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Thread: Intrinsic curvature

  1. #1 Intrinsic curvature 
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    I have a question about whether a line can be intrinsically curved.

    Is it a nonsense ? Is intrinsic curvature only a property of 2 dimensional (or higher) surfaces?

    Also is a mathematical surface the set of points where a n-dimensional object intersects with a n+1 dimensional object ?(object ="set of points"?)


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    Quote Originally Posted by geordief View Post
    I have a question about whether a line can be intrinsically curved.

    Is it a nonsense ? Is intrinsic curvature only a property of 2 dimensional (or higher) surfaces?
    A line cannot be intrinsically curved. Intrinsic curvature only exists for spaces of two or more dimensions. Also, two-dimensional spaces can only possess Ricci scalar curvature. Three-dimensional spaces can only possess Ricci tensor curvature (including Ricci scalar curvature). Spaces of at least four dimensions are required for the space to possess Weyl tensor curvature. Spaces of four or more dimensions can possess the complete Riemann curvature and there are no more types of curvature for spaces of greater than four dimensions.


    There are no paradoxes in relativity, just people's misunderstandings of it.
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    Is it more correct(or just different) to talk of an n-dimensional space rather than a n-dimensional surface?

    If they are different concepts,then is there a general relationship between the space and the surface?

    If the surface of a ball is 2-dimensional does that mean it has to be completely flat ?(as in an Earth with no hills)

    Does including hills turn this surface into a 3-dimensional surface?
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  5. #4  
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    Quote Originally Posted by geordief View Post
    Is it more correct(or just different) to talk of an n-dimensional space rather than a n-dimensional surface?

    If they are different concepts,then is there a general relationship between the space and the surface?
    One would typically use the term "surface" only for spaces that are embedded in a higher-dimensional space.


    Quote Originally Posted by geordief View Post
    If the surface of a ball is 2-dimensional does that mean it has to be completely flat ?(as in an Earth with no hills)

    Does including hills turn this surface into a 3-dimensional surface?
    No, the surface is still 2-dimensional.
    There are no paradoxes in relativity, just people's misunderstandings of it.
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  6. #5  
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    Quote Originally Posted by KJW View Post
    Quote Originally Posted by geordief View Post
    Is it more correct(or just different) to talk of an n-dimensional space rather than a n-dimensional surface?

    If they are different concepts,then is there a general relationship between the space and the surface?
    One would typically use the term "surface" only for spaces that are embedded in a higher-dimensional space.

    You say "typically". Can surfaces also not be embedded in a higher-dimensional space?


    Quote Originally Posted by geordief View Post
    If the surface of a ball is 2-dimensional does that mean it has to be completely flat ?(as in an Earth with no hills)

    Does including hills turn this surface into a 3-dimensional surface?
    No, the surface is still 2-dimensional.
    Is a 3-dimensional surface embedded in a 4-dimensional space and as a result not something we would encounter in our normal experience? (except perhaps as a mathematical representation of an aspect of Spacetime)
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  7. #6  
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    Quote Originally Posted by geordief View Post
    You say "typically". Can surfaces also not be embedded in a higher-dimensional space?
    I think it should be noted that people who are using the term "surface" for anything other than a two-dimensional surface do not have strictness in mind, and are using the term as an analogy to a two-dimensional surface. I say "typically" because I do not claim exhaustive knowledge of all the possible ways in which a person could use a two-dimensional surface as an analogy. For example, one might speak of a coordinate system being overlayed onto a space, thus using the language of a surface even though the space is not embedded in a higher-dimensional space. In this case, it is a different analogy.


    Quote Originally Posted by geordief View Post
    Is a 3-dimensional surface embedded in a 4-dimensional space and as a result not something we would encounter in our normal experience?
    A three-dimensional surface is not something that we would encounter in our normal experience. Also, some people would use the term "three-dimensional hypersurface" instead to describe the same thing.
    There are no paradoxes in relativity, just people's misunderstandings of it.
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    Quote Originally Posted by geordief View Post
    I have a question about whether a line can be intrinsically curved.
    Yes it can (in spite of what others have said). According to the so-called Riemann geometry, which he himself called "intrinsic geometry", there is no need of an embedding space. That is, curvature is a property of the "object" itself, not relative to anything else. If you want the mathematics, it's free here.

