sagnac effect*.

Unlike in Newtonian mechanics, there are no longer any universally applicable concepts of space, time and simultaneity. All of these are strictly observer-dependent notions.Relativity, the 2nd.

At this point, I would encourage you to consult a good textbook on how to actually calculate these effects, for different setups and scenarios. My personal recommendation would be [4], but pretty much any textbook on Special Relativity will do. Even a more general introductory text such as [5] ( which contains many useful exercises ) will provide a good opportunity to practice calculations.

If anyone was to ask me what the most important lesson in all of this really is, I would have to say that it is not so much the specifics of time dilation, length contraction etc, but rather the realization that the laws of physics are the same in all inertial frames. This means that the surprising fact isn’t the Lorentz transformation itself ( despite its counterintuitive manifestations ), but the empirical finding that the laws of physics are invariant under such transformations. Everything else follows from this, and it has far-reaching consequences even in such seemingly unrelated fields as quantum mechanics and quantum field theory.

The laws of physics, when formulated appropriately, are invariant under Lorentz transformations, meaning they are exactly the same in all inertial frames. This is called Lorentz invariance, and it is a fundamental symmetry of the universe.Lorentz Invariance.

This is remarkable, because it implies that one observer’s notions of time, space and simultaneity are not necessarily shared by any other observer. Time and space are therefore not absolute, immutable concepts that equally apply everywhere, but strictly depend on who is measuring them, and with what rulers and clocks. This is quite unlike the “absolute time” and “absolute space” notions employed in old Newtonian mechanics, and it thus requires a paradigm shift in the mind of the student of relativity. This shift is indeed * required* from any student of physics, because Lorentz invariance is not just some abstract, doubtful mathematical concept, but an actual property of nature itself, with very real consequences. It is also an empirically verified finding, and despite a century’s worth of increasingly sophisticated tests at every increasing levels of energy, no violations of Lorentz invariance have ever been found ( see here for details ). It does indeed seem to be a fundamental symmetry of nature, at least within the domain we can experimentally probe today. Even within our low-energy, low-velocity world of everyday human experience, Lorentz invariance has some very real manifestations, such as these for example :

- Mercury ( see your thermometer ! ) is liquid at room temperature because of Lorentz invariance
- Gold has the color it does due to Lorentz invariance
- The GPS device in your car works only because Lorentz invariance is accounted for
- Magnetism exists only due to Lorentz invariance; most electronics would not function without the principles of SR
- The corrosion resistance of certain metals ( such as gold ) is due to Lorentz invariance
- CR tubes in old TVs and computer screens have to account for relativistic effects in order to produce sharp, clear pictures
- Light would not exist without Lorentz invariance ( which, among other things, dictates that electromagnetic radiation cannot propagate instantaneously )
- Normal matter as we know it ( and hence us humans ) also would not exist without Lorentz invariance, since this symmetry plays a fundamental role in the laws governing elementary particles and their interactions ( quantum field theory )

While the actual * content* of the laws of physics is always the same in all inertial frames, their mathematical

*may not be, unless we explicitly make it so. There are three classes of objects which allow us to formulate physical laws without making explicit reference to the observer ( = coordinate basis ) : tensors and 4-vectors, certain types of spinors, and differential forms ( which are actually just a specific type of tensor ). We have already encountered tensors, and will take a closer look at spinors and differential forms in later articles. If you know that a given formulation of a law involves only the quantities mentioned above, you automatically know that it is valid in*

**form***frames of reference.*

**all**So what are some of these tensorial quantities used to formulate laws of physics in a Lorentz invariant form ? We have already encountered one of them, the * metric tensor*, which allows us to define an inner product – and hence notions of measurements of length, angles, areas, volumes etc etc – at every event in our space-time. It is, often interchangeably, associated with the line element, which gives us the infinitesimal distance between two neighbouring events.

So far so good. In order to introduce other such tensorial quantities, we take an approach that may at first seem altogether surprising – we look more closely at the * symmetries* of our space-time, specifically at

*symmetries, so we once again keep looking at geometric concepts.*

**global**Consider an astronaut in a rocket, who has a laptop with him. Let’s say at 12:00noon the astronaut is located in a region of space that is sufficiently far from any other massive body so that it can be considered empty and flat, and he is not accelerating ( i.e. he moves inertially ). What happens when he turns on his laptop ? Nothing special of course – the laptop works as normal. No surprises here. Now the astronaut waits two hours, and repeats the experiment at 2.00pm, still at the same location in space. What happens ? Of course, the laptop * still* turns on as normal, so nothing special happens at all.

