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Thread: Spacetime - A Primer

  1. #1 Spacetime - A Primer 
    Moderator Moderator Markus Hanke's Avatar
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    In light of recent discoveries, much has been talked about spacetime physics, so I thought a few words on the concept of spacetime might be helpful to the general reader here.

    Spacetime quite simply denotes the set of all events; each event needs to have a location ( 3 coordinates ) and a point in time ( 1 coordinate ) to be uniquely specified, hence we say that spacetime is 4-dimensional ( or, more accurately, 3+1 dimensional ). Mathematically, spacetime forms a manifold, i.e. a space which locally in a small neighbourhood looks like Euclidean space. In order to do any meaningful, quantifiable physics in spacetime, we will need two more essential ingredients :

    1. Some way to consistently define the notion of differentiation, i.e. being able to relate tangent spaces ( set of all tangents ) at different points on the manifold to each other. By extension, this allows us to tell what happens if we parallel-transport tangent vectors along closed curves. The mathematical object that allows us to do both of these is called an affine connection; it can be thought of as a generalisation of the partial derivative to general spaces. An affine connection has two very important invariants, being curvature and torsion, on which all observers agree. In currently accepted theories of gravitation, torsion vanishes identically, leaving only curvature to deal with.

    2. We need some way to define measurements in space-time, specifically we need to be able to tell the separation of two events in space-time. The mathematical object that allows us to do so is called a metric. Given a connection and a metric, we can define lengths, separations, areas, volumes, angles etc.

    The metric is given as a mathematical object called a tensor - this can be thought of as a type of function which is defined for each event in spacetime, and which takes an input, processes the input, and produces a real number as a result. For example, if I input two distinct vectors into the metric tensor, I will get their dot product. If I input the same vector twice, I will get that vector's (squared) length; and so on. The defining and beautiful characteristic of tensors is that they are the same for all observers, so if we formulate the laws of physics in terms of tensors, these laws will take the same form for all observers, no matter what their state of relative motion or their position in spacetime is.

    Leaving gravity aside for the minute, we want all observers to experience the same laws of physics, in order to form a consistent model of the world. A closer examination reveals that this is possible only if all observers agree as to the separation of events in spacetime, i.e. all observers should agree on the metric. If we examine this requirement mathematically, we find that the set of all geometric operations which leave the metric unchanged forms a group, the Lorentz group, and its elements are the Lorentz transformations. This immediately gives us Special Relativity, through the simple and intuitive symmetry notion that all observers should see the same laws of physics.

    To include gravity in this, we ask ourselves what gravity does to test particles. Consider two test particles, travelling parallel to each other through space. If there is no gravity, they will just continue on like this forever. However, if gravity is present, their world lines cannot remain parallel - they either diverge or converge over time. How do we express this geometrically ? It turns out the best and most straightforward way to do so is to allow the metric to vary from point to point in spacetime; the resulting theory is called General Relativity. It is a model that allows us to determine what the metric of spacetime is in the presence of sources of gravity. Those sources turn out to be a tensor quantity as well, the energy-momentum tensor ( which is in itself a result of symmetry considerations, but I will skip this for now ); all forms of energy form sources of gravity, and lead to changes in the metric.

    The metric can now be the same everywhere ( Special Relativity ), or it can vary according to location ( General Relativity ). If we allow it to also vary with time, we get changes in gravity, which, if they are periodic, yield gravitational waves / radiation. Such waves are very similar to electromagnetic waves, but they are to be understood as oscillations of the geometry of space-time itself. Also, the source of gravitational radiation are quadrupole moments ( as opposed to dipole moments in EM ), and their polarisation states are inclined at an angle of 45 degrees ( as opposed to 90 degrees in the case of EM ). Due to the latter, gravitational waves give some unique effects, such as the B-mode polarisations of the CMBR, which cannot be produced in any other way.

    This brings us full circle to the results of the south pole observatory, and why they are so important. If it passes peer-review and independent verification, it will become an essential confirmation of the predictions of the theory of relativity, as well as the standard model of particle physics.