    Is it a nonsense ? Is intrinsic curvature only a property of 2 dimensional (or higher) surfaces?
    No it's not nonsense, but it is incorrect. Worryingly, but correctly, a manifold of any dimension has the exact same properties as a space as one of any other dimension. This includes a 1-dimensional manifold, one of which is a line (the circle is a less trivial example). These have intrinsic curvature.

    Again, I can serve up the meat and two veg if you want, but the mathematics is challenging (to say the least)

    PS I admire your curiosity about science, but I would advise you to restrict yourself to an in-depth study of a single area.
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    Quote Originally Posted by Guitarist View Post
    Quote Originally Posted by geordief View Post
    I have a question about whether a line can be intrinsically curved.
    Yes it can (in spite of what others have said). According to the so-called Riemann geometry, which he himself called "intrinsic geometry", there is no need of an embedding space. That is, curvature is a property of the "object" itself, not relative to anything else. If you want the mathematics, it's free here.
    How can a 1-dimensional space have intrinsic curvature, given that the Riemann tensor is identically zero everywhere on it? The curvature of a line that is given by the covariant derivative of the tangent vector is not actually intrinsic because it does depend on the embedding space to provide the connection.
    There are no paradoxes in relativity, just people's misunderstandings of it.
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    I just want to say that this thread is getting pretty bloody cool, and that there are fascinated lurkers.
    Carry on please.
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    Lucky me. Lucky mud.
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    Quote Originally Posted by KJW View Post
    How can a 1-dimensional space have intrinsic curvature,
    So what? Either the circle is not a manifold, or that it is, but "flat"
    given that the Riemann tensor is identically zero everywhere on it {i.e the circle}?
    This is an assertion in desperate search of a proof
    The curvature of a line that is given by the covariant derivative of the tangent vector is not actually intrinsic because it does depend on the embedding space to provide the connection.
    This is not the definition that I am used to. For example spacetime is a 4-manifold with a connection, but you will struggle to find the embedding space.

    Please explain your thinking
    Last edited by Guitarist; October 27th, 2017 at 01:37 PM.
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  12. #11  
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    Quote Originally Posted by Guitarist View Post
    Quote Originally Posted by KJW View Post
    How can a 1-dimensional space have intrinsic curvature,
    So what? Either the circle is not a manifold, or that it is, but "flat"
    A circle is flat, just as a cylinder is a flat 2-dimensional surface.


    Quote Originally Posted by Guitarist View Post
    Quote Originally Posted by KJW View Post
    given that the Riemann tensor is identically zero everywhere on it {i.e the circle}?
    This is an assertion in desperate search of a proof
    It's a consequence of the algebraic properties of where . Clearly, if then which for is the entire Riemann tensor.


    Quote Originally Posted by Guitarist View Post
    Quote Originally Posted by KJW View Post
    The curvature of a line that is given by the covariant derivative of the tangent vector is not actually intrinsic because it does depend on the embedding space to provide the connection.
    So where does it end? Spacetime has a connection, but I struggle to see the embedding space.

    Please explain
    I don't follow. I said that one-dimensional spaces can't have intrinsic curvature. I didn't say that higher-dimensional spaces can't have intrinsic curvature. Indeed, I discussed intrinsic curvature for spaces of two or more dimensions.

    However, if one is considering a line in spacetime, then it is the spacetime that is the embedding space.
    There are no paradoxes in relativity, just people's misunderstandings of it.
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    Quote Originally Posted by KJW View Post

    A circle is flat, just as a cylinder is a flat 2-dimensional surface.
    What?? The circle is flat in the topological sense? That is, it is globally homeomorphic to the 1-plane? I really do not think so. Check your texts
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  14. #13  
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    Quote Originally Posted by Guitarist View Post
    Quote Originally Posted by KJW View Post
    A circle is flat, just as a cylinder is a flat 2-dimensional surface.
    What?? The circle is flat in the topological sense? That is, it is globally homeomorphic to the 1-plane? I really do not think so. Check your texts
    No one mentioned topology. I was discussing curvature, which is a local concept.
    There are no paradoxes in relativity, just people's misunderstandings of it.
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    Quote Originally Posted by KJW View Post
    No one mentioned topology.
    They didn't need to - curvature is a property of manifolds which are of course topological spaces
    curvature, which is a local concept.
    No it is not. Any manifold known to man, beast or bacterium is locally indistinguishable from some and is, of course, locally "flat".

    But the curvature of a manifold (I repeat this is a topological space with an additional property) need not be globally flat. It usually isn't

    Try this.......

