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<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><font color="#ff0000"><font face="Verdana, sans-serif"><b>E<img src="data:image/png;base64,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" name="TtsOtkCLGtr_01" align="right" hspace="5" width="175" height="60" border="0">instein’s
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general theory of relativity. </b></font></font>By showing that
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spatiomaterialism can explain the truth of Einstein’s special
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theory of relativity (STR), I have answered the first part of the
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Einsteinian reservation about using spatiomaterialism as the
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foundation for demonstrating ontologically necessary truths. In this
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section, I will answer the second part. Einstein’s general theory
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of relativity (GTR) also makes it appear that this is not a
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spatiomaterial world, and I will show how its truth can also be
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explained by spatiomaterialism.</font></font></font></p>
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<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">The
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way Einstein’s general theory of relativity explains gravitation
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does not, at first, seem compatible with spatiomaterialism. The
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foundation of the general theory is spacetime, for gravitation is
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explained as a “curvature” in spacetime, and since
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substantivalism about spacetime is incompatible with substantivalism
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about space, it seems out of the question that what the general
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theory refers to as “curved spacetime” could turn out to be an
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aspect of space and matter as substances enduring through time. (For
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a very accessible account of Einstein's general theory of relativity,
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see Clifford M. </span></font></font></font><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Will86"><font color="#0000ff"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US"><u>Will</u></span></font></font></font></a><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">'s
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</span></font></font></font><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US"><i>Was
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Einstein Right?)</i></span></font></font></font></p>
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<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">It
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is, however, possible for spatiomaterialism to explain why Einstein’s
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general theory of relativity is true. The key is what spacetime turns
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out to be in the ontological explanation of the truth of the special
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theory of relativity, for that makes it possible to explain curved
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spacetime as well. Curved spacetime is also an aspect of space and
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matter, even though as substances that endure through time, space and
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matter exist only at the present moment. </font></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><span lang="en-US">Though
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I go on in the next section to suggest an ontological explanation of
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quantum mechanics and, in the following section, take up some basic
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issues in cosmology, this explanation of the Einstein’s general
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theory of relativity pays off the second mortgage that we took out in
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order to use spatiomaterialism as the foundation for our
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philosophical argument. (See </span></font></font><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/L/LoOtgNtMort.htm" target="Lo"><font color="#0000ff"><font face="Arial, sans-serif"><span lang="en-US"><u>Necessary
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Truths</u></span></font></font></a><font color="#000000"><font face="Arial, sans-serif"><span lang="en-US">.)</span></font></font><font color="#000000"><span lang="en-US">
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</span></font><font color="#000000"><font face="Times New Roman, serif"><span lang="en-US">Quantum
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mechanics is not so crucial to this project, because there is
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continuing disagreement about its ontological implications and some
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of the possibilities are compatible with spatiomaterialism. </span></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif">We have
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already seen how the existence of consciousness can be explained in a
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spatiomaterial world (though the unity of consciousness will not be
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explained until I take up the mammalian brain in the sixth stage of
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evolution), and I have yet to take up the nature of goodness and
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holiness. But one of those four mortgages will be repaid when we see
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that Einsteinian physics provides not reason for denying that this is
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a spatiomaterial world. </font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">In
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fact, spatiomaterialism might welcome the challenge of explaining
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Einstein’s general theory of relativity, because that means it does
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not have to defend Newton’s theory of gravitation. Newton’s
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theory is <i>prima facie </i>less hospitable to spatiomaterialism
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than general relativity. If a force did act immediately at a
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distance, it would contradict the principle of local action, implying
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that spatiomaterialism is false. </font></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif">Newton’s
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theory describes an attractive force by which every material object
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acts immediately on every other material object, including those at a
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distance. Newton introduced it, in effect, as the best
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efficient-cause explanation of Kepler’s laws of planetary orbits,
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and it was confirmed by the deduction of many surprising,
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quantitatively precise predictions of measurements, becoming the
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model for the empirical method in physics. Despite its predictive
