# Planck relation, relativistic doppler effect, and relativistic mass

1. Sep 9, 2010

### alphawolf50

I've been trying to graph an idea I had, but frankly I don't understand SR well enough to ensure my assumptions are correct, and that I'm using the correct formulas. I would appreciate any input/corrections you folks could give. Here's the idea, which I'll follow with my assumptions:

Idea: Place a powerful laser in space at a location you'd like to travel to quickly. Let's say near Epsilon Eridani because Wikipedia says it has a planet and is approx. 10 light years away. Now build a space vessel with a dish to collect this laser light from the front of the vessel, and convert it into usable energy for the propulsion system. The closer the vessel approaches c, the higher it observes' the laser's frequency to be. Since the energy of light is a function of its frequency, the vessel should receive more energy the faster it goes, helping it maintain constant acceleration despite its mass also increasing.

Assumptions:
1. While the energy of the laser light doesn't actually increase, the vessel should encounter more wave fronts/sec., which I'm equating to an observed increase in energy. If we combine Planck's relation with the relativistic doppler shift, we get:

$$E=h\left(\sqrt{\frac{1+\beta}{1-\beta}}\right)f_{s}$$

(Note: I've reversed the signs above because we're heading toward the source, so I'm looking for the "blue shift" rather than the "red shift". This allowed me to use positive fractions of c rather than negative.)

2. This is correct formula for determining an object's mass at relativistic speeds?

$$m_{\mathrm{rel}} = { m \over \sqrt{1-{v^2\over c^2}}}$$

Results:
When I graphed these, the laser's observed energy was 300% of original at 0.8c, but the mass was only 166% of rest mass... something seems wrong to me. Are my assumptions just flat wrong?

Thanks again, folks. I'm new here, so please be kind :)

2. Sep 9, 2010

### Mentz114

Hi alphawolf,

your calculations could be correct. Why do think they are wrong ?

But your method of collecting energy from the laser shining from your destination isn't going to work the way you think because the light has momentum, and the beam will resist your motion.

3. Sep 9, 2010

### starthaus

The answer is that there is no relationship between the "relativistic" mass of the laser and the relativistic total energy of the light it is emitting. The two entities have no connection.

4. Sep 9, 2010

### alphawolf50

Hi Mentz,

Thanks for responding -- After I reread my post I was worried it would come off in the "speculative" category, and not be responded to. I considered re-writing it to just ask about the concepts and confirm my math, which was the true intention of the post.

I thought my calculations were wrong because I am inexperienced with the concepts involved, and because the graph of the energy increase from the beam was outpacing the mass gain of the vessel by quite a bit -- I had predicted these would either match or the beam's energy would increase at a slower rate. But, as you said, I didn't take into account the momentum of the light itself. I'll rework this with that in mind. Thanks again!

5. Sep 9, 2010

### alphawolf50

Hi Starthaus -- I was actually referring to the relativistic mass of the vessel. The laser is "fixed", and the vessel is undergoing acceleration. That being said, yes, there also isn't a connection between the relativistic mass of a vessel and the relativistic energy of a separate energy source -- but I assumed they would change at nearly the same rate. Now that I look at it again, I see that my assumption was faulty since the doppler effect has nothing to do with relativity, except that it has to be calculated differently at very high speeds.

Thank you for your input -- it helped me see another flaw in my concept :)