X-Received: by 10.13.242.134 with SMTP id b128mr2476650ywf.28.1459282301935; Tue, 29 Mar 2016 13:11:41 -0700 (PDT) X-Received: by 10.157.35.14 with SMTP id j14mr53616otb.8.1459282301870; Tue, 29 Mar 2016 13:11:41 -0700 (PDT) Path: csiph.com!usenet.blueworldhosting.com!feeder01.blueworldhosting.com!border2.nntp.dca1.giganews.com!nntp.giganews.com!b101no891650qga.1!news-out.google.com!ha2ni30igb.0!nntp.google.com!sy18no94653igc.0!postnews.google.com!glegroupsg2000goo.googlegroups.com!not-for-mail Newsgroups: sci.physics.relativity Date: Tue, 29 Mar 2016 13:11:41 -0700 (PDT) In-Reply-To: Complaints-To: groups-abuse@google.com Injection-Info: glegroupsg2000goo.googlegroups.com; posting-host=174.33.142.2; posting-account=RNDykAoAAAAw6g0YHqVINA1ypmXbii2P NNTP-Posting-Host: 174.33.142.2 References: <1814ad0d-bcd0-4e58-b114-e80930174260@googlegroups.com> <010ffb8c-ebc0-461b-bf40-7b9b1220c834@googlegroups.com> <4f8c59df-4ce5-4984-8fa7-3d423f6a579f@googlegroups.com> <0ba7ea70-9d22-4cb7-9e9a-a01f11fccb95@googlegroups.com> User-Agent: G2/1.0 MIME-Version: 1.0 Message-ID: Subject: Re: Inescapable (symmetric) twins paradox From: David Waite Injection-Date: Tue, 29 Mar 2016 20:11:41 +0000 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Lines: 100 Xref: csiph.com sci.physics.relativity:380438 On Tuesday, March 29, 2016 at 12:50:54 PM UTC-7, Maciej Wo=C5=BAniak wrote: > > > > ds=C2=B2=3D(1+=CE=B1x'/c=C2=B2)=C2=B2dct'=C2=B2-dx'=C2=B2-dy=C2=B2-= dz=C2=B2 > > > > where =CE=B1 is a function of t'. > > > > > This is a light sphere equation only > > >=20 > > No its not crackpot. Its > > ds=C2=B2=3Ddct=C2=B2-dx=C2=B2-dy=C2=B2-dz=C2=B2 > > under the transformation=20 > > ct=3D=E2=88=AB=CE=B3dct'+=CE=B3=CE=B2x'=20 > > x=3D=CE=B3x'+=E2=88=AB=CE=B3=CE=B2dct'=20 > > where =CE=B3 and =CE=B2 are to be expressed as functions of t'.=20 > > and=20 > > =CE=B1=3D=CE=B3=C2=B2(d=CE=B2/dt')c=20 > As we're talking about mathematics, let's remind that the oldest > and one of most trustful pieces of it was rejected by Great Guru, > because he found it's no fun anymore. I know this is over your crackpot head, but I=E2=80=99ll suppose I=E2=80=99= ll spell it out anyway. Lets say Al is an unaccelerated observer in the fla= t spacetime of special relativity. Coordinates appropriate for an inertial = frame observer express the spacetime line element as ds=C2=B2=3Ddct=C2=B2-dx=C2=B2-dy=C2=B2-dz=C2=B2 Al=E2=80=99s inertial frame coordinates will be t,x,y,z and that will be th= e expression for the line element according to his coordinates.=20 Now consider another observer Bert, who may even be accelerated.=20 Local to him using locally rectilinear coordinates he will find that the li= ne element reduces to ds=C2=B2=E2=86=92 dct=E2=80=99=C2=B2-dx=E2=80=99=C2=B2-dy=E2=80=99=C2=B2-dz= =E2=80=99=C2=B2 He doesn=E2=80=99t move with respect to himself so using this as a descript= ion of his own path through spacetime the spatial differentials vanish and = the path describes his wristwatch time, proper time. ds=C2=B2=E2=86=92 dc=CF=84=C2=B2 Since ds=C2=B2 is an invariant we see that the path of Bert as described by= Al=E2=80=99s inertial frame coordinates is dct=C2=B2-dx=C2=B2-dy=C2=B2-dz=C2=B2=3D dc=CF=84=C2=B2 Algebra dct=C2=B2{1-[(dx/dct)=C2=B2+(dy/dct)=C2=B2+(dz/dct)=C2=B2]}=3D dc=CF=84=C2= =B2 dt=C2=B2{1-[(dx/dct)=C2=B2+(dy/dct)=C2=B2+(dz/dct)=C2=B2]}=3D d=CF=84=C2=B2 dt=C2=B2[1-(v=C2=B2/c=C2=B2)]=3D d=CF=84=C2=B2 dt=C2=B2=3D d=CF=84=C2=B2/[1-(v=C2=B2/c=C2=B2)] dt=3D d=CF=84/sqrt[1-(v=C2=B2/c=C2=B2)] So this time dilation equation is how the inertial frame observer Al, obser= ves the other observer Bert to be time dilated, no matter whether Bert is a= ccelerated or not.