In summary, the conversation discusses a confusion regarding the meaning of a non-zero component in the momentarily comoving inertial frame. The book is correct in stating that the time component of the four-acceleration is zero in this frame. However, the statement that only one component of ##\tilde{U}## is non-zero is misleading. It would be clearer to say that ##\tilde{U}## only has a non-zero time component. This is a common issue in Schutz's writing, as the physics is sound but the explanations can be confusing.
#1
Ahmed1029
109
40
I'm wondering is I'm missing something, or this should be " a non-zero component"?
No, it means that in the MCRF the only non-zero component of ##\tilde{U}## is the zeroth one (because the three velocity is zero by definition in that frame). Hence the four acceleration (which he's just proved is orthogonal to ##\tilde{U}##) must take the form given.
I agree it's not particularly clearly written - it would be better if he'd said "##\tilde{U}## has only a ##{\tilde{U}}^0## component".
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#4
Ahmed1029
109
40
Ah yes, it's clear now. I thought he meant that it had only one component taking the value zero, which didn't make sense.