    Suppose a line (do not assume it is a straight line in the Euclidean sense). Take a vector at some arbitrary point . Take another distinct, but arbitrary point and a tangent vector there. Then if and only if the tangent vectors at and coincide i.e are parallel, then one says the curvature is zero.
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  16. #15  
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    Quote Originally Posted by Guitarist View Post
    Quote Originally Posted by KJW View Post
    No one mentioned topology.
    They didn't need to - curvature is a property of manifolds which are of course topological spaces

    curvature, which is a local concept.
    No it is not. Any manifold known to man, beast or bacterium is locally indistinguishable from some and is, of course, locally "flat".

    But the curvature of a manifold (I repeat this is a topological space with an additional property) need not be globally flat. It usually isn't
    No, because a general topological manifold is not equipped with curvature. That requires additional structure, specifically a connection. Therefore, homeomorphisms are not the appropriate mappings that define the equivalence classes relevant to the OP's question.


    Quote Originally Posted by Guitarist View Post
    Suppose a line (do not assume it is a straight line in the Euclidean sense). Take a vector at some arbitrary point . Take another distinct, but arbitrary point and a tangent vector there. Then if and only if the tangent vectors at and coincide i.e are parallel, then one says the curvature is zero.
    Since we discussing the line from an intrinsic perspective, the two tangent vectors do coincide. Thus, they are parallel and the curvature is zero. The implicit notion of parallel transport to which you refer is a property of the embedding space, not the line itself.
    There are no paradoxes in relativity, just people's misunderstandings of it.
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    Quote Originally Posted by KJW View Post
    Quote Originally Posted by geordief View Post
    I have a question about whether a line can be intrinsically curved.

    Is it a nonsense ? Is intrinsic curvature only a property of 2 dimensional (or higher) surfaces?
    A line cannot be intrinsically curved. Intrinsic curvature only exists for spaces of two or more dimensions. Also, two-dimensional spaces can only possess Ricci scalar curvature. Three-dimensional spaces can only possess Ricci tensor curvature (including Ricci scalar curvature). Spaces of at least four dimensions are required for the space to possess Weyl tensor curvature. Spaces of four or more dimensions can possess the complete Riemann curvature and there are no more types of curvature for spaces of greater than four dimensions.
    Am I right to assume that all those types of(intrinsic?)curvature are mathematical definitions and only apply to physical scenarios to the extent that they can model them successfully?

    When it comes to physical scenarios ,would I also be right to say that ,without exception every physical object actually possesses intrinsic curvature? (even in the hypothetical absence of gravitational fields)

    To reassure myself that I am getting my ducks in a row ,when discussing intrinsic curvature are we here modeling physical /and or mathematical objects as sets of points**?

    **(that's a manifold,I think)
    Last edited by geordief; October 31st, 2017 at 07:24 AM.
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    I agree with GiantEvil. However it is a pity that DrRocket is not present for this fascinating exchange.
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  19. #18  
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    Quote Originally Posted by geordief View Post
    Am I right to assume that all those types of(intrinsic?)curvature are mathematical definitions and only apply to physical scenarios to the extent that they can model them successfully?
    It should be noted that the curvature of spacetime is a fact and not just an abstract theoretical notion that only applies to mathematical models of reality rather than physical reality itself. Being mathematically defined doesn't restrict a notion to mathematical models if one can define a mapping from the mathematics to the physics. For example, distances in spacetime can be physically measured (and hence defined) by rulers and clocks. Then using the mathematical expressions for curvature in terms of the metric, one can calculate the values of the curvature of physical spacetime. But one does need to establish a correspondence between the mathematical realm and the physical realm.


    Quote Originally Posted by geordief View Post
    When it comes to physical scenarios ,would I also be right to say that ,without exception every physical object actually possesses intrinsic curvature? (even in the hypothetical absence of gravitational fields)
    Curvature is a property of spaces. But the Einstein equation does say that the mathematical quantity called the Einstein tensor field is equivalent to the physical quantity called the energy-momentum density tensor field. Even though energy-momentum is associated with physical objects, the Einstein tensor field is still a property of spacetime.


    Quote Originally Posted by geordief View Post
    To reassure myself that I am getting my ducks in a row ,when discussing intrinsic curvature are we here modeling physical /and or mathematical objects as sets of points**?