=20 Now Julio is making the crackpot assertion that even when accelerated Bert = would get the result finding Al time dilated. Relativity says he doesn=E2= =80=99t and relativity yields no contradiction, so claiming otherwise is a = strawman fallacy. To find how things go according to the accelerated spacet= ime standards of Bert, you must first assign Bert accelerated coordinates a= ppropriate to his accelerated state. So lets say Bert accelerates along the= x-axis. We=E2=80=99ll assign him accelerated coordinate of t=E2=80=99 and = x=E2=80=99 according to=20 ct=3D=E2=88=AB=CE=B3dct'+=CE=B3=CE=B2x'=20 x=3D=CE=B3x'+=E2=88=AB=CE=B3=CE=B2dct' where =CE=B3 and =CE=B2 are to be expressed as functions of t=E2=80=99. Now= he accelerates with arbitrary time dependence but two things can clearly b= e seen before we even go any further. First of all, these reduce to Lorentz= transformation only when he is not accelerating, and only then will he obs= erve Al to be time dilated, however, second of all, since he is always at x= =E2=80=99=3D0, he has to agree with Al that in any round trip when the watc= hes are compared side be side that it will be his own watch that will have = recorded less time in accordance with special relativistic time dilation of =CE=94ct=3D=E2=88=AB=CE=B3dct' as that comes directly from the first transformation equation for the coord= inates assigned him. Now under this transformation the line element expressed in accelerated Ber= t=E2=80=99s coordinates is ds=C2=B2=3D(1+=CE=B1x'/c=C2=B2)=C2=B2dct'=C2=B2-dx'=C2=B2-dy=C2=B2-dz=C2=B2 =CE=B1=3D=CE=B3=C2=B2(d=CE=B2/dt')c From a general relativity perspective, what Bert describes as going on is h= e describes inertial Al=E2=80=99s motion as a geodesic in what he perceives= as a gravitational acceleration field which yields a gravitational red or = blue shift or gravitational time dilation effect on clocks far along the x= =E2=80=99 direction corresponding to the (1+=CE=B1x'/c=C2=B2)=C2=B2 element= of the metric. According to the accelerated spacetime standards appropriat= e for Bert, he finds that when Al is far away, and he accelerates toward Al= , Al=E2=80=99s watch runs fast corresponding to a blue shift. In the end bo= th observers agree that after a round trip, Burt aged less and by how much.= =20 Now Julio just introduces another accelerated frame. It doesn=E2=80=99t mat= ter. You still have to treat that frame as accelerated, the same way we had= to treat Bert=E2=80=99s frame as accelerated and assigned him appropriate = accelerated coordinates. The kind of acceleration that Bert experiences is = absolute. It is not relative and is not treated as relative in relativity. = Saying otherwise is a strawman fallacy. You can tell who accelerates and in= the end all observers can agree on that. If the spacetime you observe is= =20 ds=C2=B2=3Ddct=C2=B2-dx=C2=B2-dy=C2=B2-dz=C2=B2 then you=E2=80=99re not accelerating. If the spacetime you observe is something like ds=C2=B2=3D(1+=CE=B1x'/c=C2=B2)=C2=B2dct'=C2=B2-dx'=C2=B2-dy=C2=B2-dz=C2=B2 then you absolutely ARE accelerating.=20 The spacetime length of the acceleration 4-vector on you in that case is no= nzero, and that is an invariant.