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success, Newton’s law of gravitation had nothing to say about how
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such forces are exerted on objects at a distance, except that they
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act instantaneously at a distance. </font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif">Action at a
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distance was puzzling to classical physicists, since it did not fit
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well with their intuitive understanding of nature as composed of
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space and matter in time. Even Newton was uncomfortable with the
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notion, and he refused to make any hypotheses about how gravitation
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worked in his <i>Principia</i>.<sup><a class="sdendnoteanc" name="sdendnote1anc" href="#sdendnote1sym"><sup>i</sup></a></sup>
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But action at a distance could not be rejected for being incompatible
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with spatiomaterialism, for that would require using space as an
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ontological cause, and Newtonian physics did not recognize the
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validity of ontological arguments. Still, when Einstein proposed an
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explanation of gravitation that implied that gravitational forces
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propagate at a finite velocity, even physicists were relieved at not
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having to believe in action at a distance. And it did remove what
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would otherwise be an insuperable objection to spatiomaterialism. </font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif">Einstein’s
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general theory of relativity was, however, another highly
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mathematical hypothesis, which predicted many quantitatively precise
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measurements, and since it implies that gravitational acceleration is
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caused by a curvature of spacetime, a realist interpretation of
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Einstein’s theory seems to imply that spacetime is a substance. But
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if the real nature of what exists in addition to mass and energy is
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spacetime, that is, a four-dimensional entity in which time is one of
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the dimensions along with space, then existence is not in time and
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“real change” is not ontologically possible. Thus, general
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relativity solved one ontological problem, but only by introducing
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another. The challenge is, therefore, to explain how curved spacetime
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can be understood as an aspect of a world constituted by space and
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matter as substances that exist only at the present moment. </font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><font color="#993366"><font face="Verdana, sans-serif"><b>C<img src="data:image/png;base64,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" name="TtsOtkCLGtr_02" align="right" hspace="5" width="225" height="36" border="0">urved
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spacetime. </b></font></font>Having discovered STR by assuming the
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local equivalence of all inertial frames, Einstein sought to use the
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same approach in explaining acceleration due to gravity, that is, by
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including reference frames that were being accelerated by
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gravitation. Thus, the main assumption of his general theory of
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relativity is the equivalence of inertial frames to reference frames
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falling freely in gravitational fields. </font></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">What
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Einstein himself called the “principle of equivalence” assumes
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that nothing can be detected within any reference frame (that is,
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locally) that would distinguish a reference frame in inertial motion
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from one in free fall. </font></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif">Or, to put
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Einstein’s equivalence principle the opposite way, a reference
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frame at rest in a gravitational field is indistinguishable from one
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being accelerated by a force; the push that we ordinarily call the
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“force” of gravity is actually the force of the earth
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accelerating us upward from what is equivalent to inertial motion. </font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">This
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further equivalence can be only local, however, because free-falling
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frames are obviously different in how they are related to the rest of
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the world, or globally. Though inertial frames simply continue in
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motion indefinitely, free-falling reference frames eventually collide
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with the center of gravity, because gravitational fields are imposed
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by matter concentrated at certain locations. Thus, what makes the
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general theory of relativity general is that it includes both
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inertial and free-falling reference frames, and Einstein’s highly
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mathematical description of how they fit together as parts of a
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single world is a theory of acceleration due to gravity. </font></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif">Einstein’s
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strategy in GTR paralleled that of his special theory. In STR,
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Einstein used his principle of relativity (implying the equivalence
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of all inertial frames) to derive a mathematical description of how
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they must be related globally (the Lorentz transformation equations).
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In his general theory, Einstein started with the assumption that
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reference frames in free fall are locally equivalent to inertial
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frames, and using the four-dimensional, spacetime mathematics from
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special relativity, he derived equations describing how all reference
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frames, inertial and free-falling frames, are related to one another.