    **(that's a manifold,I think)
    The notion of intrinsic curvature goes way beyond the notion of "sets of points". It should be noted that the debate between Guitarist and myself was a debate about the appropriate level of generality to apply to the discussion. Mathematical notions are defined in terms of satisfying a given set of axioms. However, if one has the definition of a particular mathematical notion, then one can define specialised forms of that notion by adding more axioms to the original definition. Even a notion as general as topological spaces goes beyond mere sets of points. Manifolds are more special than topological spaces, but even these do not have the axioms necessary to equip the notion with curvature. Manifolds require additional axioms that define a connection in order for them to be equipped with the notion of curvature. However, I've gone a little bit further and considered not just manifolds equipped with a connection but manifolds equipped with a metric, as it is these that correspond to macroscopic physical spacetime.
    Last edited by KJW; November 3rd, 2017 at 11:27 AM.
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    Quote Originally Posted by KJW View Post



    Curvature is a property of spaces. But the Einstein equation does say that the mathematical quantity called the Einstein tensor field is equivalent to the physical quantity called the energy-momentum density tensor field. Even though energy-momentum is associated with physical objects, the Einstein tensor field is still a property of spacetime.

    What about objects like fluids?Can they be modeled using the concept of curvature? Can the motion of an object through a variably dense medium also be modeled like this?

    Quote Originally Posted by geordief View Post
    To reassure myself that I am getting my ducks in a row ,when discussing intrinsic curvature are we here modeling physical /and or mathematical objects as sets of points**?

    **(that's a manifold,I think)

    The notion of intrinsic curvature goes way beyond the notion of "sets of points".
    It should be noted that the debate between Guitarist and myself was a debate about the appropriate level of generality to apply to the discussion. Mathematical notions are defined in terms of satisfying a given set of axioms. However, if one has the definition of a particular mathematical notion, then one can define specialised forms of that notion by adding more axioms to the original definition. Even a notion as general as topological spaces goes beyond mere sets of points. Manifolds are more special than topological spaces, but even these do not have the axioms necessary to equip the notion with curvature. Manifolds require additional axioms that define a connection in order for them to be equipped with the notion of curvature. However, I've gone a little bit further and considered not just manifolds equipped with a connection but manifolds equipped with a metric, as it is these that correspond to macroscopic physical spacetime.
    A bit over my head (interesting nonetheless) ,but do you think you could expand a tiny bit on the part I have bolded? Can intrinsic curvature be modeled in ways that are unrelated to "sets of points"?
    Last edited by geordief; November 4th, 2017 at 10:47 AM.
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  21. #20  
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    Quote Originally Posted by geordief View Post
    What about objects like fluids? Can they be modelled using the concept of curvature? Can the motion of an object through a variably dense medium also be modelled like this?
    Fluids can be modelled by specifying the energy-momentum field, then applying the Einstein equation.


    Quote Originally Posted by geordief View Post
    A bit over my head (interesting nonetheless) ,but do you think you could expand a tiny bit on the part I have bolded?
    I actually did expand on that. But if you would like further details, have a look at topological spaces.


    Quote Originally Posted by geordief View Post
    Can intrinsic curvature be modelled in ways that are unrelated to "sets of points"?
    I'm not sure of what you are expecting, but I prefer to downplay the geometrical aspect of curvature in favour of the analytical╣ aspect. The Riemann curvature tensor is the obstruction to an arbitrary connection object being coordinate-transformed to zero. This can be regarded as the necessary and sufficient condition for the existence of a solution to a particular system of differential equations.


    ╣ Calculus, as distinct from algebra.
    There are no paradoxes in relativity, just people's misunderstandings of it.
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    Quote Originally Posted by KJW View Post
    Quote Originally Posted by geordief View Post
    What about objects like fluids? Can they be modelled using the concept of curvature? Can the motion of an object through a variably dense medium also be modelled like this?
    Fluids can be modelled by specifying the energy-momentum field, then applying the Einstein equation.
    And do those models incorporate intrinsic curvature?(it sounds like they might but I don't understand the Einstein equation or how it is arrived at or applied ...)

    The Einstein equation is applicable to seemingly mundane** problems like modeling fluids and the passage of objects through non homogeneous fluids ?

    **I mean mundane compared to modeling gravitational fields which I assume is what it is most known for.
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  23. #22  
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    Quote Originally Posted by geordief View Post
    And do those models incorporate intrinsic curvature?(it sounds like they might but I don't understand the Einstein equation or how it is arrived at or applied ...)
    Yes, general relativity is based on intrinsic curvature. And given that we have no access to outside of spacetime, spacetime curvature has to be intrinsic╣. The Einstein equation is a statement of the equivalence between physical energy-momentum and mathematical Einstein curvature (which I can't explain in non-mathematical terms). Given the energy-momentum, the Einstein curvature is obtained directly. But to obtain the spacetime metric, from which the total (Riemann) curvature is calculated, one needs to solve the equation.


