# The dimensional analysis of the GR geodesic equation

The geodesic equation parametrized by the proper time contains two terms:

$${d^{2}x^{\mu } \over ds^{2}}=-\Gamma ^{\mu }{}_{{\alpha \beta }}{dx^{\alpha } \over ds}{dx^{\beta } \over ds}\$$ The dimensions of the different elements of the previous expresion are

$$[x^{\mu }]=[s]$$ Both have dimnesion of length. The metric tensor being dimensionless implies the dimensionless of the Christoffel symbols $\Gamma ^{\mu }{}_{{\alpha \beta }}$ and consequently the left and right sides of the equation of the geodesic have different dimensions. What is wrong?

The Christoffel symbols are obtained by differentiating with respect to $x^\alpha$, and since the metric is dimensionless if we write the dimensions we end up with:
$$\left[{d^{2}x^{\mu } \over ds^{2}}\right] =\left[\frac{d}{dx^\alpha}\right]\left[{dx^{\alpha } \over ds}\right]\left[{dx^{\beta } \over ds}\right]$$
{"ft_lang_label":"__label__en","ft_lang_prob":0.73615533,"math_prob":0.9999838,"size":558,"snap":"2022-05-2022-21","text_gpt3_token_len":153,"char_repetition_ratio":0.14620939,"word_repetition_ratio":0.0,"special_character_ratio":0.29749104,"punctuation_ratio":0.044444446,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99990225,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-21T00:38:44Z\",\"WARC-Record-ID\":\"<urn:uuid:53ec25f0-45a2-4c11-ac00-2a8c9bb88793>\",\"Content-Length\":\"222557\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:454e4c98-8ea4-4057-b33f-f302efcb2c5a>\",\"WARC-Concurrent-To\":\"<urn:uuid:96faa9a4-3497-4ebe-92af-7a84a2ecf2ee>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://physics.stackexchange.com/questions/289145/the-dimensional-analysis-of-the-gr-geodesic-equation\",\"WARC-Payload-Digest\":\"sha1:42EZQLW6P3ZPGXHVUURBQI6HIEL53Q4E\",\"WARC-Block-Digest\":\"sha1:NK7LUUK5XVXTW46ZTZVPFIN77EG7XZOQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662534693.28_warc_CC-MAIN-20220520223029-20220521013029-00379.warc.gz\"}"}