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In both theories, the equivalence of reference frames means that the
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laws of physics hold the same way on each of them. That means that
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there is a mathematical transformation of explanations of events
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given on any one reference frame into explanations given on the other
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in which the laws of physics have the same form. In special
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relativity, only a Lorentz transformation was required, making them
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Lorentz covariant. But in general relativity, it is a more general
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transformation, which includes both inertial frames and free-fall
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frames, called “general covariance”. How objects change their
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motion depends on centers of mass in their neighborhoods, and using
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general covariance as a constraint, Einstein was able to deduce
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equations that describe what classical physics attributed to a force
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of gravity. </font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Einstein’s
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general theory of relativity describes a spacetime world in which the
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accumulation of matter (both mass and energy) causes a “curvature”
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in the surrounding spacetime. This curvature explains the
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acceleration that Newtonian physics attributed to a force of
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gravitation, because it determines, in turn, the inertial path for
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any matter located there. (Such an inertial path though curved
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spacetime is called a “geodesic”). </font></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">GTR
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also predicts various new phenomena, including the bending (and
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slowing down) of light rays passing through gravitational fields, the
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precession of the perihelion of Mercury, and a gravitational red
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shift. These predictions all differ from classical physics, and since
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GTR entails the possibility of black holes, including rotating black
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holes, it has become the foundation of cosmology. Except for the
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precession of Mercury’s perihelion, these phenomena were not even
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expected before Einstein’s argument, much less explained, and so
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the confirmation of these predictions justified accepting the general
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theory by the empirical method of physics. </font></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Realism
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about the general theory of relativity, like realism about the
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special theory, makes it hard to avoid thinking of spacetime as a
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substance on a par with what it contains. The curvature of space­time
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is supposed to <i>cause </i>the acceleration of mater that is
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ordinarily attributed to gravity, and it would be hard to explain how
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a property of spacetime can have such an effect on what it contains,
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if spacetime did not exist independently of matter.</font></font></font></p>
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<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif">GTR is,
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like STR, a highly mathematical theory. Gravitation is described by
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the Einstein field equations, which relate the distribution of mass
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and non-gravitational energy to the curvature of spacetime.
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Currently, GTR is usually interpreted in terms of differential
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geometry. Spacetime is postulated as a four-dimensional continuous
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manifold of points (<i>M</i>), and there are two kinds of (tensor)
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equations defined everywhere on the manifold. The metric-field tensor
|
||
(<i>g</i>) defines the metric (and geometric) relations among points
|
||
in spacetime, and the stress-energy tensor (<i>T</i>) represents the
|
||
distribution of matter (mass and energy) in spacetime (and its
|
||
effects). </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Jointly, <i>M,
|
||
g, </i>and <i>T</i> are called a “model” of GTR, and even for a
|
||
world with a particular distribution of mass and energy, there are
|
||
infinitely many different, yet empirically equivalent models. They
|
||
all predict the same gravitational phenomena, but each model involves
|
||
a different coordinate system, for each is based on a different local
|
||
inertial reference frames at its location in spacetime, that is,
|
||
adapted to material objects with different free-fall trajectories.<sup><a class="sdendnoteanc" name="sdendnote2anc" href="#sdendnote2sym"><sup>ii</sup></a></sup>
|
||
Their empirical equivalence is an assumption that Einstein used to
|
||
derive his field equations, and it is one of the meanings sometimes
|
||
given to “general relativity”. On this geometrical approach, GTR
|
||
also seems to imply substantivalism about spacetime, because the
|
||
four-dimensional manifold of points (<i>M</i>) must be postulated in
|
||
order to define the metric-field tensor (g) and stress-energy tensor
|
||
(T).<sup><a class="sdendnoteanc" name="sdendnote3anc" href="#sdendnote3sym"><sup>iii</sup></a></sup></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">The
|
||
challenge that GTR poses for spatiomaterialism is that it implies
|
||
that what exists is spacetime, rather than space and matter existing
|
||
as substances in time. In a spacetime ontology, time is another
|
||
dimension of what exists on a par with the spatial dimensions (except
|
||
for a change in sign and the velocity of light as a scaling factor).
|
||
Its implications about time were used in </span></font></font></font><font color="#0000ff"><font face="Arial, sans-serif"><font size="3" style="font-size: 12pt"><span lang="en-US"><u><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/L/LoOtfSTime.htm" target="Lo">Spatiomaterialism</a><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/L/LoOtfSTime.htm">:
|
||
Time</a></u></span></font></font></font><font color="#000000"><span lang="en-US">
|
||
</span></font><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">to
|
||
show that spatiomaterialism is a better ontological explanation of
|
||
nature than spacetime ontology (or “spatiotemporalism”).