    Any questions - ask away


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    Thank you once again for the clear description of the relationship between mathematical objects and space time, Markus.
    I looked up Einsteins field equations and found the following lecture on YT: Einstein Field Equations - for beginners! - YouTube

    Is this a good breakdown of the maths? I know YT isn't too popular as an information source, so I thought I'd ask...


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    Quote Originally Posted by RobinM View Post
    Thank you once again for the clear description of the relationship between mathematical objects and space time, Markus.
    I looked up Einsteins field equations and found the following lecture on YT: Einstein Field Equations - for beginners! - YouTube

    Is this a good breakdown of the maths? I know YT isn't too popular as an information source, so I thought I'd ask...
    Ok, it's a rather lengthy video at over two hours, and I don't really have the time to watch it at the moment. I kind of scrolled through it though, and it appears to be alright. I'd say give it a go.

    Where do these equations fit into the bigger picture ? I mentioned in my OP that it is the presence of sources of energy-momentum which change the geometry of space-time, i.e. the metric tensor; and that's exactly what the field equations tell us ( cosmological constant omitted for clarity ) :



    The is the energy-momentum tensor, is a proportionality constant, and is the Einstein tensor, which is a measure for the curvature of space-time. The equations hence simply tell us that curvature is equal to energy-momentum, up to a proportionality constant. The Einstein tensor depends on the components of the metric tensor in a complicated and non-linear way; given an energy-momentum tensor, these equations thus enable us to find a corresponding metric of space-time. There is no standard procedure for solving them, and no general solution, and not many closed analytical solutions are known due to the inherent complexity. We can treat simple and highly symmetric cases though, and more cases still can be treated numerically.
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    Thanks Markus. I got about half-way through before I was overwhelmed with my lack of mathematical understanding. I feel as though I need to study derivatives and calculus in general in much greater depth before I can begin to comprehend these equations, but I was able to follow quite a bit further than I could before, so that's something.

    Back to the books!

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    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by RobinM View Post
    Back to the books!
    If you can at all afford it, I can't recommend "Gravitation" by Misner/Thorne/Wheeler highly enough; it's considered the gold standard for General Relativity for a good reason. Some of the observational data used in the book is obviously outdated, but the maths and physics are as valid as ever. You should consider it - having said that, a good knowledge of single and multivariable calculus is needed for this, as it is for any textbook on GR.
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    I had quite often heard (or misinterpreted perhaps)that Gravity doesn't really exist and that that instead SpaceTime became distorted in the presence of massive bodies.

    From what you are saying can I realign my understanding to "Gravity does exist and is poorly understood.However it has been mathematically modeled with a very high degree of accuracy" ?

    And I have also come to believe that Space Time itself is not "distorted" as a "thing" since it only exists as a mathematical model (which is indeed distorted ).

    Is my understanding right so far? Can I forget the whole "Gravity doesn't exist" thing? Is it more like Gravity =Reality / SpaceTime = Representation of Reality

    thanks and if I am confusing or incoherent I apologize.
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    Moderator Moderator Markus Hanke's Avatar
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    Quote Originally Posted by geordief View Post
    Gravity doesn't really exist
    Not sure where you heard that, but gravity definitely exists.

    and that that instead SpaceTime became distorted in the presence of massive bodies
    Essentially correct - it is the distortion ( curvature ) which manifests as gravity.

    Gravity does exist and is poorly understood
    No, it is very well understood, at least in the classical domain.

    it has been mathematically modeled with a very high degree of accuracy
    Yes, which is why it is well understood.

    And I have also come to believe that Space Time itself is not "distorted" as a "thing" since it only exists as a mathematical model (which is indeed distorted ).
    No, curvature isn't "just" a mathematical gimmick, it is an empirically measureable quantity. See for example Pound, or Shapiro delay - Wikipedia, the free encyclopedia. And space-time definitely exists, or else we wouldn't be here to discuss this
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    thanks - I hope that will give me a bit more to get my teeth into.
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