We now shift the rocket with our astronaut to a different location in space, still far away from any outside influences. We repeat the experiment. Will the laptop still turn on as normal ? Of course it will – once again, no surprises here. Even if, at the new location, we let some time pass and try it again, the laptop will never somehow work in any different way. What about if we rotate the rocket, and let it be oriented into a different direction ? Will that somehow affect how the laptop works ? Obviously it won’t, as an appropriate experiment readily confirms.

The conclusion is that our system – the laptop, and by extension the rocket and the astronaut himself – are * invariant* under certain transformations such as translations in time and space, rotations, and others. There seem to be

*associated with our space-time, which means we can perform certain transformations of a physical system without affecting certain aspects of it.*

**symmetries**If we investigate this surprisingly simple insight more closely ( refer to appropriate textbooks for details ), we eventually find that specific transformations of a given system correspond to the invariance of specific aspects of it; in other words, symmetries of space-time can be associated with conserved quantities of physical systems. This is known as

Every global symmetry of space-time is associated with a conserved quantity, and its corresponding conservation law.Noether’s Theorem.

In particular, a mathematical investigation leads to the following conserved quantities when Noether’s theorem is applied to space-time :

lead to the**Translations in space****conservation of linear momentum**lead to the**Translations in time****conservation of energy**lead to the**Rotational translations****conservation of angular momentum**lead to**Mirror translations****parity invariance**

The theorem can also be applied to more abstract field theories – the same principles apply, and we once again get conserved quantities. For example, the gauge invariance of the electromagnetic potential yields electric charge and its conservation.

This is a truly remarkable result, the significance of which can hardly be overstated. Noether’s theorem has profound consequences both for classical physics as well as for quantum field theory – the paradigm of associating geometric symmetries with conserved quantities is certainly on par with Einstein’s paradigm of associating geometry with gravity, yet it is an irony of science history that, while everyone knows Albert Einstein, too many people have never heard of Emmy Noether, who was a truly brilliant scientist who made groundbreaking contributions to both theoretical physics as well as pure mathematics ( below diagram taken from [6] ).

Let us come back to our laptop. As we have seen, it doesn’t matter at what location in space, and at what time we turn on our laptop – it will always function the exact same way. To put this differently – our system ( the laptop ) is not affected by space-time translations, which corresponds to the combination of points (1) and (2) in the list above. If we formalise this in terms of the maths, we find that the global symmetry of space-time translations invariance corresponds to a conserved tensorial quantity, which describes the distribution of energy and momentum in space-time; this quantity is called the * stress-energy-momentum tensor*, and it is commonly denoted by . We will take a closer look at this tensor now, since it is of great importance within the theory of relativity.

The energy-momentum tensor is a symmetric ( i.e. you can swap the indices ) rank-2 tensor; it is a linear machine defined at each event in space-time, and it describes energy density, momentum density, and stresses as measured by an observer at that event. As being a rank-2 tensor, it has two slots for us to pass an input, and it will produce a corresponding output. In order to understand how the energy-momentum tensor processes a given input, and what result we get, it helps to think of energy-momentum as a “river” in space-time, or as a flow or energy and/or momentum through space [7] :

Suppose we have an observer at a specific event in space-time, as well as an energy-momentum tensor defined at that same event. If we insert the 4-velocity vector of that observer into one of the slots of the tensor, and leave the other slot empty, the tensor will produce for us the 4-momentum per unit volume, i.e. the density of 4-momentum as measured by that observer, which is again a 4-vector [8] :

Or, written in index notation :

(6)

If you insert the observer’s 4-velocity into one slot, and an arbitrary * unit* (!) vector into the other slot, you get the projection of the 4-momentum density described above into the unit vector direction [8] :

In index notation :

(7)

If you insert the observer’s 4-velocity into * both* slots of the tensor, you get as a result the density of mass-energy that the observer sees in his frame [8] :

Now let our observer choose two space-like basis vectors and of his Lorentz frame, and insert those into the two slots of the tensor. The result will be the *j,k* component of the stress as measured by the observer :

With the above, we can attribute a physical meaning to the various components of the stress-energy-momentum tensor, the reason for the name of which should have become sufficiently clear by now :

- is mass-energy density
- is 4-momentum per unit volume
- is the k-component of energy flux
- is the
*j,k*component of stress

And the same thing as a graphic [9] :