|
||
Substantivalism about spacetime makes it impossible to explain “real
|
||
change”, because if what exists is a four-dimensional entity, and
|
||
time is part of its structure, then nothing can be coming into
|
||
existence or going out of existence as time passes. </span></font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">As we saw
|
||
in <font face="Arial, sans-serif">Spatiomaterialism, </font>there is
|
||
no way for spacetime substantivalism to avoid refutation by the fact
|
||
that our experience of change itself take place through time and we
|
||
are parts of nature, except by postulating an additional, subjective
|
||
substance, for whom spacetime and the events it contains have the
|
||
appearance of real change. Not only does the addition of such a
|
||
subjective substance make spacetime ontology more complex, but it
|
||
also poses the problem of relating eternal and enduring substances as
|
||
parts of the same world, a problem that Plato never solved. And even
|
||
if it could be solved, this modification would be <i>ad hoc</i>, for
|
||
it would explain nothing but the <i>appearance </i>that change takes
|
||
place through time. There is, therefore, no question that
|
||
spatiomaterialism is a better ontology, if it is possible. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">In
|
||
order to show that spatiomaterialism is possible is to show that it
|
||
can explain why GTR appears to be true, and that means explaining all
|
||
the relevant phenomena on the assumption that nothing exists but
|
||
space and matter enduring through time. This is to describe a model
|
||
or solution of the Einstein field equations that differs from the
|
||
prevailing geometrical interpretation because, instead of postulating
|
||
a four-dimensional manifold and defining geometrical objects on it,
|
||
spatiomaterialism postulates space and matter as substances enduring
|
||
through time. Nothing exists in a spatiomaterial world but what
|
||
exists at present, and thus, the interaction of space and matter must
|
||
somehow have an aspect that explains what Einsteinians are referring
|
||
to when they talk about “curved spacetime” and that aspect must
|
||
explain all the phenomena predicted by the general theory. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">We can tell
|
||
that not in principle impossible for a world of substances that exist
|
||
only at the present moment to explain the truth of GTR, because even
|
||
on the received geometrical interpretation, there is a standard of
|
||
simultaneity implicit in each model’s assignment of space and time
|
||
coordinates to every event in the universe. All the spacetime events
|
||
with the same temporal coordinates that we now have in some model for
|
||
our universe (a certain “simultaneity hypersurface” in curved
|
||
spacetime) <i>could </i>be all that actually exists at the present
|
||
moment, and their spatial coordinates <i>could </i>be referring to
|
||
parts of a three dimensional Euclidean substance. Of course, this
|
||
could be true of only one model, for although every model assigns
|
||
some coordinates to us now, different models entail different
|
||
standards of simultaneity, and if different models were <i>ontologically
|
||
</i>equivalent, the substances constituting the world would have to
|
||
include spacetime. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Moreover,
|
||
in order to hold, in effect, that one of all possible models
|
||
represents absolute space and time, spatiomaterialism would have to
|
||
show that there is a law of gravitation that explains not only the
|
||
approximate truth of the Newtonian theory in it, but also all the new
|
||
phenomena predicted by GTR. We can also tell that such a law is not
|
||
in principle impossible, because GTR itself implies that the relevant
|
||
events in that model are all related in a regular way. Still, the
|
||
regularity would have to be described without referring to space­time
|
||
or spacetime curvature, that is, explained as constituted by
|
||
(Euclidean) space and matter enduring through time. And there would
|
||
be problem about the regularity, only if its description turned out
|
||
to be very complex. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Finally,
|
||
since spatiomaterialism would take reality to be equivalent to what
|
||
exists in a single model of GTR according to the received geometrical
|
||
approach, we should also expect the spatiomaterialist law of
|
||
gravitation to explain why different models are observationally
|
||
equivalent, that is, to explain “general relativity”, in the
|
||
sense that enabled Einstein to derive his mathematical representation
|
||
of gravitation. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">This
|
||
is a tall order, but it is possible, as I will show here by giving an
|
||
ontological explanation of why Einstein’s GTR is true. It is an
|
||
intuitively intelligible explanation, rather than a mathematical
|
||
explanation, because what is required to explain the truth of any
|
||
theory ontologically is showing that there are aspects of the
|
||
substances postulated by the ontology that correspond to the theory.
