# Classical mechanics gravitational waves

Would there be gravitational waves even if general relativity was wrong? For example imagine there was a theory of gravity that was consistent with special relativity. How different could gravitational waves be? In analogy with electromagnetism one would still expect a particle carrying gravitational force at the speed of light and the acceleration of objects creating some kind of gravitational radiation. Would the difference be qualitative or just quantitative?

• Short answer: no. You ask if general relativity could be wrong, then basically list all the postulates that directly lead to general relativity. – Señor O Mar 7 '18 at 21:53
• Just to clarify, I am not questioning general relativity what I am questioning is the claim that gravitational waves are surprising. I don't believe it's true but I don't see any fundamental reason why gravity couldn't be a property of fields like electromagnetism instead of a geometric property. – ericf Mar 7 '18 at 21:57
• What I am saying is that if gravity has the property of fields and affects everything equally, that's all it takes to lead to general relativity. If the elctric field affected all things equally, then there would be a general relativity associated with it – Señor O Mar 7 '18 at 22:02

Yes, In fact we'd expect most theories that have causality baked into them to have something like gravitational waves.

But, even beyond that, if we have the Einstein-Hilbert action (in the absence of matter):

$$S = \int \sqrt{-g}\,R$$

we already know that we can ultimately rearrange the equations of motion:

$$R_{ab} - \frac{1}{2}Rg_{ab} = 0$$

in such a way that they become:

$$g_{ab},{}_{c}{}^{c} + {\rm first\;order\;in\;g}= 0$$

So, so long as you only modify the original action by adding terms that depend only on auxiliary fields and that are at most linear in first derivatives of the metric, you will end up with a theory that is not general relativity but that also will contain gravitational waves, because it could not possibly change anything about this second derivative term you get out of the EOM.

• Thanks for the answer. But in that case why would gravitational waves be considered proof of general relativity? Could there be some quantitative difference such as in the orbit of Mercury? – ericf Mar 7 '18 at 23:09
• @ericf: general relativity predicts that particular gravitatioanl waves get generated in particular ways. IN particular, coinciding them with ns-ns mergers that are seen optiocally with predicted waveforms IS a specific prediction of GR, and not these other theories. – Jerry Schirmer Mar 7 '18 at 23:49
• Thanks again. Are there any only mildly technical papers on this subject for someone who took GR and field theory many years ago. – ericf Mar 8 '18 at 0:02
• @ericf: I don't know about papers, but this type of thing is one of the places where MTW shines. – Jerry Schirmer Mar 8 '18 at 7:05