Now consider what happens when we place an infinitesimal test cube into our “river” of energy-momentum; what we will find ( proof omitted here ) is that the amount of energy-momentum flowing into the cube is the same as the amount of energy-momentum flowing back out of the cube. This is true at every event in space-time, and implies that the energy-momentum tensor is a conserved quantity, just as Noether’s theorem demands. In mathematical terms, consider a 4D region of space-time *V* bounded by some 3D surface. Then, the 4-momentum flowing into V through the surface must flow out again [10] :

(8)

Stoke’s theorem allows us to “pull” the boundary operator up from the boundary and into the integral itself ( remember that “|” denotes the ordinary derivative ), and integrate over the entire region instead of just the boundary :

(9)

We can now write this in differential form :

(10)

Remember that the energy-momentum tensor is symmetric, so it does not matter which of the two slots we form the divergence on.

. This is a symmetric, rank-2 tensor which describes the densities and fluxes of energy, momentum and stresses in space-time. It is a quantity that is conserved at every event in space-time.Energy-Momentum Tensor

Physically, the vanishing of the divergence (10) means that there are * no* sources or sinks of energy-momentum anywhere in space-time. You cannot create or destroy energy and momentum, you can only change them into different forms.

Every physical system has an energy-momentum tensor associated with it. This is true for material objects ( stars, planets etc etc ), just as it is true for things like electromagnetic fields, quantum fields and so on. Without derivation or proof, I will give you two specific expressions for special but very important cases; the first one is the energy-momentum tensor for a * perfect fluid*, i.e. a fluid that is completely described by mass density and isotropic pressure

*. Its energy-momentum tensor is*

**only**(11)

In this, is the mass-density, *p *is the isotropic pressure, is the 4-velocity of the perfect fluid ( which need not be constant ), and is the metric tensor. One can obtain the various quantities associated with such fluids from the energy-momentum tensor as per the prescription above.

The second special case I would like to mention is the one of an electromagnetic field; the stress-energy tensor of such fields is completely specified by the electromagnetic field tensor and the metric tensor :

(12)

Just like we obtained the stress-energy-momentum tensor from global symmetries of space-time, we can also obtain other such quantities, for example a tensor representing angular momentum. However, I will not discuss this here, since those quantities do not directly appear in what we are aiming for at the moment, being General Relativity. I refer the interested reader to appropriate textbooks such as Misner/Thorne/Wheeler.

And this concludes our quick overview on Special Relativity. All of this may seem a bit overwhelming right now, but do remember that the purpose of this article is only to introduce you to the concepts; I can * not* function as a substitute for actual in-depth study of these concepts. You likely won’t be able to perform any actual calculations using the energy-momentum tensor etc just by reading this; however, I hope that you come out of it with a better understanding of what Special Relativity is

*. The main ideas, in a nutshell, are :*

**about**- The classical world is described as a 4-dimensional space-time, being the collection of all events ( points in space at a given time ).
- The relationship between events is given by the geometry of space-time
- In the absence of tidal gravity, space-time is flat ( Minkowski space-time )
- The global symmetries of space-time correspond to locally conserved quantities, such as energy-momentum ( Noether’s theorem )
- All inertial observers experience the same laws of physics; this is the principle of Lorentz invariance, which is a fundamental symmetry of nature

Everything we have done so far was done under the assumption that there are no sources of gravity anywhere in space-time. In the next article, we will take a closer look at the mathematical description of space-time and its geometry; this will provide us with the tools we need to think about how to incorporate gravity into our model of the classical world, and hence to progress onwards into General Relativity.

Stay tuned !

### References :

[1] https://en.wikipedia.org/wiki/Lorentz_transformation

[2] http://www.physicsoftheuniverse.com/images/relativity_time_dilation.jpg

[3] https://www.bnl.gov/rhic/physics.asp

[4] Taylor/Wheeler, “*Space-Time Physics – An Introduction to Special Relativity*“, 2nd Edition

[5] Young/Freedman, “*University Physics, With Modern Physics*“, 13th Edition

[6] http://quantum-bits.org/wp-content/uploads/2010/05/noether.png

[7] Misner/Thorne/Wheeler, “*Gravitation*“, Fig 5.1, page 133

[8] Misner/Thorne/Wheeler, “*Gravitation*“, Box 5.1, page 131

[9] https://en.wikipedia.org/wiki/Stress–energy_tensor

[10] http://gregegan.customer.netspace.net.au/FOUNDATIONS/03/found03.html

Further Reading : Manifolds and Curvature

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