|
||
That requires a qualitative argument, which identifies the kinds of
|
||
regularities and how they are related according to the theory, and
|
||
then shows that they can all correspond to aspects of the same world.
|
||
To be sure, the aspects of the substances pointed out must be
|
||
quantitatively adequate as well. But that is rather trivial, once the
|
||
qualitative argument has shown what the parameters are, how they are
|
||
related to one another, and the signs and order of magnitude of their
|
||
quantities, because substances can be postulated as having whatever
|
||
quantitative aspects are required to make the measurements come out
|
||
correctly. Thus, I will leave it as a challenge to those who would
|
||
disprove spatiomaterialism to show that the aspects identified here
|
||
cannot all be quantitatively accurate.</font></font></font></p>
|
||
<div id="sdendnote1">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdendnotesym" name="sdendnote1sym" href="#sdendnote1anc">i</a><span lang="en-US">
|
||
Newton later suggested various mechanisms to account for
|
||
gravitation. See </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Burtt"><font color="#0000ff"><span lang="en-US"><u>Burtt</u></span></font></a><span lang="en-US">
|
||
(1980, pp. 264ff).</span></p>
|
||
</div>
|
||
<div id="sdendnote2">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdendnotesym" name="sdendnote2sym" href="#sdendnote2anc">ii</a><span lang="en-US">
|
||
This equivalence can also be put mathematically, as </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Burtt"><font color="#0000ff"><span lang="en-US"><u>Hoefer</u></span></font></a><span lang="en-US">
|
||
(1996) does: “By taking one model <</span><span lang="en-US"><i>M,
|
||
g, T</i></span><span lang="en-US">> and applying a diffeomorphism
|
||
</span><span lang="en-US"><i>h</i></span><span lang="en-US">
|
||
(essentially, a permutation of the points in </span><span lang="en-US"><i>M</i></span><span lang="en-US">
|
||
satisfying certain restrictions), one can generate a ‘new’ model
|
||
of the theory <</span><span lang="en-US"><i>M, g, T</i></span><span lang="en-US">>
|
||
which is qualitatively identical, but which has the material
|
||
contents and the metric field distributed differently over the point
|
||
manifold of </span><span lang="en-US"><i>M</i></span><span lang="en-US">”
|
||
(7-8). </span>
|
||
</p>
|
||
</div>
|
||
<div id="sdendnote3">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdendnotesym" name="sdendnote3sym" href="#sdendnote3anc">iii</a><span lang="en-US">
|
||
This is the orthodox approach, represented by Michael </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Burtt"><font color="#0000ff"><span lang="en-US"><u>Friedman</u></span></font></a><span lang="en-US">
|
||
(1983), and John </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Burtt"><font color="#0000ff"><span lang="en-US"><u>Earman</u></span></font></a><span lang="en-US">
|
||
(1989). But Earman and John Norton use the “hole argument” to
|
||
raise doubts about the four-dimension­al manifold of points
|
||
being a substance are raised by the “hole argument”. See </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Burtt"><font color="#0000ff"><span lang="en-US"><u>Earman
|
||
and Norton</u></span></font></a><span lang="en-US"> (1987). But
|
||
spacetime substantivalism has its defenders, such as Hoefer (1996).
|
||
Though the spatiomaterialist theory does not need to answer the hole
|
||
argument to defend its substantivalism about space, it may be
|
||
relevant to mention that it sees the “hole argument” as an
|
||
artifact of the mathematical formulation of GTR. Instead of seeing
|
||
the models as different (locally) inertial frames used to assign
|
||
coordinates throughout the universe with a certain standard of
|
||
simultaneity, the hole argument interprets their observational
|
||
equivalence as a mere mathematical operation (a diffeomorphism; see
|
||
previous footnote), and that makes it possible to hold that there
|
||
can be “holes”, or regions where, in effect, different standards
|
||
of simultaneity hold. The spatiomaterialist ontological explanation
|
||
of the observational equivalence of different models of GTR will be
|
||
given at the end of this explanation of the general theory itself.</span></p>
|
||
</div>
|
||
</body>
|
||
</html> |