    Quote Originally Posted by geordief View Post
    The Einstein equation is applicable to seemingly mundane** problems like modelling fluids and the passage of objects through non homogeneous fluids ?

    **I mean mundane compared to modelling gravitational fields which I assume is what it is most known for.
    Once one has a specification of the energy-momentum, then the Einstein equation can be applied, though solving the equation might not be so easy.


    ╣ Two interesting question are: (1) Given the metric of a space of some dimension, what higher-dimensional spaces can this space be embedded in? (2) Given the metric of a space of some dimension, what lower-dimensional spaces can be embedded in this space? In the case where the two spaces have the same dimension, embedding one into the other is the same as a coordinate transformation and can only be done if the Riemann curvature is the same (in the appropriate sense) for both spaces.
    There are no paradoxes in relativity, just people's misunderstandings of it.
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    Quote Originally Posted by KJW View Post
    Yes, general relativity is based on intrinsic curvature. And given that we have no access to outside of spacetime, spacetime curvature has to be intrinsic╣. The Einstein equation is a statement of the equivalence between physical energy-momentum and mathematical Einstein curvature (which I can't explain in non-mathematical terms).
    Is it the equivalence that is hard to explain or the way curvature is described in the spacetime model?

    Does the measurement process bear any comparison with,say how one might measure the curvature of a simple sphere?

    Does one measure distances in various directions in spacetime and define curvature as some kind of a function of them?(with sources of energy-momentum causing particular changes to the distances)
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  25. #24  
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    Quote Originally Posted by geordief View Post
    Is it the equivalence that is hard to explain or the way curvature is described in the spacetime model?
    I apologise for the ambiguity. What is hard to explain in non-mathematical terms is "Einstein curvature" (and distinguishing between the different curvature tensors).


    Quote Originally Posted by geordief View Post
    Does the measurement process bear any comparison with, say how one might measure the curvature of a simple sphere?
    Apart from the differences associated with the increased number of dimensions, the curvature of spacetime is essentially the same as the curvature of a sphere. However, the curvature of a sphere can be viewed extrinsically because the embedding in three-dimensional space is easy to visualise, whereas we don't visualise spacetime as embedded in anything. But note that a sphere only has simple scalar curvature, whereas spacetime has the full curvature that only exists in four or more dimensions.


    Quote Originally Posted by geordief View Post
    Does one measure distances in various directions in spacetime and define curvature as some kind of a function of them?
    Yes. The metric is a generalised Pythagorean expression that specifies the distance between infinitesimally separated points specified by their coordinates. The coefficients of the dxudxv terms╣ collectively form the metric tensor and are functions of the coordinates. The curvature tensors contain second-order partial derivatives of the metric tensor.


    ╣ The superscripts simply specify the components of a vector and are not exponents.
    There are no paradoxes in relativity, just people's misunderstandings of it.
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    Quote Originally Posted by KJW View Post


    Quote Originally Posted by geordief View Post
    Does the measurement process bear any comparison with, say how one might measure the curvature of a simple sphere?
    Apart from the differences associated with the increased number of dimensions, the curvature of spacetime is essentially the same as the curvature of a sphere. However, the curvature of a sphere can be viewed extrinsically because the embedding in three-dimensional space is easy to visualise, whereas we don't visualise spacetime as embedded in anything. But note that a sphere only has simple scalar curvature, whereas spacetime has the full curvature that only exists in four or more dimensions.
    If we choose a distance in spacetime such as 1 lightyear and make a map of all the events in space that would have been judged(ie potentially observed) by someone on (as an example) the Earth as having occured then **,would that set of events constitute a surface in spacetime?

    And if the original spacetime distance (1 ly) was reduced or increased by an infinitesimal amount ,would the change in that surface be a method of calculating the curvature in spacetime produced by a massive object in the vicinity ?


    **the observer would make an inventory of all events that had taken place and only select those which had apparently occurred "simultaneously" with the arrival of a beam of light originating 1 year in the past(a hypersurface of simultaneity?)

    EDIT: How many dimensions are needed to model/describe that surface? (the set of all events appearing to an observer as having occurred at the same time)
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