Re: A Flaw of General Relativity, a New Metric and Cosmological Implications

Hi Tom,


You are implicitly assuming a single coordinate system can be used from
infinity all the way in to r=0.



No, look carefully; the analysis in section 2 is wholly above an event
horizon. The flaw can be shown between any two altitudes above an event
horizon.



Consider this construction at
points at successively smaller values of r -- in the limit as r->2M, the
speed of the infalling object measured by these successive inertial
frames will approach c.



The speed approaches c, but does not approach a limit of c, and that's a
huge difference. Rather, the speed approaches a limit of infinity just like
escape velocity does as r tends to zero. Look at eq. 4 in fig. 2. Above an
event horizon, eq. 4 is Einstein's equation for the speed to which you
refer. What limit does it approach as r tends to zero? It approaches
infinity.

So indeed, in the above sense the limit of the velocity of an infalling
object is indeed c. You are correct, and the speed of that infalling
object is the escape velocity from the point it is measured. Note that
this limit of c is reached as r->2M, and there _IS_ a black hole
present. <shrug>

At and below an event horizon all objects must fall, establishing the limit
to which you refer. But that limit is a different animal than the limit the
speed approaches above an event horizon, which is infinity. Section 2 uses
the latter limit to show a flaw of GR; the former limit is immaterial to
that.


But your section 2 is quoted in its entirety above (except some dialog),
and it does not support this claim at all. Indeed, the limit of c is
reached at r=2M, not r=0, and that is fully consistent with the a priori
requirement that infalling velocity = escape velocity.



As noted above, you're misusing your limits. There are two limits involved.
When the applicable one is used, the claim is supported.



Yes. But these flaws are quite different from the flaws in your
argument. Note that all other physical theories have such singularities
and unphysical solutions, so this is not at all unique to GR. GR's
incompatibility with QM is far more serious, but your argument does not
touch upon it at all.

The comment you quoted is just saying that we already know the theory is
flawed in one way, so it should not be a surprise that it is flawed in
another way.





"Tom Roberts" <tjroberts137@xxxxxxxxxxxxx> wrote in message
news:XKksg.63443$fb2.30315@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Zanket wrote:
A Flaw of General Relativity, a New Metric and Cosmological Implications
http://zanket.home.att.net/

The flaw is in your assumptions, and lack of recognizing what
assumptions you are actually making, not in GR.

I'll discuss the exterior Schwarzschild solution in GR, using the usual
radial variable r. I'll use the usual definition and call it a black hole.


Your article says:
Section 1 shows that directly measured free-fall velocity approaches a
limit of c in a uniform gravitational field. This limit applies
everywhere since a gravitational field is everywhere uniform locally.
Then the directly measured free-fall velocity of an object falling
freely from rest at infinity approaches a limit of c. This was inferred
by means general relativity allows. In general relativity, above an event
horizon of a black hole, an object falling freely from
rest at infinity passes each altitude at a directly measured velocity
equal to the escape velocity there (3). If this velocity approached a
limit of c then so would escape velocity, in which case
escape velocity would always be less than c and then there would be
no black holes. But general relativity predicts black holes. Then
the theory is inconsistent.

You are implicitly assuming a single coordinate system can be used from
infinity all the way in to r=0. And you assume that this coordinate
system can be used to measure meaningful velocities with a limit of c as
predicted by SR. This is a blatantly false assumption, because the
manifold is curved. It is indeed true that at each point along the
falling object's path you can find LOCAL coordinates with that property,
but there is no such SINGLE coordinate system throughout, as you
implicitly assumed. See below for how to do this.

Let me discuss this particular statement there in more detail:
If this velocity approached a limit of c then so would escape velocity,
in which case escape velocity would always be less than c and then there
would be no black holes.

Consider an object infalling from r=infinity. At every point along its
trajectory one can construct a LOCAL inertial frame by using standard
clocks and rulers, holding them at rest relative to the black hole, and
releasing them into freefall just as the infalling object arrives (one
must pre-arrange to set the clocks so they will be synchronized
immediately after the frame is released). Consider this construction at
points at successively smaller values of r -- in the limit as r->2M, the
speed of the infalling object measured by these successive inertial
frames will approach c.

So indeed, in the above sense the limit of the velocity of an infalling
object is indeed c. You are correct, and the speed of that infalling
object is the escape velocity from the point it is measured. Note that
this limit of c is reached as r->2M, and there _IS_ a black hole
present. <shrug>


In Section 3 you say:
an object falling freely from rest at infinity passes each altitude at a
directly measured velocity equal to the escape velocity there, in which
case section 2 shows that escape velocity must approach a limit of c
(unity) as r tends to zero.

But your section 2 is quoted in its entirety above (except some dialog),
and it does not support this claim at all. Indeed, the limit of c is
reached at r=2M, not r=0, and that is fully consistent with the a priori
requirement that infalling velocity = escape velocity.

You are implicitly assuming that the object can escape from any point
with r>0 (and therefore must have speed <c relative to any
locally-inertial frame). But you have no basis for this assumption, and
did not even mention it at all. In fact, in GR for the Schwarzschild
spacetime, a timelike object cannot escape from any point with r<2M --
this is not an assumption, but is a _conclusion_ based on the geometry
of the manifold.


It has long been known that general relativity predicts its own demise by
predicting central singularities where its equations fail and where it is
incompatible with quantum mechanics. Then it should not be a surprise
that the theory is flawed.

Yes. But these flaws are quite different from the flaws in your
argument. Note that all other physical theories have such singularities
and unphysical solutions, so this is not at all unique to GR. GR's
incompatibility with QM is far more serious, but your argument does not
touch upon it at all.


Your argument is based on basic misunderstandings, no more.


Tom Roberts


.

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