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1

Depending on the shape of the universe the luminosity distance is given by : \begin{equation} d_L(z) = \left\{ \begin{array}{rl} \frac{(1 + z) c}{H_0 \sqrt{|\Omega_k|}} \sin \left[ \sqrt{|\Omega_k|} \int _0 ^z \frac{dz'}{H(z')/H_0} \right] & \mbox{for $k = 1$} \\ \frac{(1 + z) c}{H_0} \int _0 ^z \frac{dz'}{H(z')/H_0} & \mbox{for $k = 0$} \\ ...


7

Since you only mention acceleration to 0.5c, we'll assume we're dealing with special relativity alone. In this case, your accelerating computer 'loses time' -- its clock moves slower. Computers ultimately work on clock cycles. Thus it is fair to say that, as its clocking is ticking slower -- from your point of view -- the computer on your desk will finish ...


3

You're thinking about gravitational time dilation. Time machines do exists. If you go in a space ship and travel around the supermassive blackhole in the center of Milky Way, close enough to not fall in it, and then come back to Earth, you just traveled to the future (relative to the space further from you). So in that thinking line, if you want to make a ...


14

1) No, because it's actually going slower from your perspective. In special relativity, "the fastest wristwatch is always your own". 2) Yes, but remember that it's farther away from us now, so it will take some time to get to us (if it was travelling at 0.5c it will take 50% longer to get to us). 3) Mostly in that as an observer the redshift effect would ...


4

First, note that we are quite sure what the overall nuclear spin is; we are not sure how to obtain it mathematically from available models. Due to the phenomenon of color confinement, there are no gluons at low energies in QCD (the theory underlying nuclear physics). Importantly, you can't say there are this or that many gluons in any proton or neutron. ...


0

The order of the events never changes in Special Relativity (as Jerry Schirmer stated above) to preserve causality. If you are thinking of Boltzmann entropy then I would guess its still increasing as the number of states after evolution (no matter how slow compared to a different reference frame) goes up just like in any other system.


1

Then A will observe that B's time is elapsing more slowly than its own. It will also notice that B is shorter in the direction of travel than A is. But, it is also true that B will observe that A's time is elapsing more slowly than its own and B will also notice that A is shorter in the direction of travel than B is. This is because motion is ...


-1

Acceleration need not be relative to any object. It is relative to space (/spacetime), which is like a universal coordinate system. Whenever you accelerate, there are signs. For example, imagine you are in an elevator in deep space. If you begin to accelerate upwards (in the direction normal to the top of the elevator car) at 9.8 m/s^2, you will begin to ...


2

It's better to think of the deflection of the photon as an effect of its travel through curved spacetime. You can generally choose to analyze the problem in the rest frame of the massive object. In that case spacetime is curved in all directions around the object, and so the photon's path is deflected both as it approaches and as it recedes. If you want to ...


1

Any object that has mass just has a gravitational field around it and light (or particles, photons) just get refracted when it enters this field.


2

The transformation law for time intervals is given by $$c\Delta t'=\gamma(c\Delta t-\beta\Delta x),$$ with $\gamma=1/\sqrt{1-\beta^2}$ and $\beta=v/c.$ This coincides with your second formula only if $\Delta x=0$.


1

This is a general property of (pseudo)Riemannian geometry. I do not think there is anything specifically physical about it beyond the geometry. It holds even if $\varphi$ in not Lorentz invariant. In (pseudo)Riemannian geometry the covariant derivative $\nabla_i$ replaces partials $\partial_i$. The Laplace-Beltrami operator $$\Delta \triangleq ...


1

They still can't communicate. The horizon you are talking about is the event horizon. Assuming a spacetime event happens here and now, and let light signal propagate forward in time. The wavefront of the light signal for $t = \infty$ or collapse time is the event horizon. For standard Friedman cosmology, \begin{equation} ds^2 = -dt^2 + R^2(t) ...



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